Abstract group 744.7: $C_{124}.C_6$

• Label: 744.7
• Order: \( 2^{3} \cdot 3 \cdot 31 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 31 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3 \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{4} \cdot 3 \cdot 5 \cdot 31 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \cdot 5 \)
• Perm deg.: $39$
• Trans deg.: $248$
• Rank: $2$

Group information

Description:	$C_{124}.C_6$
Order:	\(744\)\(\medspace = 2^{3} \cdot 3 \cdot 31 \)
Exponent:	\(372\)\(\medspace = 2^{2} \cdot 3 \cdot 31 \)
Automorphism group:	$D_4\times F_{31}$, of order \(7440\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \cdot 31 \)
Composition factors:	$C_2$ x 3, $C_3$, $C_{31}$
Derived length:	$2$

This group is nonabelian and metacyclic (hence solvable, supersolvable, monomial, and metabelian).

Group statistics

Order	1	2	3	4	6	12	31	62	124
Elements	1	1	62	126	62	372	30	30	60	744
Conjugacy classes	1	1	2	3	2	6	5	5	10	35
Divisions	1	1	1	3	1	3	1	1	1	13
Autjugacy classes	1	1	2	2	2	4	1	1	1	15

Dimension	1	2	4	6	30	60
Irr. complex chars.	12	3	0	20	0	0	35
Irr. rational chars.	4	5	1	0	2	1	13

Minimal presentations

Permutation degree:	$39$
Transitive degree:	$248$
Rank:	$2$
Inequivalent generating pairs:	$24$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	6	12	120
Arbitrary	6	10	34

Constructions

Presentation:	$\langle a, b \mid b^{124}=1, a^{6}=b^{62}, b^{a}=b^{99} \rangle$
Permutation group:	Degree $39$ $\langle(2,3)(4,6)(5,9)(7,8)(10,14)(11,17)(12,20)(13,19)(15,22)(16,23)(18,24)(21,27) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 32 & 0 \\ 0 & 32 \end{array}\right), \left(\begin{array}{rr} 1 & 27 \\ 0 & 25 \end{array}\right), \left(\begin{array}{rr} 63 & 92 \\ 31 & 30 \end{array}\right), \left(\begin{array}{rr} 1 & 3 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 62 \\ 62 & 32 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/93\Z)$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_{31}:Q_8)$ $\,\rtimes\,$ $C_3$	$(C_{31}:C_3)$ $\,\rtimes\,$ $Q_8$	$C_{31}$ $\,\rtimes\,$ $(C_3\times Q_8)$		more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{124}$ . $C_6$	$(C_{31}:C_4)$ . $C_6$ (2)	$C_{62}$ . $(C_2\times C_6)$	$C_4$ . $(C_{31}:C_6)$	all 8

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

Homology

Abelianization:	$C_{2} \times C_{6} \simeq C_{2}^{2} \times C_{3}$
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 294 subgroups in 24 conjugacy classes, 15 normal (11 characteristic).

Characteristic subgroups are shown in this color. Normal (but not characteristic) subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_{62}:C_6$
Commutator:	$G' \simeq$ $C_{62}$	$G/G' \simeq$ $C_2\times C_6$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_{62}:C_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_{124}$	$G/\operatorname{Fit} \simeq$ $C_6$
Radical:	$R \simeq$ $C_{124}.C_6$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{62}$	$G/\operatorname{soc} \simeq$ $C_2\times C_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $Q_8$
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 744: $C_{124}.C_6$

Order 372: $C_{31}:C_{12}$ x 2, $C_{124}:C_3$

Order 248: $C_{31}:Q_8$

Order 186: $C_{31}:C_6$

Order 124: $C_{31}:C_4$ x 2, $C_{124}$

Order 93: $C_{31}:C_3$

Order 62: $C_{62}$

Order 31: $C_{31}$

Order 24: $C_3\times Q_8$

Order 12: $C_{12}$ x 3

Order 8: $Q_8$

Order 6: $C_6$

Order 4: $C_4$ x 3

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 744: $C_{124}.C_6$

Order 372: $C_{124}:C_3$, $C_{31}:C_{12}$

Order 248: $C_{31}:Q_8$

Order 186: $C_{31}:C_6$

Order 124: $C_{124}$, $C_{31}:C_4$

Order 93: $C_{31}:C_3$

Order 62: $C_{62}$

Order 31: $C_{31}$

Order 24: $C_3\times Q_8$

Order 12: $C_{12}$ x 2

Order 8: $Q_8$

Order 6: $C_6$

Order 4: $C_4$ x 2

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 744: $C_{124}.C_6$ ($C_1$)

