Abstract group 18872.b: $C_{1348}:C_{14}$

• Label: 18872.b
• Order: \( 2^{3} \cdot 7 \cdot 337 \)
• Exponent: \( 2^{2} \cdot 7 \cdot 337 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 7 \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{6} \cdot 3 \cdot 7 \cdot 337 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \cdot 3 \)
• Perm deg.: $341$
• Trans deg.: $1348$
• Rank: $2$

Group information

Description:	$C_{1348}:C_{14}$
Order:	\(18872\)\(\medspace = 2^{3} \cdot 7 \cdot 337 \)
Exponent:	\(9436\)\(\medspace = 2^{2} \cdot 7 \cdot 337 \)
Automorphism group:	$C_{337}.C_{168}.C_2^3$, of order \(452928\)\(\medspace = 2^{6} \cdot 3 \cdot 7 \cdot 337 \)
Composition factors:	$C_2$ x 3, $C_7$, $C_{337}$
Derived length:	$2$

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

Group statistics

Order	1	2	4	7	14	28	337	674	1348
Elements	1	675	676	2022	6066	8088	336	336	672	18872
Conjugacy classes	1	3	4	6	18	24	24	24	48	152
Divisions	1	3	2	1	3	2	1	1	1	15
Autjugacy classes	1	2	2	6	12	12	1	1	1	38

Dimension	1	2	6	12	14	336	672
Irr. complex chars.	56	0	0	0	96	0	0	152
Irr. rational chars.	4	2	4	2	0	2	1	15

Minimal presentations

Permutation degree:	$341$
Transitive degree:	$1348$
Rank:	$2$
Inequivalent generating pairs:	$288$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	14	28	672
Arbitrary	14	16	338

Constructions

Presentation:	$\langle a, b \mid b^{1348}=1, a^{14}=b^{674}, b^{a}=b^{273} \rangle$
Permutation group:	Degree $341$ $\langle(2,3)(4,8)(5,7)(6,9)(10,23)(11,22)(12,24)(13,21)(14,20)(15,19)(16,26)(17,25) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 150 & 0 \\ 0 & 173 \end{array}\right), \left(\begin{array}{rr} 1 & 0 \\ 0 & 336 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{337})$
Direct product:	$C_4$ $\, \times\, $ $(C_{337}:C_{14})$
Semidirect product:	$D_{337}$ $\,\rtimes\,$ $C_{28}$ (2)	$C_{1348}$ $\,\rtimes\,$ $C_{14}$	$(C_4\times D_{337})$ $\,\rtimes\,$ $C_7$	$(C_{1348}:C_7)$ $\,\rtimes\,$ $C_2$	all 8
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$D_{674}$ . $C_{14}$	$(C_{674}:C_{14})$ . $C_2$	$C_{674}$ . $(C_2\times C_{14})$	$C_2$ . $(C_{674}:C_{14})$	all 5

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

Homology

Abelianization:	$C_{2} \times C_{28} \simeq C_{2} \times C_{4} \times C_{7}$
Schur multiplier:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 4400 subgroups in 32 conjugacy classes, 19 normal (15 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_4$	$G/Z \simeq$ $C_{337}:C_{14}$
Commutator:	$G' \simeq$ $C_{337}$	$G/G' \simeq$ $C_2\times C_{28}$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_{674}:C_{14}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{1348}$	$G/\operatorname{Fit} \simeq$ $C_{14}$
Radical:	$R \simeq$ $C_{1348}:C_{14}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{674}$	$G/\operatorname{soc} \simeq$ $C_2\times C_{14}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times C_4$
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: $C_{1348}:C_{14}$

Order 9436: $C_{337}:C_{28}$, $C_{674}:C_{14}$, $C_{1348}:C_7$

Order 4718: $C_{337}:C_{14}$ x 2, $C_{337}:C_{14}$

Order 2696: $C_4\times D_{337}$

Order 2359: $C_{337}:C_7$

Order 1348: $D_{674}$, $C_{1348}$, $C_{337}:C_4$

Order 674: $D_{337}$ x 2, $C_{674}$

Order 337: $C_{337}$

Order 56: $C_2\times C_{28}$

Order 28: $C_{28}$ x 2, $C_2\times C_{14}$

Order 14: $C_{14}$ x 3

Order 8: $C_2\times C_4$

Order 7: $C_7$

Order 4: $C_4$ x 2, $C_2^2$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 18872: $C_{1348}:C_{14}$

Order 9436: $C_{337}:C_{28}$, $C_{674}:C_{14}$, $C_{1348}:C_7$

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

Order 2696: $C_4\times D_{337}$

Order 2359: $C_{337}:C_7$

Order 1348: $D_{674}$, $C_{1348}$, $C_{337}:C_4$

Order 674: $C_{674}$, $D_{337}$

Order 337: $C_{337}$

Order 56: $C_2\times C_{28}$

Order 28: $C_{28}$ x 2, $C_2\times C_{14}$

Order 14: $C_{14}$ x 2

Order 8: $C_2\times C_4$

Order 7: $C_7$

Order 4: $C_4$ x 2, $C_2^2$

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 18872: $C_{1348}:C_{14}$ ($C_1$)

Order 9436: $C_{337}:C_{28}$ ($C_2$), $C_{674}:C_{14}$ ($C_2$), $C_{1348}:C_7$ ($C_2$)

