Abstract group 1536.408633421: $(C_2^2\times C_4^2).S_4$

• Label: 1536.408633421
• Order: \( 2^{9} \cdot 3 \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{12} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \)
• Perm deg.: $24$
• Trans deg.: $24$
• Rank: $2$

Group information

Description:	$(C_2^2\times C_4^2).S_4$
Order:	\(1536\)\(\medspace = 2^{9} \cdot 3 \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$C_4^2:A_4.D_4.C_2^3$, of order \(12288\)\(\medspace = 2^{12} \cdot 3 \)
Composition factors:	$C_2$ x 9, $C_3$
Derived length:	$4$

This group is nonabelian and monomial (hence solvable).

Group statistics

Order	1	2	3	4	6	8
Elements	1	111	128	336	384	576	1536
Conjugacy classes	1	11	1	16	3	16	48
Divisions	1	11	1	14	3	8	38
Autjugacy classes	1	9	1	11	3	2	27

Dimension	1	2	3	6	12	24
Irr. complex chars.	8	4	24	4	8	0	48
Irr. rational chars.	4	6	12	10	4	2	38

Minimal presentations

Permutation degree:	$24$
Transitive degree:	$24$
Rank:	$2$
Inequivalent generating pairs:	$18$

Minimal degrees of faithful linear representations

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

Constructions

Presentation:	${\langle a, b, c, d, e, f \mid a^{4}=b^{6}=e^{4}=f^{4}=[a,e]=[c,d]=[e,f]= \!\cdots\! \rangle}$
Permutation group:	Degree $24$ $\langle(1,2)(5,8,6,7)(9,18)(10,17)(11,19)(12,20)(13,24,14,23)(15,21,16,22), (1,17,9) \!\cdots\! \rangle$
Transitive group:	24T3507	24T3654	24T3665		more information
Direct product:	$C_2$ $\, \times\, $ $((C_2\times C_4^2).S_4)$
Semidirect product:	$(C_2^5:A_4)$ $\,\rtimes\,$ $C_4$	$((C_2\times C_4^2):A_4)$ $\,\rtimes\,$ $C_4$	$(C_4^2:A_4)$ $\,\rtimes\,$ $(C_2\times C_4)$	$(D_4^2:C_2)$ $\,\rtimes\,$ $(C_3:C_4)$	all 6
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_2^3:D_4)$ . $S_4$ (2)	$C_2^5$ . $(A_4:C_4)$ (2)	$(D_4^2:C_2)$ . $D_6$	$(D_4^2:C_2^2)$ . $S_3$	all 17

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

Homology

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

Subgroups

There are 12760 subgroups in 1379 conjugacy classes, 35 normal (23 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $(C_2\times C_4^2).S_4$
Commutator:	$G' \simeq$ $C_4^2:A_4$	$G/G' \simeq$ $C_2\times C_4$
Frattini:	$\Phi \simeq$ $C_2^3$	$G/\Phi \simeq$ $C_2^3:S_4$
Fitting:	$\operatorname{Fit} \simeq$ $D_4^2:C_2^2$	$G/\operatorname{Fit} \simeq$ $S_3$
Radical:	$R \simeq$ $(C_2^2\times C_4^2).S_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^3$	$G/\operatorname{soc} \simeq$ $C_2^3.S_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times C_2^2.C_2^5.C_2$
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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Classes of subgroups up to conjugation

Order 1536: $(C_2^2\times C_4^2).S_4$

Order 768: $(C_2\times C_4^2).S_4$ x 2, $(C_2^2\times C_4^2):A_4$

Order 512: $C_2\times C_2^2.C_2^5.C_2$

Order 384: $C_2^4.S_4$ x 2, $(C_2\times C_4^2):A_4$ x 2, $C_2^4.S_4$, $C_2^5:A_4$

Order 256: $D_4^2:C_4$ x 8, $(D_4\times C_2^3):C_4$ x 2, $C_2\times \OD_{16}:D_4$ x 2, $D_4^2:C_2^2$, $C_4^2.(C_2^2\times C_4)$, $C_2\times \OD_{16}:D_4$

Order 192: $C_2^3.S_4$ x 4, $C_2^3.S_4$ x 2, $C_2^4:A_4$ x 2, $C_2^4.A_4$, $C_4^2:A_4$, $C_2^4.A_4$

