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Note: Search results may be incomplete due to uncomputed quantities: Automorphism group order (4830 objects)

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Order Exponent Nilpotency class Nilpotent
Center Commutator Central quotient Abelianization
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Perfect Direct product Semidirect product Wreath product
Automorphism group Automorphisms Transitive degree Permutation degree
Minimal degree of $\C$-irrep Minimal degree of $\Q$-irrep Minimal degree of $\C$-rep Minimal degree of $\Q$-rep
Outer aut. group Outer automorphisms Metabelian Metacyclic
Almost simple Quasisimple A-group Z-group
Frattini subgroup Derived length Rank Schur multiplier
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Results (1-50 of 1102 matches)

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Label Name Order Exponent $\card{\mathrm{conj}(G)}$ Center Central quotient Commutator Abelianization Type - length
54.14 $C_3^2:S_3$ $2 \cdot 3^{3}$ $2 \cdot 3$ $15$ $C_1$ $C_3^2:S_3$ $C_3^3$ $C_{2}$ Solvable - 2
162.16 $C_9:D_9$ $2 \cdot 3^{4}$ $2 \cdot 3^{2}$ $42$ $C_1$ $C_9:D_9$ $C_9^2$ $C_{2}$ Solvable - 2
162.45 $C_3^2:D_9$ $2 \cdot 3^{4}$ $2 \cdot 3^{2}$ $42$ $C_1$ $C_3^2:D_9$ $C_3^2\times C_9$ $C_{2}$ Solvable - 2
162.51 $S_3\times C_3^3$ $2 \cdot 3^{4}$ $2 \cdot 3$ $81$ $C_3^3$ $S_3$ $C_3$ $C_{3}^{2} \times C_{6}$ Solvable - 2
162.53 $C_3^3:C_6$ $2 \cdot 3^{4}$ $2 \cdot 3$ $45$ $C_3$ $C_3^2:S_3$ $C_3^3$ $C_{6}$ Solvable - 2
162.54 $C_3^3:S_3$ $2 \cdot 3^{4}$ $2 \cdot 3$ $42$ $C_1$ $C_3^3:S_3$ $C_3^4$ $C_{2}$ Solvable - 2
176.41 $C_2^2\times D_{22}$ $2^{4} \cdot 11$ $2 \cdot 11$ $56$ $C_2^3$ $D_{11}$ $C_{11}$ $C_{2}^{4}$ Solvable - 2
208.50 $C_2^2\times D_{26}$ $2^{4} \cdot 13$ $2 \cdot 13$ $64$ $C_2^3$ $D_{13}$ $C_{13}$ $C_{2}^{4}$ Solvable - 2
250.7 $C_5:D_{25}$ $2 \cdot 5^{3}$ $2 \cdot 5^{2}$ $64$ $C_1$ $C_5:D_{25}$ $C_5\times C_{25}$ $C_{2}$ Solvable - 2
250.14 $C_5^3:C_2$ $2 \cdot 5^{3}$ $2 \cdot 5$ $64$ $C_1$ $C_5^3:C_2$ $C_5^3$ $C_{2}$ Solvable - 2
272.53 $C_2^2\times D_{34}$ $2^{4} \cdot 17$ $2 \cdot 17$ $80$ $C_2^3$ $D_{17}$ $C_{17}$ $C_{2}^{4}$ Solvable - 2
304.35 $C_2^2.D_{38}$ $2^{4} \cdot 19$ $2^{2} \cdot 19$ $88$ $C_2^3$ $D_{19}$ $C_{19}$ $C_{2}^{2} \times C_{4}$ Solvable - 2
