Abstract group 18872.a: $C_{337}:C_{56}$

• Label: 18872.a
• Order: \( 2^{3} \cdot 7 \cdot 337 \)
• Exponent: \( 2^{3} \cdot 7 \cdot 337 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 7 \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{5} \cdot 3 \cdot 7 \cdot 337 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 3 \)
• Perm deg.: $345$
• Trans deg.: $2696$
• Rank: $2$

Group information

Description:	$C_{337}:C_{56}$
Order:	\(18872\)\(\medspace = 2^{3} \cdot 7 \cdot 337 \)
Exponent:	\(18872\)\(\medspace = 2^{3} \cdot 7 \cdot 337 \)
Automorphism group:	$C_2\times F_{337}$, of order \(226464\)\(\medspace = 2^{5} \cdot 3 \cdot 7 \cdot 337 \)
Composition factors:	$C_2$ x 3, $C_7$, $C_{337}$
Derived length:	$2$

This group is nonabelian and a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group).

Group statistics

Order	1	2	4	7	8	14	28	56	337	674
Elements	1	1	674	2022	1348	2022	4044	8088	336	336	18872
Conjugacy classes	1	1	2	6	4	6	12	24	12	12	80
Divisions	1	1	1	1	1	1	1	1	1	1	10
Autjugacy classes	1	1	2	6	2	6	12	12	1	1	44

Dimension	1	2	4	6	12	24	28	336
Irr. complex chars.	56	0	0	0	0	0	24	0	80
Irr. rational chars.	2	1	1	2	1	1	0	2	10

Minimal presentations

Permutation degree:	$345$
Transitive degree:	$2696$
Rank:	$2$
Inequivalent generating pairs:	$1152$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	28	56	672
Arbitrary	28	30	340

Constructions

Presentation:	$\langle a, b \mid a^{56}=b^{337}=1, b^{a}=b^{55} \rangle$
Permutation group:	Degree $345$ $\langle(2,4,12,40,82,26,7)(3,9,31,92,99,35,10)(5,17,56,150,182,68,20)(6,14,41,111,69,75,23) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 214 & 0 \\ 0 & 289 \end{array}\right), \left(\begin{array}{rr} 85 & 0 \\ 0 & 226 \end{array}\right), \left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{337})$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$C_{337}$ $\,\rtimes\,$ $C_{56}$	$(C_{337}:C_8)$ $\,\rtimes\,$ $C_7$	$(C_{337}:C_7)$ $\,\rtimes\,$ $C_8$		more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{674}$ . $C_{28}$	$(C_{337}:C_{28})$ . $C_2$	$(C_{337}:C_{14})$ . $C_4$	$(C_{337}:C_4)$ . $C_{14}$	all 5

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

Homology

Abelianization:	$C_{56} \simeq C_{8} \times C_{7}$
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 2032 subgroups in 16 conjugacy classes, 10 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_{337}:C_{28}$
Commutator:	$G' \simeq$ $C_{337}$	$G/G' \simeq$ $C_{56}$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_{337}:C_{28}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{674}$	$G/\operatorname{Fit} \simeq$ $C_{28}$
Radical:	$R \simeq$ $C_{337}:C_{56}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{674}$	$G/\operatorname{soc} \simeq$ $C_{28}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_8$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7$
337-Sylow subgroup:	$P_{ 337 } \simeq$ $C_{337}$

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 18872: $C_{337}:C_{56}$

Order 9436: $C_{337}:C_{28}$

Order 4718: $C_{337}:C_{14}$

Order 2696: $C_{337}:C_8$

Order 2359: $C_{337}:C_7$

Order 1348: $C_{337}:C_4$

Order 674: $C_{674}$

Order 337: $C_{337}$

Order 56: $C_{56}$

Order 28: $C_{28}$

Order 14: $C_{14}$

Order 8: $C_8$

Order 7: $C_7$

Order 4: $C_4$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 18872: $C_{337}:C_{56}$