Order 128: $(C_2^2\times D_4):C_4$ x 12, $\OD_{16}:D_4$ x 12, $D_4^2:C_2$ x 10, $(C_2\times C_4^2).C_4$ x 6, $\OD_{16}:D_4$ x 6, $C_2^5:C_4$ x 4, $C_4^2:C_2^3$ x 3, $C_2^3.C_4^2$ x 2, $C_4^2:C_2^3$ x 2, $C_2\times D_4^2$ x 2, $\OD_{16}.C_2^3$ x 2, $\OD_{16}:C_2^3$ x 2, $C_2\times C_4^2.C_4$, $C_2\times C_4^2.C_4$, $C_2^4.D_4$, $C_8.C_2^4$, $C_4^2.C_2^3$, $C_4^2.C_2^3$, $C_4^2:C_2^3$, $C_2^3.C_2^4$

Order 96: $C_2^3:A_4$ x 4, $C_4^2:C_6$ x 4, $C_4^2:C_6$ x 2, $C_2^3:A_4$ x 2, $C_2^2.S_4$

Order 64: $C_4^2:C_2^2$ x 40, $D_4^2$ x 20, $\OD_{16}:C_2^2$ x 18, $C_2\wr C_4$ x 16, $C_4^2.C_2^2$ x 16, $C_4^2:C_2^2$ x 15, $C_4^2.C_2^2$ x 12, $\OD_{16}.C_2^2$ x 10, $C_2^3:D_4$ x 10, $\OD_{16}:C_4$ x 8, $C_8.C_2^3$ x 7, $C_4^2:C_2^2$ x 6, $C_2^3:D_4$ x 6, $C_4^2:C_2^2$ x 6, $C_2^3.C_2^3$ x 5, $C_2^2\times \OD_{16}$ x 5, $C_4^2.C_4$ x 4, $C_4^2.C_4$ x 4, $C_4^2:C_4$ x 4, $C_4^2:C_2^2$ x 4, $D_4\times C_2^3$ x 3, $C_4^2:C_2^2$ x 3, $\OD_{16}.C_2^2$ x 2, $C_2^4:C_4$ x 2, $D_4:C_2^3$ x 2, $C_8.C_2^3$ x 2, $C_4^2:C_2^2$ x 2, $C_4^2:C_2^2$ x 2, $C_4^2.C_2^2$, $C_2^2\times C_4^2$

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

Order 32: $C_2^2\wr C_2$ x 60, $C_4:D_4$ x 48, $C_4^2:C_2$ x 44, $C_2^2\times D_4$ x 43, $C_2\times \OD_{16}$ x 39, $C_4^2:C_2$ x 25, $\OD_{16}:C_2$ x 22, $D_4:C_2^2$ x 22, $C_4:D_4$ x 18, $\OD_{16}:C_2$ x 16, $C_4\times D_4$ x 16, $C_2^3:C_4$ x 11, $C_2\times C_4^2$ x 9, $C_4\wr C_2$ x 8, $C_4.D_4$ x 6, $C_2^3:C_4$ x 6, $C_4:Q_8$ x 6, $C_2^2\times C_8$ x 5, $C_2^5$ x 4, $C_2^3\times C_4$ x 4, $C_2^2.D_4$ x 3, $C_2^2\times Q_8$ x 2

Order 24: $C_2\times A_4$ x 5, $C_6:C_4$

Order 16: $C_2\times D_4$ x 109, $C_2^2:C_4$ x 39, $C_2^2\times C_4$ x 38, $\OD_{16}$ x 34, $D_4:C_2$ x 28, $C_2^4$ x 26, $C_2\times C_8$ x 22, $C_4^2$ x 14, $C_4:C_4$ x 11, $C_2\times Q_8$ x 11

Order 12: $A_4$ x 3, $C_3:C_4$ x 2, $C_2\times C_6$

Order 8: $C_2\times C_4$ x 58, $C_2^3$ x 57, $D_4$ x 51, $C_8$ x 8, $Q_8$ x 7

Order 6: $C_6$ x 3

Order 4: $C_2^2$ x 43, $C_4$ x 14

Order 3: $C_3$

Order 2: $C_2$ x 11

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1536: $(C_2^2\times C_4^2).S_4$