304.41 $C_2^2\times D_{38}$ $2^{4} \cdot 19$ $2 \cdot 19$ $88$ $C_2^3$ $D_{19}$ $C_{19}$ $C_{2}^{4}$ Solvable - 2
351.13 $C_{13}:C_3^3$ $3^{3} \cdot 13$ $3 \cdot 13$ $63$ $C_3^2$ $C_{13}:C_3$ $C_{13}$ $C_{3}^{3}$ Solvable - 2
368.35 $C_2^2.D_{46}$ $2^{4} \cdot 23$ $2^{2} \cdot 23$ $104$ $C_2^3$ $D_{23}$ $C_{23}$ $C_{2}^{2} \times C_{4}$ Solvable - 2
368.41 $C_2^2\times D_{46}$ $2^{4} \cdot 23$ $2 \cdot 23$ $104$ $C_2^3$ $D_{23}$ $C_{23}$ $C_{2}^{4}$ Solvable - 2
405.12 $C_{15}\times \He_3$ $3^{4} \cdot 5$ $3 \cdot 5$ $165$ $C_3\times C_{15}$ $C_3^2$ $C_3$ $C_{3}^{2} \times C_{15}$ Nilpotent - 2
424.13 $C_2\times D_{106}$ $2^{3} \cdot 53$ $2 \cdot 53$ $112$ $C_2^2$ $D_{53}$ $C_{53}$ $C_{2}^{3}$ Solvable - 2
464.43 $C_2^2.D_{58}$ $2^{4} \cdot 29$ $2^{2} \cdot 29$ $128$ $C_2^3$ $D_{29}$ $C_{29}$ $C_{2}^{2} \times C_{4}$ Solvable - 2
464.50 $C_2^2\times D_{58}$ $2^{4} \cdot 29$ $2 \cdot 29$ $128$ $C_2^3$ $D_{29}$ $C_{29}$ $C_{2}^{4}$ Solvable - 2
472.11 $C_2\times D_{118}$ $2^{3} \cdot 59$ $2 \cdot 59$ $124$ $C_2^2$ $D_{59}$ $C_{59}$ $C_{2}^{3}$ Solvable - 2
488.13 $C_2\times D_{122}$ $2^{3} \cdot 61$ $2 \cdot 61$ $128$ $C_2^2$ $D_{61}$ $C_{61}$ $C_{2}^{3}$ Solvable - 2
496.35 $C_2^2.D_{62}$ $2^{4} \cdot 31$ $2^{2} \cdot 31$ $136$ $C_2^3$ $D_{31}$ $C_{31}$ $C_{2}^{2} \times C_{4}$ Solvable - 2
496.41 $C_2^2\times D_{62}$ $2^{4} \cdot 31$ $2 \cdot 31$ $136$ $C_2^3$ $D_{31}$ $C_{31}$ $C_{2}^{4}$ Solvable - 2
513.14 $C_{19}:C_3^3$ $3^{3} \cdot 19$ $3 \cdot 19$ $81$ $C_3^2$ $C_{19}:C_3$ $C_{19}$ $C_{3}^{3}$ Solvable - 2
536.11 $C_2\times D_{134}$ $2^{3} \cdot 67$ $2 \cdot 67$ $140$ $C_2^2$ $D_{67}$ $C_{67}$ $C_{2}^{3}$ Solvable - 2
567.41 $C_{21}.C_3^3$ $3^{4} \cdot 7$ $3^{2} \cdot 7$ $135$ $C_3^3$ $C_7:C_3$ $C_7$ $C_{3}^{2} \times C_{9}$ Solvable - 2
567.43 $C_{21}:\He_3$ $3^{4} \cdot 7$ $3 \cdot 7$ $87$ $C_3^2$ $C_{21}:C_3$ $C_{21}$ $C_{3}^{3}$ Solvable - 2
567.45 $C_{21}\times \He_3$ $3^{4} \cdot 7$ $3 \cdot 7$ $231$ $C_3\times C_{21}$ $C_3^2$ $C_3$ $C_{3}^{2} \times C_{21}$ Nilpotent - 2
567.48 $C_{21}:C_3^3$ $3^{4} \cdot 7$ $3 \cdot 7$ $135$ $C_3^3$ $C_7:C_3$ $C_7$ $C_{3}^{4}$ Solvable - 2
568.11 $C_2\times D_{142}$ $2^{3} \cdot 71$ $2 \cdot 71$ $148$ $C_2^2$ $D_{71}$ $C_{71}$ $C_{2}^{3}$ Solvable - 2
