Abstract group 57344.v: $C_2^{12}.C_{14}$

• Label: 57344.v
• Order: \( 2^{13} \cdot 7 \)
• Exponent: \( 2 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2 \cdot 7 \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{19} \cdot 3^{5} \cdot 7^{4} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{7} \cdot 3^{5} \cdot 7^{3} \)
• Perm deg.: $28$
• Trans deg.: $28$
• Rank: $3$

Group information

Description:	$C_2^{12}.C_{14}$
Order:	\(57344\)\(\medspace = 2^{13} \cdot 7 \)
Exponent:	\(14\)\(\medspace = 2 \cdot 7 \)
Automorphism group:	$F_5^3$, of order \(305892163584\)\(\medspace = 2^{19} \cdot 3^{5} \cdot 7^{4} \)
Composition factors:	$C_2$ x 13, $C_7$
Derived length:	$2$

This group is nonabelian, metabelian (hence solvable), and an A-group. Whether it is monomial has not been computed.

Group statistics

Order	1	2	7	14
Elements	1	8191	24576	24576	57344
Conjugacy classes	1	1171	6	6	1184
Divisions	1	1171	1	1	1174
Autjugacy classes	1	5	1	1	8

Minimal presentations

Permutation degree:	$28$
Transitive degree:	$28$
Rank:	$3$
Inequivalent generating triples:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	not computed	none
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h, i, j, k, l, m \mid a^{14}=b^{2}=c^{2}=d^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $28$ $\langle(17,18)(19,20)(25,26)(27,28), (1,7,10,14,18,24,26,3,5,11,15,20,22,27)(2,8,9,13,17,23,25,4,6,12,16,19,21,28) \!\cdots\! \rangle$
Transitive group:	28T591				more information
Direct product:	$C_2$ $\, \times\, $ $(C_2^{12}:C_7)$
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$C_2^{10}$ . $F_8$	$C_2^{12}$ . $C_{14}$	$C_2^9$ . $(C_2\times F_8)$	$C_2$ . $(C_2^9.F_8)$	all 11

Elements of the group are displayed as permutations of degree 28.

Homology

Abelianization:	$C_{14} \simeq C_{2} \times C_{7}$
Schur multiplier:	$C_{2}^{12}$
Commutator length:	$1$

Subgroups

There are 244 normal subgroups (6 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_2^9.F_8$
Commutator:	$G' \simeq$ $C_2^{12}$	$G/G' \simeq$ $C_{14}$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_2^{12}.C_{14}$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^{13}$	$G/\operatorname{Fit} \simeq$ $C_7$
Radical:	$R \simeq$ $C_2^{12}.C_{14}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^{13}$	$G/\operatorname{soc} \simeq$ $C_7$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^{13}$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: inclusions not computed

Classes of subgroups up to automorphism

No diagram available: inclusions not computed

Normal subgroups

No diagram available: inclusions not computed

Normal subgroups up to automorphism

No diagram available: inclusions not computed

Classes of subgroups up to conjugation

All subgroups of index up to 7 (order at least 8192) are shown, as well as all normal subgroups of any index.

Order 57344: $C_2^{12}.C_{14}$

Order 28672: $C_2^{12}:C_7$

Order 8192: $C_2^{13}$

Order 7168: $C_2^{10}:C_7$ x 18

Order 4096: $C_2^{12}$

Order 1024: $C_2^{10}$ x 18

Order 512: $C_2^9$ x 18

Order 128: $C_2^7$ x 83

Order 64: $C_2^6$ x 83

Order 16: $C_2^4$ x 18

Order 8: $C_2^3$ x 18

Order 7: $C_7$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 7 (order at least 8192) are shown, as well as all normal subgroups of any index.

Order 57344: $C_2^{12}.C_{14}$

Order 28672: $C_2^{12}:C_7$

Order 8192: $C_2^{13}$

Order 7168: $C_2^{10}:C_7$

Order 4096: $C_2^{12}$

Order 1024: $C_2^{10}$

Order 512: $C_2^9$

Order 128: $C_2^7$ x 2

Order 64: $C_2^6$ x 2

Order 16: $C_2^4$

Order 8: $C_2^3$

Order 7: $C_7$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 57344: $C_2^{12}.C_{14}$ ($C_1$)

Order 28672: $C_2^{12}:C_7$ ($C_2$)

Order 8192: $C_2^{13}$ ($C_7$)

Order 4096: $C_2^{12}$ ($C_{14}$)

Order 1024: $C_2^{10}$ ($F_8$) x 18

Order 512: $C_2^9$ ($C_2\times F_8$) x 18

Order 128: $C_2^7$ ($C_2^3:F_8$) x 81, $C_2^7$ ($C_2^3:F_8$) x 2

Order 64: $C_2^6$ ($C_2\wr C_7$) x 81, $C_2^6$ ($C_2^4:F_8$) x 2

Order 16: $C_2^4$ ($C_2^9:C_7$) x 18

Order 8: $C_2^3$ ($C_2^{10}:C_7$) x 18

Order 2: $C_2$ ($C_2^9.F_8$)

Order 1: $C_1$ ($C_2^{12}.C_{14}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 57344: $C_2^{12}.C_{14}$ ($C_1$)

Order 28672: $C_2^{12}:C_7$ ($C_2$)

Order 8192: $C_2^{13}$ ($C_7$)

Order 4096: $C_2^{12}$ ($C_{14}$)

Order 1024: $C_2^{10}$ ($F_8$)

Order 512: $C_2^9$ ($C_2\times F_8$)

Order 128: $C_2^7$ ($C_2^3:F_8$), $C_2^7$ ($C_2^3:F_8$)

Order 64: $C_2^6$ ($C_2\wr C_7$), $C_2^6$ ($C_2^4:F_8$)

Order 16: $C_2^4$ ($C_2^9:C_7$)

Order 8: $C_2^3$ ($C_2^{10}:C_7$)

Order 2: $C_2$ ($C_2^9.F_8$)

Order 1: $C_1$ ($C_2^{12}.C_{14}$)

Series

Derived series

$C_2^{12}.C_{14}$

$\rhd$

$C_2^{12}$

$\rhd$

$C_1$

Chief series

$C_2^{12}.C_{14}$

$\rhd$

$C_2^{12}:C_7$

$\rhd$

$C_2^{12}$

$\rhd$

$C_2^9$

$\rhd$

$C_2^6$

$\rhd$

$C_2^3$

$\rhd$

$C_1$

Lower central series

$C_2^{12}.C_{14}$

$\rhd$

$C_2^{12}$

Upper central series

$C_1$

$\lhd$

$C_2$

Supergroups

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

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

Character theory

Complex character table

The $1184 \times 1184$ character table is not available for this group.

Rational character table

The $1174 \times 1174$ rational character table is not available for this group.