Order 768: $(C_2\times C_4^2).S_4$, $(C_2^2\times C_4^2):A_4$

Order 512: $C_2\times C_2^2.C_2^5.C_2$

Order 384: $(C_2\times C_4^2):A_4$ x 2, $C_2^4.S_4$, $C_2^4.S_4$, $C_2^5:A_4$

Order 256: $D_4^2:C_4$ x 3, $D_4^2:C_2^2$, $C_4^2.(C_2^2\times C_4)$, $(D_4\times C_2^3):C_4$, $C_2\times \OD_{16}:D_4$, $C_2\times \OD_{16}:D_4$

Order 192: $C_2^4.A_4$, $C_2^3.S_4$, $C_2^3.S_4$, $C_4^2:A_4$, $C_2^4.A_4$, $C_2^4:A_4$

Order 128: $D_4^2:C_2$ x 8, $(C_2\times C_4^2).C_4$ x 4, $(C_2^2\times D_4):C_4$ x 4, $\OD_{16}:D_4$ x 4, $\OD_{16}:D_4$ x 4, $C_2^5:C_4$ x 2, $C_4^2:C_2^3$ x 2, $C_4^2:C_2^3$ x 2, $C_2\times C_4^2.C_4$, $C_2\times C_4^2.C_4$, $C_2^3.C_4^2$, $C_2^4.D_4$, $C_8.C_2^4$, $C_4^2.C_2^3$, $C_2\times D_4^2$, $C_4^2.C_2^3$, $C_4^2:C_2^3$, $\OD_{16}.C_2^3$, $\OD_{16}:C_2^3$, $C_2^3.C_2^4$

Order 96: $C_4^2:C_6$ x 4, $C_4^2:C_6$ x 2, $C_2^3:A_4$ x 2, $C_2^3:A_4$, $C_2^2.S_4$

Order 64: $C_4^2:C_2^2$ x 24, $C_4^2:C_2^2$ x 11, $C_4^2.C_2^2$ x 10, $D_4^2$ x 10, $C_4^2.C_2^2$ x 10, $\OD_{16}:C_2^2$ x 5, $C_4^2:C_2^2$ x 5, $C_2^3:D_4$ x 5, $\OD_{16}.C_2^2$ x 4, $C_2\wr C_4$ x 4, $C_8.C_2^3$ x 4, $C_2^3.C_2^3$ x 3, $C_2^2\times \OD_{16}$ x 3, $C_4^2:C_2^2$ x 3, $C_2^3:D_4$ x 3, $C_4^2:C_2^2$ x 3, $C_4^2:C_2^2$ x 3, $C_4^2.C_4$ x 2, $C_4^2.C_4$ x 2, $D_4:C_2^3$ x 2, $D_4\times C_2^3$ x 2, $\OD_{16}:C_4$ x 2, $C_4^2:C_4$ x 2, $\OD_{16}.C_2^2$, $C_2^4:C_4$, $C_8.C_2^3$, $C_4^2.C_2^2$, $C_4^2:C_2^2$, $C_2^2\times C_4^2$, $C_4^2:C_2^2$

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

Order 32: $C_4^2:C_2$ x 30, $C_2^2\wr C_2$ x 30, $C_2^2\times D_4$ x 28, $C_4:D_4$ x 24, $D_4:C_2^2$ x 20, $C_2\times \OD_{16}$ x 17, $C_4^2:C_2$ x 16, $C_4:D_4$ x 15, $C_2\times C_4^2$ x 9, $C_4\times D_4$ x 8, $C_2^3:C_4$ x 6, $C_4:Q_8$ x 5, $\OD_{16}:C_2$ x 4, $C_2^3\times C_4$ x 3, $\OD_{16}:C_2$ x 3, $C_2^2\times C_8$ x 3, $C_4.D_4$ x 2, $C_2^3:C_4$ x 2, $C_2^5$ x 2, $C_2^2\times Q_8$ x 2, $C_2^2.D_4$ x 2, $C_4\wr C_2$ x 2

Order 24: $C_2\times A_4$ x 4, $C_6:C_4$

Order 16: $C_2\times D_4$ x 70, $C_2^2\times C_4$ x 29, $D_4:C_2$ x 24, $C_2^2:C_4$ x 20, $C_2^4$ x 14, $C_4^2$ x 12, $\OD_{16}$ x 10, $C_2\times Q_8$ x 10, $C_2\times C_8$ x 8, $C_4:C_4$ x 6

