Abstract group 2368.54: $C_{37}\times Q_{64}$

• Label: 2368.54
• Order: \( 2^{6} \cdot 37 \)
• Exponent: \( 2^{5} \cdot 37 \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 37 \)
• $\card{Z(G)}$: \( 2 \cdot 37 \)
• $\card{\Aut(G)}$: \( 2^{11} \cdot 3^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \cdot 3^{2} \)
• Perm deg.: $101$
• Trans deg.: $2368$
• Rank: $2$

Group information

Description:	$C_{37}\times Q_{64}$
Order:	\(2368\)\(\medspace = 2^{6} \cdot 37 \)
Exponent:	\(1184\)\(\medspace = 2^{5} \cdot 37 \)
Automorphism group:	$C_{36}\times C_8.(C_8\times D_4)$, of order \(18432\)\(\medspace = 2^{11} \cdot 3^{2} \)
Composition factors:	$C_2$ x 6, $C_{37}$
Nilpotency class:	$5$
Derived length:	$2$

This group is nonabelian, elementary for $p = 2$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metacyclic (hence metabelian).

Group statistics

Order	1	2	4	8	16	32	37	74	148	296	592	1184
Elements	1	1	34	4	8	16	36	36	1224	144	288	576	2368
Conjugacy classes	1	1	3	2	4	8	36	36	108	72	144	288	703
Divisions	1	1	3	1	1	1	1	1	3	1	1	1	16
Autjugacy classes	1	1	2	1	1	1	1	1	2	1	1	1	14

Minimal presentations

Permutation degree:	$101$
Transitive degree:	$2368$
Rank:	$2$
Inequivalent generating pairs:	$114$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	2	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b \mid b^{1184}=1, a^{2}=b^{592}, b^{a}=b^{223} \rangle$
Permutation group:	Degree $101$ $\langle(1,34,2,33)(3,36,4,35)(5,38,6,37)(7,40,8,39)(9,42,10,41)(11,44,12,43)(13,46,14,45) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 57 & 68 \\ 126 & 166 \end{array}\right), \left(\begin{array}{rr} 4 & 29 \\ 84 & 4 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{223})$
Direct product:	$C_{37}$ $\, \times\, $ $Q_{64}$
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$Q_{32}$ . $C_{74}$ (2)	$C_{296}$ . $D_4$	$C_{148}$ . $D_8$	$C_{74}$ . $D_{16}$	all 12

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

Homology

Abelianization:	$C_{2} \times C_{74} \simeq C_{2}^{2} \times C_{37}$
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 74 subgroups in 30 conjugacy classes, 18 normal (14 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_{74}$	$G/Z \simeq$ $D_{16}$
Commutator:	$G' \simeq$ $C_{16}$	$G/G' \simeq$ $C_2\times C_{74}$
Frattini:	$\Phi \simeq$ $C_{16}$	$G/\Phi \simeq$ $C_2\times C_{74}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{37}\times Q_{64}$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_{37}\times Q_{64}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{74}$	$G/\operatorname{soc} \simeq$ $D_{16}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $Q_{64}$
37-Sylow subgroup:	$P_{ 37 } \simeq$ $C_{37}$

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 2368: $C_{37}\times Q_{64}$

Order 1184: $Q_{32}\times C_{37}$ x 2, $C_{1184}$

Order 592: $Q_{16}\times C_{37}$ x 2, $C_{592}$

Order 296: $Q_8\times C_{37}$ x 2, $C_{296}$

Order 148: $C_{148}$ x 3

Order 74: $C_{74}$

Order 64: $Q_{64}$

Order 37: $C_{37}$

Order 32: $Q_{32}$ x 2, $C_{32}$

Order 16: $Q_{16}$ x 2, $C_{16}$

Order 8: $Q_8$ x 2, $C_8$

Order 4: $C_4$ x 3

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 2368: $C_{37}\times Q_{64}$

Order 1184: $Q_{32}\times C_{37}$, $C_{1184}$

Order 592: $Q_{16}\times C_{37}$, $C_{592}$

Order 296: $C_{296}$, $Q_8\times C_{37}$

Order 148: $C_{148}$ x 2

Order 74: $C_{74}$

Order 64: $Q_{64}$

Order 37: $C_{37}$

Order 32: $Q_{32}$, $C_{32}$

Order 16: $Q_{16}$, $C_{16}$

Order 8: $Q_8$, $C_8$

Order 4: $C_4$ x 2

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 2368: $C_{37}\times Q_{64}$ ($C_1$)

Order 1184: $Q_{32}\times C_{37}$ ($C_2$) x 2, $C_{1184}$ ($C_2$)

Order 592: $C_{592}$ ($C_2^2$)

Order 296: $C_{296}$ ($D_4$)

Order 148: $C_{148}$ ($D_8$)

Order 74: $C_{74}$ ($D_{16}$)

Order 64: $Q_{64}$ ($C_{37}$)

Order 37: $C_{37}$ ($Q_{64}$)

Order 32: $Q_{32}$ ($C_{74}$) x 2, $C_{32}$ ($C_{74}$)

Order 16: $C_{16}$ ($C_2\times C_{74}$)

Order 8: $C_8$ ($D_4\times C_{37}$)

Order 4: $C_4$ ($D_8\times C_{37}$)

Order 2: $C_2$ ($D_{16}\times C_{37}$)

Order 1: $C_1$ ($C_{37}\times Q_{64}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 2368: $C_{37}\times Q_{64}$ ($C_1$)

Order 1184: $Q_{32}\times C_{37}$ ($C_2$), $C_{1184}$ ($C_2$)

Order 592: $C_{592}$ ($C_2^2$)

Order 296: $C_{296}$ ($D_4$)

Order 148: $C_{148}$ ($D_8$)

Order 74: $C_{74}$ ($D_{16}$)

Order 64: $Q_{64}$ ($C_{37}$)

Order 37: $C_{37}$ ($Q_{64}$)

Order 32: $Q_{32}$ ($C_{74}$), $C_{32}$ ($C_{74}$)

Order 16: $C_{16}$ ($C_2\times C_{74}$)

Order 8: $C_8$ ($D_4\times C_{37}$)

Order 4: $C_4$ ($D_8\times C_{37}$)

Order 2: $C_2$ ($D_{16}\times C_{37}$)

Order 1: $C_1$ ($C_{37}\times Q_{64}$)

Series

Derived series

$C_{37}\times Q_{64}$

$\rhd$

$C_{16}$

$\rhd$

$C_1$

Chief series

$C_{37}\times Q_{64}$

$\rhd$

$Q_{32}\times C_{37}$

$\rhd$

$C_{592}$

$\rhd$

$C_{296}$

$\rhd$

$C_{148}$

$\rhd$

$C_{74}$

$\rhd$

$C_{37}$

$\rhd$

$C_1$

Lower central series

$C_{37}\times Q_{64}$

$\rhd$

$C_{16}$

$\rhd$

$C_8$

$\rhd$

$C_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_{74}$

$\lhd$

$C_{148}$

$\lhd$

$C_{296}$

$\lhd$

$C_{592}$

$\lhd$

$C_{37}\times Q_{64}$

Supergroups

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

This group is a maximal quotient of 2 larger groups in the database.

Character theory

Complex character table

The $703 \times 703$ character table is not available for this group.

Rational character table

The $16 \times 16$ rational character table is not available for this group.