Abstract group $C_{33}$

• Label: 33.1
• Order: \( 3 \cdot 11 \)
• Exponent: \( 3 \cdot 11 \)
• Abelian: yes
• $\card{\operatorname{Aut}(G)}$: \( 2^{2} \cdot 5 \)
• Perm deg.: $14$
• Trans deg.: $33$
• Rank: $1$

Group information

Description:	$C_{33}$
Order:	\(33\)\(\medspace = 3 \cdot 11 \)
Exponent:	\(33\)\(\medspace = 3 \cdot 11 \)
Automorphism group:	$C_2\times C_{10}$, of order \(20\)\(\medspace = 2^{2} \cdot 5 \) (generators)
Outer automorphisms:	$C_2\times C_{10}$, of order \(20\)\(\medspace = 2^{2} \cdot 5 \)
Composition factors:	$C_3$, $C_{11}$
Nilpotency class:	$1$
Derived length:	$1$

This group is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 3,11$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).

Group statistics

Order	1	3	11	33
Elements	1	2	10	20	33
Conjugacy classes	1	2	10	20	33
Divisions	1	1	1	1	4
Autjugacy classes	1	1	1	1	4

Dimension	1	2	10	20
Irr. complex chars.	33	0	0	0	33
Irr. rational chars.	1	1	1	1	4

Minimal Presentations

Permutation degree:	$14$
Transitive degree:	$33$
Rank:	$1$
Inequivalent generators:	$1$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	1	2	20
Arbitrary	1	2	12

Constructions

Presentation:	$\langle a \mid a^{33}=1 \rangle$
Permutation group:	Degree $14$ $\langle(1,3,2), (4,14,13,12,11,10,9,8,7,6,5)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 7 & 22 \\ 9 & 7 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{23})$
Transitive group:	33T1				more information
Direct product:	$C_3$ $\, \times\, $ $C_{11}$
Semidirect product:	not isomorphic to a non-trivial semidirect product
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

Elements of the group are displayed as words in the generators from the presentation given above.

Homology

Primary decomposition:	$C_{3} \times C_{11}$
Schur multiplier:	$C_1$
Commutator length:	$0$

Subgroups

There are 4 subgroups, all normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_{33}$	$G/Z \simeq$ $C_1$
Commutator:	$G' \simeq$ $C_1$	$G/G' \simeq$ $C_{33}$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_{33}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{33}$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_{33}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{33}$	$G/\operatorname{soc} \simeq$ $C_1$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
11-Sylow subgroup:	$P_{ 11 } \simeq$ $C_{11}$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

Classes of subgroups up to automorphism

Normal subgroups

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 33: $C_{33}$

Order 11: $C_{11}$

Order 3: $C_3$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 33: $C_{33}$

Order 11: $C_{11}$

Order 3: $C_3$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 33: $C_{33}$ ($C_1$)

Order 11: $C_{11}$ ($C_3$)

Order 3: $C_3$ ($C_{11}$)

Order 1: $C_1$ ($C_{33}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 33: $C_{33}$ ($C_1$)

Order 11: $C_{11}$ ($C_3$)

Order 3: $C_3$ ($C_{11}$)

Order 1: $C_1$ ($C_{33}$)

Series

Derived series	$C_{33}$	$\rhd$	$C_1$
Chief series	$C_{33}$	$\rhd$	$C_{11}$	$\rhd$	$C_1$
Lower central series	$C_{33}$	$\rhd$	$C_1$
Upper central series	$C_1$	$\lhd$	$C_{33}$

Supergroups

This group is a maximal subgroup of 59 larger groups in the database.

This group is a maximal quotient of 56 larger groups in the database.

Character theory

Complex character table

See the $33 \times 33$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

		1A	3A	11A	33A
	Size	1	2	10	20
	3 P	1A	3A	11A	33A
	11 P	1A	1A	11A	11A
33.1.1a		$1$	$1$	$1$	$1$
33.1.1b		$2$	$-1$	$2$	$-1$
33.1.1c		$10$	$10$	$-1$	$-1$
33.1.1d		$20$	$-10$	$-2$	$1$