584.14 $C_2\times D_{146}$ $2^{3} \cdot 73$ $2 \cdot 73$ $152$ $C_2^2$ $D_{73}$ $C_{73}$ $C_{2}^{3}$ Solvable - 2
592.35 $C_{74}:Q_8$ $2^{4} \cdot 37$ $2^{2} \cdot 37$ $154$ $C_2^2$ $D_{74}$ $C_{74}$ $C_{2}^{3}$ Solvable - 2
592.37 $C_2\times D_{148}$ $2^{4} \cdot 37$ $2^{2} \cdot 37$ $154$ $C_2^2$ $D_{74}$ $C_{74}$ $C_{2}^{3}$ Solvable - 2
592.43 $C_2^2.D_{74}$ $2^{4} \cdot 37$ $2^{2} \cdot 37$ $160$ $C_2^3$ $D_{37}$ $C_{37}$ $C_{2}^{2} \times C_{4}$ Solvable - 2
592.50 $C_2^2\times D_{74}$ $2^{4} \cdot 37$ $2 \cdot 37$ $160$ $C_2^3$ $D_{37}$ $C_{37}$ $C_{2}^{4}$ Solvable - 2
632.11 $C_2\times D_{158}$ $2^{3} \cdot 79$ $2 \cdot 79$ $164$ $C_2^2$ $D_{79}$ $C_{79}$ $C_{2}^{3}$ Solvable - 2
656.36 $C_{82}:Q_8$ $2^{4} \cdot 41$ $2^{2} \cdot 41$ $170$ $C_2^2$ $D_{82}$ $C_{82}$ $C_{2}^{3}$ Solvable - 2
656.38 $C_2\times D_{164}$ $2^{4} \cdot 41$ $2^{2} \cdot 41$ $170$ $C_2^2$ $D_{82}$ $C_{82}$ $C_{2}^{3}$ Solvable - 2
656.42 $Q_8\times D_{41}$ $2^{4} \cdot 41$ $2^{2} \cdot 41$ $110$ $C_2$ $C_2\times D_{82}$ $C_{82}$ $C_{2}^{3}$ Solvable - 2
656.43 $D_{164}:C_2$ $2^{4} \cdot 41$ $2^{2} \cdot 41$ $110$ $C_2$ $C_2\times D_{82}$ $C_{82}$ $C_{2}^{3}$ Solvable - 2
656.44 $C_2^2.D_{82}$ $2^{4} \cdot 41$ $2^{2} \cdot 41$ $176$ $C_2^3$ $D_{41}$ $C_{41}$ $C_{2}^{2} \times C_{4}$ Solvable - 2
656.52 $C_2^2\times D_{82}$ $2^{4} \cdot 41$ $2 \cdot 41$ $176$ $C_2^3$ $D_{41}$ $C_{41}$ $C_{2}^{4}$ Solvable - 2
664.11 $C_2\times D_{166}$ $2^{3} \cdot 83$ $2 \cdot 83$ $172$ $C_2^2$ $D_{83}$ $C_{83}$ $C_{2}^{3}$ Solvable - 2
686.1 $D_{343}$ $2 \cdot 7^{3}$ $2 \cdot 7^{3}$ $173$ $C_1$ $D_{343}$ $C_{343}$ $C_{2}$ Solvable - 2
686.7 $C_7:D_{49}$ $2 \cdot 7^{3}$ $2 \cdot 7^{2}$ $173$ $C_1$ $C_7:D_{49}$ $C_7\times C_{49}$ $C_{2}$ Solvable - 2
686.8 $C_7^2:D_7$ $2 \cdot 7^{3}$ $2 \cdot 7$ $38$ $C_7$ $C_7:D_7$ $\He_7$ $C_{2}$ Solvable - 3
686.10 $C_2\times \He_7$ $2 \cdot 7^{3}$ $2 \cdot 7$ $110$ $C_{14}$ $C_7^2$ $C_7$ $C_{7} \times C_{14}$ Nilpotent - 2
686.12 $D_7\times C_7^2$ $2 \cdot 7^{3}$ $2 \cdot 7$ $245$ $C_7^2$ $D_7$ $C_7$ $C_{7} \times C_{14}$ Solvable - 2
686.13 $C_7^2:C_{14}$ $2 \cdot 7^{3}$ $2 \cdot 7$ $182$ $C_7$ $C_7:D_7$ $C_7^2$ $C_{14}$ Solvable - 2
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