Order 4718: $C_{337}:C_{14}$ ($C_4$) x 2, $C_{337}:C_{14}$ ($C_2^2$)

Order 2696: $C_4\times D_{337}$ ($C_7$)

Order 2359: $C_{337}:C_7$ ($C_2\times C_4$)

Order 1348: $D_{674}$ ($C_{14}$), $C_{1348}$ ($C_{14}$), $C_{337}:C_4$ ($C_{14}$)

Order 674: $D_{337}$ ($C_{28}$) x 2, $C_{674}$ ($C_2\times C_{14}$)

Order 337: $C_{337}$ ($C_2\times C_{28}$)

Order 4: $C_4$ ($C_{337}:C_{14}$)

Order 2: $C_2$ ($C_{674}:C_{14}$)

Order 1: $C_1$ ($C_{1348}:C_{14}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 18872: $C_{1348}:C_{14}$ ($C_1$)

Order 9436: $C_{337}:C_{28}$ ($C_2$), $C_{674}:C_{14}$ ($C_2$), $C_{1348}:C_7$ ($C_2$)

Order 4718: $C_{337}:C_{14}$ ($C_2^2$), $C_{337}:C_{14}$ ($C_4$)

Order 2696: $C_4\times D_{337}$ ($C_7$)

Order 2359: $C_{337}:C_7$ ($C_2\times C_4$)

Order 1348: $D_{674}$ ($C_{14}$), $C_{1348}$ ($C_{14}$), $C_{337}:C_4$ ($C_{14}$)

Order 674: $C_{674}$ ($C_2\times C_{14}$), $D_{337}$ ($C_{28}$)

Order 337: $C_{337}$ ($C_2\times C_{28}$)

Order 4: $C_4$ ($C_{337}:C_{14}$)

Order 2: $C_2$ ($C_{674}:C_{14}$)

Order 1: $C_1$ ($C_{1348}:C_{14}$)

Series

Derived series

$C_{1348}:C_{14}$

$\rhd$

$C_{337}$

$\rhd$

$C_1$

Chief series

$C_{1348}:C_{14}$

$\rhd$

$C_{1348}:C_7$

$\rhd$

$C_{1348}$

$\rhd$

$C_{674}$

$\rhd$

$C_{337}$

$\rhd$

$C_1$

Lower central series

$C_{1348}:C_{14}$

$\rhd$

$C_{337}$

Upper central series

$C_1$

$\lhd$

$C_4$

Supergroups

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

This group is a maximal quotient of 1 larger groups in the database.

Character theory

Complex character table

See the $152 \times 152$ 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	2B	2C	4A	4B	7A	14A	14B	14C	28A	28B	337A	674A	1348A
	Size	1	1	337	337	2	674	2022	2022	2022	2022	4044	4044	336	336	672
	2 P	1A	1A	1A	1A	2A	2A	7A	7A	7A	7A	14B	14B	337A	337A	674A
	7 P	1A	2A	2B	2C	4A	4B	7A	14A	14B	14C	28A	28B	337A	674A	1348A
	337 P	1A	2A	2B	2C	4A	4B	1A	2B	2A	2C	4A	4B	337A	674A	1348A
18872.b.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
18872.b.1b		$1$	$1$	$-1$	$-1$	$-1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$1$	$1$	$-1$
18872.b.1c		$1$	$1$	$-1$	$-1$	$1$	$-1$	$1$	$-1$	$-1$	$1$	$-1$	$1$	$1$	$1$	$1$
18872.b.1d		$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$-1$
18872.b.1e		$2$	$-2$	$-2$	$2$	$0$	$0$	$2$	$-2$	$2$	$-2$	$0$	$0$	$2$	$-2$	$0$
18872.b.1f		$2$	$-2$	$2$	$-2$	$0$	$0$	$2$	$2$	$-2$	$-2$	$0$	$0$	$2$	$-2$	$0$
18872.b.1g		$6$	$6$	$6$	$6$	$6$	$6$	$-1$	$-1$	$-1$	$-1$	$-1$	$-1$	$6$	$6$	$6$
18872.b.1h		$6$	$6$	$-6$	$-6$	$-6$	$6$	$-1$	$1$	$1$	$-1$	$-1$	$1$	$6$	$6$	$-6$
18872.b.1i		$6$	$6$	$-6$	$-6$	$6$	$-6$	$-1$	$1$	$1$	$-1$	$1$	$-1$	$6$	$6$	$6$
18872.b.1j		$6$	$6$	$6$	$6$	$-6$	$-6$	$-1$	$-1$	$-1$	$-1$	$1$	$1$	$6$	$6$	$-6$
18872.b.1k		$12$	$-12$	$-12$	$12$	$0$	$0$	$-2$	$2$	$-2$	$2$	$0$	$0$	$12$	$-12$	$0$
18872.b.1l		$12$	$-12$	$12$	$-12$	$0$	$0$	$-2$	$-2$	$2$	$2$	$0$	$0$	$12$	$-12$	$0$
18872.b.14a		$336$	$336$	$0$	$0$	$336$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$-1$	$-1$	$-1$
18872.b.14b		$336$	$336$	$0$	$0$	$-336$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$-1$	$-1$	$1$
18872.b.14c		$672$	$-672$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$-2$	$2$	$0$