Abstract group 292.2: $C_{292}$

• Label: 292.2
• Order: \( 2^{2} \cdot 73 \)
• Exponent: \( 2^{2} \cdot 73 \)
• Abelian: yes
• $\card{\Aut(G)}$: \( 2^{4} \cdot 3^{2} \)
• Perm deg.: $77$
• Trans deg.: $292$
• Rank: $1$

Group information

Description:	$C_{292}$
Order:	\(292\)\(\medspace = 2^{2} \cdot 73 \)
Exponent:	\(292\)\(\medspace = 2^{2} \cdot 73 \)
Automorphism group:	$C_2\times C_{72}$, of order \(144\)\(\medspace = 2^{4} \cdot 3^{2} \)
Composition factors:	$C_2$ x 2, $C_{73}$
Nilpotency class:	$1$
Derived length:	$1$

This group is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,73$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).

Group statistics

Order	1	2	4	73	146	292
Elements	1	1	2	72	72	144	292
Conjugacy classes	1	1	2	72	72	144	292
Divisions	1	1	1	1	1	1	6
Autjugacy classes	1	1	1	1	1	1	6

Dimension	1	2	72	144
Irr. complex chars.	292	0	0	0	292
Irr. rational chars.	2	1	2	1	6

Minimal presentations

Permutation degree:	$77$
Transitive degree:	$292$
Rank:	$1$
Inequivalent generators:	$1$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	1	2	144
Arbitrary	1	2	74

Constructions

Presentation:	$\langle a \mid a^{292}=1 \rangle$
Permutation group:	Degree $77$ $\langle(1,4,2,3), (5,77,76,75,74,73,72,71,70,69,68,67,66,65,64,63,62,61,60,59,58,57,56,55,54,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 72 & 0 \\ 0 & 72 \end{array}\right), \left(\begin{array}{rr} 27 & 0 \\ 0 & 27 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{73})$
Direct product:	$C_4$ $\, \times\, $ $C_{73}$
Semidirect product:	not isomorphic to a non-trivial semidirect product
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{146}$ . $C_2$	$C_2$ . $C_{146}$			more information
Aut. group:	$\Aut(C_{293})$	$\Aut(C_{586})$

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

Homology

Primary decomposition:	$C_{4} \times C_{73}$
Schur multiplier:	$C_1$
Commutator length:	$0$

Subgroups

There are 6 subgroups, all normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_{292}$	$G/Z \simeq$ $C_1$
Commutator:	$G' \simeq$ $C_1$	$G/G' \simeq$ $C_{292}$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_{146}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{292}$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_{292}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{146}$	$G/\operatorname{soc} \simeq$ $C_2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4$
73-Sylow subgroup:	$P_{ 73 } \simeq$ $C_{73}$

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 292: $C_{292}$

Order 146: $C_{146}$

Order 73: $C_{73}$

Order 4: $C_4$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 292: $C_{292}$

Order 146: $C_{146}$

Order 73: $C_{73}$

Order 4: $C_4$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 292: $C_{292}$ ($C_1$)

Order 146: $C_{146}$ ($C_2$)

Order 73: $C_{73}$ ($C_4$)

Order 4: $C_4$ ($C_{73}$)

Order 2: $C_2$ ($C_{146}$)

Order 1: $C_1$ ($C_{292}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 292: $C_{292}$ ($C_1$)

Order 146: $C_{146}$ ($C_2$)

Order 73: $C_{73}$ ($C_4$)

Order 4: $C_4$ ($C_{73}$)

Order 2: $C_2$ ($C_{146}$)

Order 1: $C_1$ ($C_{292}$)

Series

Derived series	$C_{292}$	$\rhd$	$C_1$
Chief series	$C_{292}$	$\rhd$	$C_{146}$	$\rhd$	$C_{73}$	$\rhd$	$C_1$
Lower central series	$C_{292}$	$\rhd$	$C_1$
Upper central series	$C_1$	$\lhd$	$C_{292}$

Supergroups

This group is a maximal subgroup of 26 larger groups in the database.

This group is a maximal quotient of 19 larger groups in the database.

Character theory

Complex character table

See the $292 \times 292$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

		1A	2A	4A	73A	146A	292A
	Size	1	1	2	72	72	144
	2 P	1A	1A	2A	73A	73A	146A
	73 P	1A	2A	4A	73A	146A	292A
292.2.1a		$1$	$1$	$1$	$1$	$1$	$1$
292.2.1b		$1$	$1$	$-1$	$1$	$1$	$-1$
292.2.1c		$2$	$-2$	$0$	$2$	$-2$	$0$
292.2.1d		$72$	$72$	$72$	$-1$	$-1$	$-1$
292.2.1e		$72$	$72$	$-72$	$-1$	$-1$	$1$
292.2.1f		$144$	$-144$	$0$	$-2$	$2$	$0$