Abstract group 688128.bd: $C_2^{12}.(C_7\times S_4)$

• Label: 688128.bd
• Order: \( 2^{15} \cdot 3 \cdot 7 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2 \cdot 7 \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{16} \cdot 3^{2} \cdot 7^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2 \cdot 3 \cdot 7 \)
• Perm deg.: $28$
• Trans deg.: $28$
• Rank: $2$

Group information

Description:	$C_2^{12}.(C_7\times S_4)$
Order:	\(688128\)\(\medspace = 2^{15} \cdot 3 \cdot 7 \)
Exponent:	\(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \)
Automorphism group:	Group of order \(28901376\)\(\medspace = 2^{16} \cdot 3^{2} \cdot 7^{2} \)
Composition factors:	$C_2$ x 15, $C_3$, $C_7$
Derived length:	$3$

This group is nonabelian and solvable. Whether it is monomial has not been computed.

Group statistics

Order	1	2	3	4	7	14	21	28
Elements	1	16767	32768	48768	24576	221184	196608	147456	688128
Conjugacy classes	1	401	1	19	6	12	6	6	452
Divisions	1	401	1	19	1	2	1	1	427
Autjugacy classes	1	24	1	5	1	2	1	1	36

Minimal presentations

Permutation degree:	$28$
Transitive degree:	$28$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	21	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p \mid b^{3}=c^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $28$ $\langle(1,12,18,28,6,14,24,2,9,17,25,5,15,23,3,10,20,26,8,16,22,4,11,19,27,7,13,21), (1,3,4,2)(5,6,8,7)(9,10,12,11)(14,15)(17,20)(21,22,24,23)(25,27,28,26)\rangle$
Transitive group:	28T1142	42T1805	42T1806		more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$(C_2^{12}:C_7)$ . $S_4$	$C_2^{14}$ . $(S_3\times C_7)$	$C_2^{12}$ . $(C_7\times S_4)$	$(C_2^{14}.C_{21})$ . $C_2$	all 9

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

Homology

Abelianization:	$C_{14} \simeq C_{2} \times C_{7}$
Schur multiplier:	$C_{2}^{4}$
Commutator length:	$1$

Subgroups

There are 14 normal subgroups (10 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $C_2^{12}.(C_7\times S_4)$
Commutator:	$G' \simeq$ $C_2^{14}.C_3$	$G/G' \simeq$ $C_{14}$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_2^{12}.(C_7\times S_4)$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^{14}$	$G/\operatorname{Fit} \simeq$ $S_3\times C_7$
Radical:	$R \simeq$ $C_2^{12}.(C_7\times S_4)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^{14}$	$G/\operatorname{soc} \simeq$ $S_3\times C_7$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^8.C_2^6.C_2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
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 24 (order at least 28672) are shown, as well as all normal subgroups of any index.

Order 688128: $C_2^{12}.(C_7\times S_4)$

Order 344064: $C_2^{14}.C_{21}$

Order 229376: $\ASigmaL(1,16384)$

Order 172032: $C_2^{12}.(S_3\times C_7)$

Order 114688: $C_2^7.C_2\wr C_7$, $C_2\times C_2^{12}.C_{14}$, $C_2^{12}.C_{28}$

Order 98304: $C_2^{14}.S_3$

Order 86016: $C_2^{12}:C_{21}$

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

Order 49152: $C_2^{14}.C_3$

Order 32768: $C_2^8.C_2^6.C_2$

Order 28672: $C_2^6.(D_4\times F_8)$ x 2, $C_2^{12}:C_7$

Order 16384: $C_2^{14}$

Order 10752: $C_2^6.(C_7\times S_4)$ x 2

Order 4096: $C_2^{12}$

Order 256: $C_2^8$ x 2

Order 64: $C_2^6$ x 2

Order 7: $C_7$

Order 4: $C_2^2$

Order 3: $C_3$

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 24 (order at least 28672) are shown, as well as all normal subgroups of any index.