Order 9436: $C_{337}:C_{28}$

Order 4718: $C_{337}:C_{14}$

Order 2696: $C_{337}:C_8$

Order 2359: $C_{337}:C_7$

Order 1348: $C_{337}:C_4$

Order 674: $C_{674}$

Order 337: $C_{337}$

Order 56: $C_{56}$

Order 28: $C_{28}$

Order 14: $C_{14}$

Order 8: $C_8$

Order 7: $C_7$

Order 4: $C_4$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 18872: $C_{337}:C_{56}$ ($C_1$)

Order 9436: $C_{337}:C_{28}$ ($C_2$)

Order 4718: $C_{337}:C_{14}$ ($C_4$)

Order 2696: $C_{337}:C_8$ ($C_7$)

Order 2359: $C_{337}:C_7$ ($C_8$)

Order 1348: $C_{337}:C_4$ ($C_{14}$)

Order 674: $C_{674}$ ($C_{28}$)

Order 337: $C_{337}$ ($C_{56}$)

Order 2: $C_2$ ($C_{337}:C_{28}$)

Order 1: $C_1$ ($C_{337}:C_{56}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 18872: $C_{337}:C_{56}$ ($C_1$)

Order 9436: $C_{337}:C_{28}$ ($C_2$)

Order 4718: $C_{337}:C_{14}$ ($C_4$)

Order 2696: $C_{337}:C_8$ ($C_7$)

Order 2359: $C_{337}:C_7$ ($C_8$)

Order 1348: $C_{337}:C_4$ ($C_{14}$)

Order 674: $C_{674}$ ($C_{28}$)

Order 337: $C_{337}$ ($C_{56}$)

Order 2: $C_2$ ($C_{337}:C_{28}$)

Order 1: $C_1$ ($C_{337}:C_{56}$)

Series

Derived series

$C_{337}:C_{56}$

$\rhd$

$C_{337}$

$\rhd$

$C_1$

Chief series

$C_{337}:C_{56}$

$\rhd$

$C_{337}:C_{28}$

$\rhd$

$C_{337}:C_{14}$

$\rhd$

$C_{337}:C_7$

$\rhd$

$C_{337}$

$\rhd$

$C_1$

Lower central series

$C_{337}:C_{56}$

$\rhd$

$C_{337}$

Upper central series

$C_1$

$\lhd$

$C_2$

Supergroups

This group is a maximal subgroup of 4 larger groups in the database.

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

Character theory

Complex character table

See the $80 \times 80$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

		1A	2A	4A	7A	8A	14A	28A	56A	337A	674A
	Size	1	1	674	2022	1348	2022	4044	8088	336	336
	2 P	1A	1A	2A	7A	4A	7A	14A	28A	337A	337A
	7 P	1A	2A	4A	7A	8A	14A	28A	56A	337A	674A
	337 P	1A	2A	4A	1A	8A	2A	4A	8A	337A	674A
	Schur
18872.a.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
18872.a.1b		$1$	$1$	$1$	$1$	$-1$	$1$	$1$	$-1$	$1$	$1$
18872.a.1c		$2$	$2$	$-2$	$2$	$0$	$2$	$-2$	$0$	$2$	$2$
18872.a.1d		$4$	$-4$	$0$	$4$	$0$	$-4$	$0$	$0$	$4$	$-4$
18872.a.1e		$6$	$6$	$6$	$-1$	$6$	$-1$	$-1$	$-1$	$6$	$6$
18872.a.1f		$6$	$6$	$6$	$-1$	$-6$	$-1$	$-1$	$1$	$6$	$6$
18872.a.1g		$12$	$12$	$-12$	$-2$	$0$	$-2$	$2$	$0$	$12$	$12$
18872.a.1h		$24$	$-24$	$0$	$-4$	$0$	$4$	$0$	$0$	$24$	$-24$
18872.a.28a		$336$	$336$	$0$	$0$	$0$	$0$	$0$	$0$	$-1$	$-1$
18872.a.28b	2	$336$	$-336$	$0$	$0$	$0$	$0$	$0$	$0$	$-1$	$1$