Abstract group 1536.408641060: $C_2^6:S_4$

• Label: 1536.408641060
• Order: \( 2^{9} \cdot 3 \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \)
• $\card{Z(G)}$: \( 2^{4} \)
• $\card{\Aut(G)}$: \( 2^{16} \cdot 3^{4} \cdot 5 \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{11} \cdot 3^{3} \cdot 5 \cdot 7 \)
• Perm deg.: not computed
• Trans deg.: $48$
• Rank: $5$

Group information

Description:	$C_2^6:S_4$
Order:	\(1536\)\(\medspace = 2^{9} \cdot 3 \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$A_4^2.C_2^6.A_8$, of order \(185794560\)\(\medspace = 2^{16} \cdot 3^{4} \cdot 5 \cdot 7 \)
Composition factors:	$C_2$ x 9, $C_3$
Derived length:	$3$

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

Group statistics

Order	1	2	3	4	6
Elements	1	447	32	576	480	1536
Conjugacy classes	1	95	1	48	15	160
Divisions	1	95	1	48	15	160
Autjugacy classes	1	6	1	1	1	10

Dimension	1	2	3	6
Irr. complex chars.	32	16	96	16	160
Irr. rational chars.	32	16	96	16	160

Minimal presentations

Permutation degree:	not computed
Transitive degree:	$48$
Rank:	$5$
Inequivalent generating 5-tuples:	$13385800$

Minimal degrees of faithful linear representations

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

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h, i \mid a^{2}=b^{6}=c^{2}=d^{2}=e^{2}=f^{2}= \!\cdots\! \rangle}$
Direct product:	$C_2$ ${}^4$ $\, \times\, $ $(C_2^2:S_4)$
Semidirect product:	$C_2^8$ $\,\rtimes\,$ $S_3$	$C_2^6$ $\,\rtimes\,$ $S_4$	$C_2^7$ $\,\rtimes\,$ $D_6$	$C_2^5$ $\,\rtimes\,$ $(C_2\times S_4)$	all 15
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Aut. group:	$\Aut(Q_8:D_4)$	$\Aut(Q_8\times C_{24})$

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

Homology

Abelianization:	$C_{2}^{5} $
Schur multiplier:	$C_{2}^{13}$
Commutator length:	$1$

Subgroups

There are 782110 subgroups in 144837 conjugacy classes, 709 normal (7 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^4$	$G/Z \simeq$ $C_2^2:S_4$
Commutator:	$G' \simeq$ $C_2^2:A_4$	$G/G' \simeq$ $C_2^5$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_2^6:S_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^8$	$G/\operatorname{Fit} \simeq$ $S_3$
Radical:	$R \simeq$ $C_2^6:S_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^8$	$G/\operatorname{soc} \simeq$ $S_3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times C_2^7.C_2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: subgroups only stored up to automorphism

Classes of subgroups up to automorphism

No diagram available

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

Order 1536: $C_2^6:S_4$

Order 768: $C_2^5:S_4$ x 30, $C_2^8.C_3$

Order 512: $C_2\times C_2^7.C_2$

Order 384: $C_2^2\wr S_3$ x 140, $C_2^5:A_4$ x 15, $C_2^4\times S_4$ x 3

Order 256: $C_2^5:D_4$ x 120, $D_4\times C_2^5$ x 3, $C_2^6:C_4$ x 3, $C_2^8$

Order 192: $C_2^3:S_4$ x 120, $C_2^3\times S_4$ x 90, $C_2^2\wr C_3$ x 35, $A_4\times C_2^4$ x 4

Order 128: $C_2^4:D_4$ x 2240, $D_4\times C_2^4$ x 276, $C_2^5:C_4$ x 180, $C_2^7$ x 80, $C_2^5\times C_4$ x 3

Order 96: $C_2^2\times S_4$ x 420, $C_2^3\times A_4$ x 60, $C_2^2:S_4$ x 16, $C_2^3:A_4$ x 15, $C_2^3\times D_6$

Order 64: $C_2^3:D_4$ x 7680, $D_4\times C_2^3$ x 4560, $C_2^6$ x 2264, $C_2^4:C_4$ x 1680, $C_2^4\times C_4$ x 138

Order 48: $C_2\times S_4$ x 360, $C_2^2\times A_4$ x 140, $C_2^2\times D_6$ x 30, $C_2^3\times C_6$, $C_2^2:A_4$

Order 32: $C_2^5$ x 18780, $C_2^2\times D_4$ x 16320, $C_2^2\wr C_2$ x 4096, $C_2^3:C_4$ x 2880, $C_2^3\times C_4$ x 1140

Order 24: $C_2\times D_6$ x 140, $C_2\times A_4$ x 60, $S_4$ x 48, $C_2^2\times C_6$ x 15

Order 16: $C_2^4$ x 39557, $C_2\times D_4$ x 12288, $C_2^2\times C_4$ x 2040, $C_2^2:C_4$ x 768

Order 12: $D_6$ x 120, $C_2\times C_6$ x 35, $A_4$ x 4

Order 8: $C_2^3$ x 20640, $D_4$ x 1536, $C_2\times C_4$ x 768

Order 6: $S_3$ x 16, $C_6$ x 15

Order 4: $C_2^2$ x 2714, $C_4$ x 48

Order 3: $C_3$

Order 2: $C_2$ x 95

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1536: $C_2^6:S_4$

Order 768: $C_2^5:S_4$, $C_2^8.C_3$

Order 512: $C_2\times C_2^7.C_2$

Order 384: $C_2^5:A_4$, $C_2^2\wr S_3$, $C_2^4\times S_4$

Order 256: $C_2^5:D_4$ x 2, $C_2^8$, $D_4\times C_2^5$, $C_2^6:C_4$

Order 192: $A_4\times C_2^4$ x 2, $C_2^2\wr C_3$, $C_2^3:S_4$, $C_2^3\times S_4$

Order 128: $C_2^7$ x 6, $D_4\times C_2^4$ x 6, $C_2^4:D_4$ x 3, $C_2^5:C_4$ x 2, $C_2^5\times C_4$