Order 688128: $C_2^{12}.(C_7\times S_4)$

Order 344064: $C_2^{14}.C_{21}$

Order 229376: $\ASigmaL(1,16384)$

Order 172032: $C_2^{12}.(S_3\times C_7)$

Order 114688: $C_2^7.C_2\wr C_7$, $C_2\times C_2^{12}.C_{14}$, $C_2^{12}.C_{28}$

Order 98304: $C_2^{14}.S_3$

Order 86016: $C_2^{12}:C_{21}$

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

Order 49152: $C_2^{14}.C_3$

Order 32768: $C_2^8.C_2^6.C_2$

Order 28672: $C_2^{12}:C_7$, $C_2^6.(D_4\times F_8)$

Order 16384: $C_2^{14}$

Order 10752: $C_2^6.(C_7\times S_4)$

Order 4096: $C_2^{12}$

Order 256: $C_2^8$

Order 64: $C_2^6$

Order 7: $C_7$

Order 4: $C_2^2$

Order 3: $C_3$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 688128: $C_2^{12}.(C_7\times S_4)$ ($C_1$)

Order 344064: $C_2^{14}.C_{21}$ ($C_2$)

Order 114688: $C_2^7.C_2\wr C_7$ ($S_3$)

Order 98304: $C_2^{14}.S_3$ ($C_7$)

Order 49152: $C_2^{14}.C_3$ ($C_{14}$)

Order 28672: $C_2^{12}:C_7$ ($S_4$)

Order 16384: $C_2^{14}$ ($S_3\times C_7$)

Order 4096: $C_2^{12}$ ($C_7\times S_4$)

Order 256: $C_2^8$ ($C_2^6:(S_3\times C_7)$) x 2

Order 64: $C_2^6$ ($C_2^6.(C_7\times S_4)$) x 2

Order 4: $C_2^2$ ($C_2^{12}.(S_3\times C_7)$)

Order 1: $C_1$ ($C_2^{12}.(C_7\times S_4)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 688128: $C_2^{12}.(C_7\times S_4)$ ($C_1$)

Order 344064: $C_2^{14}.C_{21}$ ($C_2$)

Order 114688: $C_2^7.C_2\wr C_7$ ($S_3$)

Order 98304: $C_2^{14}.S_3$ ($C_7$)

Order 49152: $C_2^{14}.C_3$ ($C_{14}$)

Order 28672: $C_2^{12}:C_7$ ($S_4$)

Order 16384: $C_2^{14}$ ($S_3\times C_7$)

Order 4096: $C_2^{12}$ ($C_7\times S_4$)

Order 256: $C_2^8$ ($C_2^6:(S_3\times C_7)$)

Order 64: $C_2^6$ ($C_2^6.(C_7\times S_4)$)

Order 4: $C_2^2$ ($C_2^{12}.(S_3\times C_7)$)

Order 1: $C_1$ ($C_2^{12}.(C_7\times S_4)$)

Series

Derived series

$C_2^{12}.(C_7\times S_4)$

$\rhd$

$C_2^{14}.C_3$

$\rhd$

$C_2^{14}$

$\rhd$

$C_1$

Chief series

$C_2^{12}.(C_7\times S_4)$

$\rhd$

$C_2^{14}.C_{21}$

$\rhd$

$C_2^{14}.C_3$

$\rhd$

$C_2^{14}$

$\rhd$

$C_2^8$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Lower central series

$C_2^{12}.(C_7\times S_4)$

$\rhd$

$C_2^{14}.C_3$

Upper central series

$C_1$

Supergroups

This group is a maximal subgroup of 5 larger groups in the database.

This group is a maximal quotient of 0 larger groups in the database.

Character theory

Complex character table

The $452 \times 452$ character table is not available for this group.

Rational character table

The $427 \times 427$ rational character table is not available for this group.