Order 372: $C_{31}:C_{12}$ ($C_2$) x 2, $C_{124}:C_3$ ($C_2$)

Order 248: $C_{31}:Q_8$ ($C_3$)

Order 186: $C_{31}:C_6$ ($C_2^2$)

Order 124: $C_{31}:C_4$ ($C_6$) x 2, $C_{124}$ ($C_6$)

Order 93: $C_{31}:C_3$ ($Q_8$)

Order 62: $C_{62}$ ($C_2\times C_6$)

Order 31: $C_{31}$ ($C_3\times Q_8$)

Order 4: $C_4$ ($C_{31}:C_6$)

Order 2: $C_2$ ($C_{62}:C_6$)

Order 1: $C_1$ ($C_{124}.C_6$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 744: $C_{124}.C_6$ ($C_1$)

Order 372: $C_{124}:C_3$ ($C_2$), $C_{31}:C_{12}$ ($C_2$)

Order 248: $C_{31}:Q_8$ ($C_3$)

Order 186: $C_{31}:C_6$ ($C_2^2$)

Order 124: $C_{124}$ ($C_6$), $C_{31}:C_4$ ($C_6$)

Order 93: $C_{31}:C_3$ ($Q_8$)

Order 62: $C_{62}$ ($C_2\times C_6$)

Order 31: $C_{31}$ ($C_3\times Q_8$)

Order 4: $C_4$ ($C_{31}:C_6$)

Order 2: $C_2$ ($C_{62}:C_6$)

Order 1: $C_1$ ($C_{124}.C_6$)

Series

Derived series

$C_{124}.C_6$

$\rhd$

$C_{62}$

$\rhd$

$C_1$

Chief series

$C_{124}.C_6$

$\rhd$

$C_{124}:C_3$

$\rhd$

$C_{124}$

$\rhd$

$C_{62}$

$\rhd$

$C_{31}$

$\rhd$

$C_1$

Lower central series

$C_{124}.C_6$

$\rhd$

$C_{62}$

$\rhd$

$C_{31}$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_4$

Supergroups

This group is a maximal subgroup of 8 larger groups in the database.

This group is a maximal quotient of 3 larger groups in the database.

Character theory

Complex character table

See the $35 \times 35$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

		1A	2A	3A	4A	4B	4C	6A	12A	12B	12C	31A	62A	124A
	Size	1	1	62	2	62	62	62	124	124	124	30	30	60
	2 P	1A	1A	3A	2A	2A	2A	3A	6A	6A	6A	31A	31A	62A
	3 P	1A	2A	1A	4A	4B	4C	2A	4B	4A	4C	31A	62A	124A
	31 P	1A	2A	3A	4A	4B	4C	6A	12A	12B	12C	31A	62A	124A
	Schur
744.7.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
744.7.1b		$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$-1$
744.7.1c		$1$	$1$	$1$	$-1$	$1$	$-1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$-1$
744.7.1d		$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$-1$	$1$	$-1$	$1$	$1$	$1$
744.7.1e		$2$	$2$	$-1$	$2$	$2$	$2$	$-1$	$-1$	$-1$	$-1$	$2$	$2$	$2$
744.7.1f		$2$	$2$	$-1$	$-2$	$-2$	$2$	$-1$	$1$	$1$	$-1$	$2$	$2$	$-2$
744.7.1g		$2$	$2$	$-1$	$-2$	$2$	$-2$	$-1$	$-1$	$1$	$1$	$2$	$2$	$-2$
744.7.1h		$2$	$2$	$-1$	$2$	$-2$	$-2$	$-1$	$1$	$-1$	$1$	$2$	$2$	$2$
744.7.2a	2	$2$	$-2$	$2$	$0$	$0$	$0$	$-2$	$0$	$0$	$0$	$2$	$-2$	$0$
744.7.2b		$4$	$-4$	$-2$	$0$	$0$	$0$	$2$	$0$	$0$	$0$	$4$	$-4$	$0$
744.7.6a		$30$	$30$	$0$	$30$	$0$	$0$	$0$	$0$	$0$	$0$	$-1$	$-1$	$-1$
744.7.6b		$30$	$30$	$0$	$-30$	$0$	$0$	$0$	$0$	$0$	$0$	$-1$	$-1$	$1$
744.7.6c	2	$60$	$-60$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$-2$	$2$	$0$