Order 96: $C_2^3\times A_4$ x 2, $C_2^3\times D_6$, $C_2^3:A_4$, $C_2^2:S_4$, $C_2^2\times S_4$

Order 64: $C_2^6$ x 25, $D_4\times C_2^3$ x 9, $C_2^4\times C_4$ x 3, $C_2^3:D_4$ x 3, $C_2^4:C_4$ x 2

Order 48: $C_2^2\times A_4$ x 2, $C_2^3\times C_6$, $C_2^2\times D_6$, $C_2^2:A_4$, $C_2\times S_4$

Order 32: $C_2^5$ x 46, $C_2^2\times D_4$ x 9, $C_2^3\times C_4$ x 3, $C_2^2\wr C_2$ x 3, $C_2^3:C_4$ x 2

Order 24: $C_2\times A_4$ x 2, $C_2^2\times C_6$, $C_2\times D_6$, $S_4$

Order 16: $C_2^4$ x 57, $C_2\times D_4$ x 9, $C_2^2\times C_4$ x 3, $C_2^2:C_4$ x 2

Order 12: $A_4$ x 2, $C_2\times C_6$, $D_6$

Order 8: $C_2^3$ x 46, $D_4$ x 4, $C_2\times C_4$ x 3

Order 6: $C_6$, $S_3$

Order 4: $C_2^2$ x 25, $C_4$

Order 3: $C_3$

Order 2: $C_2$ x 6

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1536: $C_2^6:S_4$ ($C_1$)

Order 768: $C_2^5:S_4$ ($C_2$) x 30, $C_2^8.C_3$ ($C_2$)

Order 384: $C_2^2\wr S_3$ ($C_2^2$) x 140, $C_2^5:A_4$ ($C_2^2$) x 15

Order 256: $C_2^8$ ($S_3$)

Order 192: $C_2^3:S_4$ ($C_2^3$) x 120, $C_2^2\wr C_3$ ($C_2^3$) x 35

Order 128: $C_2^7$ ($D_6$) x 15

Order 96: $C_2^2:S_4$ ($C_2^4$) x 16, $C_2^3:A_4$ ($C_2^4$) x 15

Order 64: $C_2^6$ ($C_2\times D_6$) x 35, $C_2^6$ ($S_4$) x 3

Order 48: $C_2^2:A_4$ ($C_2^5$)

Order 32: $C_2^5$ ($C_2\times S_4$) x 45, $C_2^5$ ($C_2^2\times D_6$) x 15

Order 16: $C_2^4$ ($C_2^2\times S_4$) x 105, $C_2^4$ ($C_2^3\times D_6$), $C_2^4$ ($C_2^2:S_4$)

Order 8: $C_2^3$ ($C_2^3\times S_4$) x 45, $C_2^3$ ($C_2^3:S_4$) x 15

Order 4: $C_2^2$ ($C_2^2\wr S_3$) x 35, $C_2^2$ ($C_2^4\times S_4$) x 3

Order 2: $C_2$ ($C_2^5:S_4$) x 15

Order 1: $C_1$ ($C_2^6:S_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1536: $C_2^6:S_4$ ($C_1$)

Order 768: $C_2^5:S_4$ ($C_2$), $C_2^8.C_3$ ($C_2$)

Order 384: $C_2^5:A_4$ ($C_2^2$), $C_2^2\wr S_3$ ($C_2^2$)

Order 256: $C_2^8$ ($S_3$)

Order 192: $C_2^2\wr C_3$ ($C_2^3$), $C_2^3:S_4$ ($C_2^3$)

Order 128: $C_2^7$ ($D_6$)

Order 96: $C_2^3:A_4$ ($C_2^4$), $C_2^2:S_4$ ($C_2^4$)

Order 64: $C_2^6$ ($C_2\times D_6$), $C_2^6$ ($S_4$)

Order 48: $C_2^2:A_4$ ($C_2^5$)

Order 32: $C_2^5$ ($C_2^2\times D_6$), $C_2^5$ ($C_2\times S_4$)

Order 16: $C_2^4$ ($C_2^3\times D_6$), $C_2^4$ ($C_2^2:S_4$), $C_2^4$ ($C_2^2\times S_4$)

Order 8: $C_2^3$ ($C_2^3:S_4$), $C_2^3$ ($C_2^3\times S_4$)

Order 4: $C_2^2$ ($C_2^2\wr S_3$), $C_2^2$ ($C_2^4\times S_4$)

Order 2: $C_2$ ($C_2^5:S_4$)

Order 1: $C_1$ ($C_2^6:S_4$)

Series

Derived series

$C_2^6:S_4$

$\rhd$

$C_2^2:A_4$

$\rhd$

$C_2^4$

$\rhd$

$C_1$

Chief series

$C_2^6:S_4$

$\rhd$

$C_2^5:S_4$

$\rhd$

$C_2^5:A_4$

$\rhd$

$C_2^7$

$\rhd$

$C_2^6$

$\rhd$

$C_2^5$

$\rhd$

$C_2^3$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Lower central series

$C_2^6:S_4$

$\rhd$

$C_2^2:A_4$

Upper central series

$C_1$

$\lhd$

$C_2^4$

Supergroups

This group is a maximal subgroup of 11 larger groups in the database.

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

Character theory

Complex character table

Every character has rational values, so the complex character table is the same as the rational character table below.

Rational character table

See the $160 \times 160$ rational character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.