Abstract group 768.1088652: $C_2^5.S_4$

• Label: 768.1088652
• Order: \( 2^{8} \cdot 3 \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{11} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \)
• Perm deg.: $20$
• Trans deg.: $48$
• Rank: $2$

Group information

Description:	$C_2^5.S_4$
Order:	\(768\)\(\medspace = 2^{8} \cdot 3 \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$C_4^3:(C_2^2\times S_4)$, of order \(6144\)\(\medspace = 2^{11} \cdot 3 \)
Composition factors:	$C_2$ x 8, $C_3$
Derived length:	$4$

This group is nonabelian and monomial (hence solvable).

Group statistics

Order	1	2	3	4	6	8
Elements	1	55	128	360	128	96	768
Conjugacy classes	1	7	1	12	1	2	24
Divisions	1	7	1	8	1	1	19
Autjugacy classes	1	5	1	7	1	1	16

Dimension	1	2	3	6	12	24
Irr. complex chars.	4	2	12	2	4	0	24
Irr. rational chars.	2	3	6	5	2	1	19

Minimal presentations

Permutation degree:	$20$
Transitive degree:	$48$
Rank:	$2$
Inequivalent generating pairs:	$18$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	12	24	24
Arbitrary	12	14	14

Constructions

Presentation:	${\langle a, b, c, d, e, f \mid a^{4}=b^{3}=c^{2}=d^{2}=e^{4}=f^{4}=[a,c]= \!\cdots\! \rangle}$
Permutation group:	Degree $20$ $\langle(2,6)(4,13)(7,16)(8,15)(9,14)(10,12)(17,18,19,20), (17,19)(18,20), (2,3,6) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_4^2:A_4)$ $\,\rtimes\,$ $C_4$	$C_2^4$ $\,\rtimes\,$ $(A_4:C_4)$ (2)	$C_4^2$ $\,\rtimes\,$ $(A_4:C_4)$	$(C_4^2:C_2^2)$ $\,\rtimes\,$ $(C_3:C_4)$	more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2^5$ . $S_4$ (2)	$(C_2\times C_4^2)$ . $S_4$	$C_2$ . $(C_4^2:S_4)$	$(C_2^5:A_4)$ . $C_2$	all 7

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

Homology

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

Subgroups

There are 3140 subgroups in 308 conjugacy classes, 15 normal (11 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_4^2:S_4$
Commutator:	$G' \simeq$ $C_4^2:A_4$	$G/G' \simeq$ $C_4$
Frattini:	$\Phi \simeq$ $C_2^3$	$G/\Phi \simeq$ $C_2^2:S_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_4^2:C_2^3$	$G/\operatorname{Fit} \simeq$ $S_3$
Radical:	$R \simeq$ $C_2^5.S_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^3$	$G/\operatorname{soc} \simeq$ $C_2^2:S_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^5.D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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Classes of subgroups up to automorphism

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Normal subgroups

Normal subgroups up to automorphism

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Subgroup information Click on a subgroup in the diagram to see information about it.

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Classes of subgroups up to conjugation

Order 768: $C_2^5.S_4$

Order 384: $C_2^5:A_4$

Order 256: $C_2^5.D_4$

Order 192: $C_2^3.S_4$ x 2, $C_2^3.S_4$, $C_4^2:A_4$

Order 128: $C_2^5:C_4$ x 2, $C_2^4.D_4$ x 2, $C_2^4.D_4$, $C_2^4.D_4$, $C_4^2:C_2^3$

Order 96: $C_4^2:C_6$ x 2, $C_2^3:A_4$ x 2

Order 64: $C_4^2:C_2^2$ x 5, $C_2\wr C_4$ x 4, $C_2^4:C_4$ x 2, $C_2^4:C_4$ x 2, $C_4^2:C_4$ x 2, $C_2^2.C_4^2$ x 2, $C_4^2:C_2^2$ x 2, $C_2^3:D_4$ x 2, $\OD_{16}:C_2^2$, $C_4^2:C_4$, $C_4^2:C_2^2$, $C_2^3.D_4$

Order 48: $A_4:C_4$ x 5, $C_2^2:A_4$ x 2, $C_4^2:C_3$ x 2

Order 32: $C_4^2:C_2$ x 14, $C_2^2\wr C_2$ x 12, $C_2^3:C_4$ x 12, $C_2^3:C_4$ x 4, $\OD_{16}:C_2$ x 2, $C_2^5$ x 2, $C_2^2\times D_4$ x 2, $C_4\times D_4$ x 2, $C_2\times C_4^2$ x 2, $Q_8:C_4$ x 2, $C_2^2\times Q_8$, $C_2\times \OD_{16}$, $C_4^2:C_2$

Order 24: $C_2\times A_4$ x 7

Order 16: $C_2^2:C_4$ x 29, $C_2\times D_4$ x 12, $C_2^4$ x 10, $C_2^2\times C_4$ x 8, $C_4^2$ x 5, $C_2\times Q_8$ x 5, $\OD_{16}$ x 2, $C_4:C_4$ x 2, $C_2\times C_8$

