Abstract group 372.2: $C_{124}:C_3$

• Label: 372.2
• Order: \( 2^{2} \cdot 3 \cdot 31 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 31 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3 \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{2} \cdot 3 \cdot 5 \cdot 31 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \cdot 5 \)
• Perm deg.: $35$
• Trans deg.: $124$
• Rank: $2$

Group information

Description:	$C_{124}:C_3$
Order:	\(372\)\(\medspace = 2^{2} \cdot 3 \cdot 31 \)
Exponent:	\(372\)\(\medspace = 2^{2} \cdot 3 \cdot 31 \)
Automorphism group:	$C_2\times F_{31}$, of order \(1860\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 31 \)
Composition factors:	$C_2$ x 2, $C_3$, $C_{31}$
Derived length:	$2$

This group is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 3$.

Group statistics

Order	1	2	3	4	6	12	31	62	124
Elements	1	1	62	2	62	124	30	30	60	372
Conjugacy classes	1	1	2	2	2	4	10	10	20	52
Divisions	1	1	1	1	1	1	1	1	1	9
Autjugacy classes	1	1	2	1	2	2	1	1	1	12

Dimension	1	2	3	4	30	60
Irr. complex chars.	12	0	40	0	0	0	52
Irr. rational chars.	2	3	0	1	2	1	9

Minimal presentations

Permutation degree:	$35$
Transitive degree:	$124$
Rank:	$2$
Inequivalent generating pairs:	$48$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	3	6	60
Arbitrary	3	6	32

Constructions

Presentation:	$\langle a, b \mid a^{3}=b^{124}=1, b^{a}=b^{25} \rangle$
Permutation group:	Degree $35$ $\langle(32,33,34,35), (2,3,5)(4,7,12)(6,10,9)(8,14,21)(11,17,19)(13,15,16)(18,24,28) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrr} 3 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \end{array}\right), \left(\begin{array}{rrr} 2 & 3 & 1 \\ 1 & 4 & 4 \\ 4 & 0 & 4 \end{array}\right), \left(\begin{array}{rrr} 1 & 2 & 0 \\ 4 & 2 & 4 \\ 3 & 0 & 2 \end{array}\right), \left(\begin{array}{rrr} 4 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 4 \end{array}\right) \right\rangle \subseteq \GL_{3}(\F_{5})$
Direct product:	$C_4$ $\, \times\, $ $(C_{31}:C_3)$
Semidirect product:	$C_{124}$ $\,\rtimes\,$ $C_3$	$C_{31}$ $\,\rtimes\,$ $C_{12}$			more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{62}$ . $C_6$	$(C_{31}:C_6)$ . $C_2$	$C_2$ . $(C_{31}:C_6)$		more information

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

Homology

Abelianization:	$C_{12} \simeq C_{4} \times C_{3}$
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 102 subgroups in 12 conjugacy classes, 9 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_4$	$G/Z \simeq$ $C_{31}:C_3$
Commutator:	$G' \simeq$ $C_{31}$	$G/G' \simeq$ $C_{12}$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_{31}:C_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_{124}$	$G/\operatorname{Fit} \simeq$ $C_3$
Radical:	$R \simeq$ $C_{124}:C_3$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{62}$	$G/\operatorname{soc} \simeq$ $C_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
31-Sylow subgroup:	$P_{ 31 } \simeq$ $C_{31}$

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 372: $C_{124}:C_3$

Order 186: $C_{31}:C_6$

Order 124: $C_{124}$

Order 93: $C_{31}:C_3$

Order 62: $C_{62}$

Order 31: $C_{31}$

Order 12: $C_{12}$

Order 6: $C_6$

Order 4: $C_4$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 372: $C_{124}:C_3$

Order 186: $C_{31}:C_6$

Order 124: $C_{124}$

Order 93: $C_{31}:C_3$

Order 62: $C_{62}$

Order 31: $C_{31}$

Order 12: $C_{12}$

Order 6: $C_6$

Order 4: $C_4$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 372: $C_{124}:C_3$ ($C_1$)

Order 186: $C_{31}:C_6$ ($C_2$)

Order 124: $C_{124}$ ($C_3$)

Order 93: $C_{31}:C_3$ ($C_4$)

Order 62: $C_{62}$ ($C_6$)

Order 31: $C_{31}$ ($C_{12}$)

Order 4: $C_4$ ($C_{31}:C_3$)

Order 2: $C_2$ ($C_{31}:C_6$)

Order 1: $C_1$ ($C_{124}:C_3$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 372: $C_{124}:C_3$ ($C_1$)

Order 186: $C_{31}:C_6$ ($C_2$)

Order 124: $C_{124}$ ($C_3$)

Order 93: $C_{31}:C_3$ ($C_4$)

Order 62: $C_{62}$ ($C_6$)

Order 31: $C_{31}$ ($C_{12}$)

Order 4: $C_4$ ($C_{31}:C_3$)

Order 2: $C_2$ ($C_{31}:C_6$)

Order 1: $C_1$ ($C_{124}:C_3$)

Series

Derived series	$C_{124}:C_3$	$\rhd$	$C_{31}$	$\rhd$	$C_1$
Chief series	$C_{124}:C_3$	$\rhd$	$C_{124}$	$\rhd$	$C_{62}$	$\rhd$	$C_{31}$	$\rhd$	$C_1$
Lower central series	$C_{124}:C_3$	$\rhd$	$C_{31}$
Upper central series	$C_1$	$\lhd$	$C_4$

Supergroups

This group is a maximal subgroup of 17 larger groups in the database.

This group is a maximal quotient of 9 larger groups in the database.

Character theory

Complex character table

See the $52 \times 52$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

		1A	2A	3A	4A	6A	12A	31A	62A	124A
	Size	1	1	62	2	62	124	30	30	60
	2 P	1A	1A	3A	2A	3A	6A	31A	31A	62A
	3 P	1A	2A	1A	4A	2A	4A	31A	62A	124A
	31 P	1A	2A	3A	4A	6A	12A	31A	62A	124A
372.2.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
372.2.1b		$1$	$1$	$1$	$-1$	$1$	$-1$	$1$	$1$	$-1$
372.2.1c		$2$	$2$	$-1$	$2$	$-1$	$-1$	$2$	$2$	$2$
372.2.1d		$2$	$-2$	$2$	$0$	$-2$	$0$	$2$	$-2$	$0$
372.2.1e		$2$	$2$	$-1$	$-2$	$-1$	$1$	$2$	$2$	$-2$
372.2.1f		$4$	$-4$	$-2$	$0$	$2$	$0$	$4$	$-4$	$0$
372.2.3a		$30$	$30$	$0$	$30$	$0$	$0$	$-1$	$-1$	$-1$
372.2.3b		$30$	$30$	$0$	$-30$	$0$	$0$	$-1$	$-1$	$1$
372.2.3c		$60$	$-60$	$0$	$0$	$0$	$0$	$-2$	$2$	$0$