Order 12: $A_4$ x 7, $C_3:C_4$

Order 8: $C_2^3$ x 31, $C_2\times C_4$ x 20, $D_4$ x 8, $Q_8$ x 3, $C_8$

Order 6: $C_6$

Order 4: $C_2^2$ x 29, $C_4$ x 8

Order 3: $C_3$

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 768: $C_2^5.S_4$

Order 384: $C_2^5:A_4$

Order 256: $C_2^5.D_4$

Order 192: $C_2^3.S_4$, $C_2^3.S_4$, $C_4^2:A_4$

Order 128: $C_2^4.D_4$, $C_2^5:C_4$, $C_2^4.D_4$, $C_2^4.D_4$, $C_4^2:C_2^3$

Order 96: $C_4^2:C_6$ x 2, $C_2^3:A_4$

Order 64: $C_4^2:C_2^2$ x 4, $C_4^2:C_4$ x 2, $C_2\wr C_4$ x 2, $C_4^2:C_2^2$ x 2, $\OD_{16}:C_2^2$, $C_2^4:C_4$, $C_2^4:C_4$, $C_2^2.C_4^2$, $C_2^3:D_4$, $C_4^2:C_4$, $C_4^2:C_2^2$, $C_2^3.D_4$

Order 48: $A_4:C_4$ x 2, $C_4^2:C_3$ x 2, $C_2^2:A_4$

Order 32: $C_4^2:C_2$ x 11, $C_2^2\wr C_2$ x 6, $C_2^3:C_4$ x 6, $C_2^3:C_4$ x 2, $C_2^2\times D_4$ x 2, $C_4\times D_4$ x 2, $C_2\times C_4^2$ x 2, $Q_8:C_4$ x 2, $\OD_{16}:C_2$, $C_2^5$, $C_2^2\times Q_8$, $C_2\times \OD_{16}$, $C_4^2:C_2$

Order 24: $C_2\times A_4$ x 3

Order 16: $C_2^2:C_4$ x 13, $C_2\times D_4$ x 10, $C_4^2$ x 5, $C_2^4$ x 5, $C_2\times Q_8$ x 5, $C_2^2\times C_4$ x 5, $\OD_{16}$ x 2, $C_4:C_4$ x 2, $C_2\times C_8$

Order 12: $A_4$ x 3, $C_3:C_4$

Order 8: $C_2\times C_4$ x 15, $C_2^3$ x 14, $D_4$ x 6, $Q_8$ x 3, $C_8$

Order 6: $C_6$

Order 4: $C_2^2$ x 14, $C_4$ x 7

Order 3: $C_3$

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 768: $C_2^5.S_4$ ($C_1$)

Order 384: $C_2^5:A_4$ ($C_2$)

Order 192: $C_4^2:A_4$ ($C_4$)

Order 128: $C_4^2:C_2^3$ ($S_3$)

Order 64: $C_4^2:C_2^2$ ($C_3:C_4$)

Order 32: $C_2^5$ ($S_4$) x 2, $C_2\times C_4^2$ ($S_4$)

Order 16: $C_2^4$ ($A_4:C_4$) x 2, $C_4^2$ ($A_4:C_4$)

Order 8: $C_2^3$ ($C_2^2:S_4$)

Order 4: $C_2^2$ ($C_2^3.S_4$)

Order 2: $C_2$ ($C_4^2:S_4$)

Order 1: $C_1$ ($C_2^5.S_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 768: $C_2^5.S_4$ ($C_1$)

Order 384: $C_2^5:A_4$ ($C_2$)

Order 192: $C_4^2:A_4$ ($C_4$)

Order 128: $C_4^2:C_2^3$ ($S_3$)

Order 64: $C_4^2:C_2^2$ ($C_3:C_4$)

Order 32: $C_2^5$ ($S_4$), $C_2\times C_4^2$ ($S_4$)

Order 16: $C_4^2$ ($A_4:C_4$), $C_2^4$ ($A_4:C_4$)

Order 8: $C_2^3$ ($C_2^2:S_4$)

Order 4: $C_2^2$ ($C_2^3.S_4$)

Order 2: $C_2$ ($C_4^2:S_4$)

Order 1: $C_1$ ($C_2^5.S_4$)

Series

Derived series

$C_2^5.S_4$

$\rhd$

$C_4^2:A_4$

$\rhd$

$C_4^2:C_2^2$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Chief series

$C_2^5.S_4$

$\rhd$

$C_2^5:A_4$

$\rhd$

$C_4^2:A_4$

$\rhd$

$C_4^2:C_2^2$

$\rhd$

$C_2^4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Lower central series

$C_2^5.S_4$

$\rhd$

$C_4^2:A_4$

Upper central series

$C_1$

$\lhd$

$C_2$

Supergroups

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

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

Character theory

Complex character table

See the $24 \times 24$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $19 \times 19$ rational character table.