Order 12: $A_4$ x 2, $C_2\times C_6$, $C_3:C_4$

Order 8: $C_2\times C_4$ x 45, $C_2^3$ x 32, $D_4$ x 32, $Q_8$ x 6, $C_8$ x 2

Order 6: $C_6$ x 3

Order 4: $C_2^2$ x 26, $C_4$ x 11

Order 3: $C_3$

Order 2: $C_2$ x 9

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1536: $(C_2^2\times C_4^2).S_4$ ($C_1$)

Order 768: $(C_2\times C_4^2).S_4$ ($C_2$) x 2, $(C_2^2\times C_4^2):A_4$ ($C_2$)

Order 384: $(C_2\times C_4^2):A_4$ ($C_2^2$), $(C_2\times C_4^2):A_4$ ($C_4$), $C_2^5:A_4$ ($C_4$)

Order 256: $D_4^2:C_2^2$ ($S_3$)

Order 192: $C_4^2:A_4$ ($C_2\times C_4$)

Order 128: $D_4^2:C_2$ ($D_6$), $D_4^2:C_2$ ($C_3:C_4$), $C_4^2:C_2^3$ ($C_3:C_4$)

Order 64: $C_2^3:D_4$ ($S_4$) x 2, $C_4^2:C_2^2$ ($C_6:C_4$), $C_2^2\times C_4^2$ ($S_4$)

Order 32: $C_2^5$ ($A_4:C_4$) x 2, $C_2^2\wr C_2$ ($C_2\times S_4$) x 2, $C_2^2\wr C_2$ ($A_4:C_4$) x 2, $C_2\times C_4^2$ ($A_4:C_4$) x 2, $C_2\times C_4^2$ ($C_2\times S_4$)

Order 16: $C_2^4$ ($C_2^2.S_4$) x 2, $C_4^2$ ($C_2^2.S_4$), $C_2^4$ ($C_2^2:S_4$)

Order 8: $C_2^3$ ($C_2^3.S_4$) x 2, $C_2^3$ ($C_2^3:S_4$)

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

Order 2: $C_2$ ($(C_2\times C_4^2).S_4$)

Order 1: $C_1$ ($(C_2^2\times C_4^2).S_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1536: $(C_2^2\times C_4^2).S_4$ ($C_1$)

Order 768: $(C_2\times C_4^2).S_4$ ($C_2$), $(C_2^2\times C_4^2):A_4$ ($C_2$)

Order 384: $(C_2\times C_4^2):A_4$ ($C_2^2$), $(C_2\times C_4^2):A_4$ ($C_4$), $C_2^5:A_4$ ($C_4$)

Order 256: $D_4^2:C_2^2$ ($S_3$)

Order 192: $C_4^2:A_4$ ($C_2\times C_4$)

Order 128: $D_4^2:C_2$ ($D_6$), $D_4^2:C_2$ ($C_3:C_4$), $C_4^2:C_2^3$ ($C_3:C_4$)

Order 64: $C_4^2:C_2^2$ ($C_6:C_4$), $C_2^3:D_4$ ($S_4$), $C_2^2\times C_4^2$ ($S_4$)

Order 32: $C_2\times C_4^2$ ($A_4:C_4$) x 2, $C_2^5$ ($A_4:C_4$), $C_2^2\wr C_2$ ($C_2\times S_4$), $C_2^2\wr C_2$ ($A_4:C_4$), $C_2\times C_4^2$ ($C_2\times S_4$)

Order 16: $C_4^2$ ($C_2^2.S_4$), $C_2^4$ ($C_2^2:S_4$), $C_2^4$ ($C_2^2.S_4$)

Order 8: $C_2^3$ ($C_2^3.S_4$) x 2, $C_2^3$ ($C_2^3:S_4$)

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

Order 2: $C_2$ ($(C_2\times C_4^2).S_4$)

Order 1: $C_1$ ($(C_2^2\times C_4^2).S_4$)

Series

Derived series

$(C_2^2\times C_4^2).S_4$

$\rhd$

$C_4^2:A_4$

$\rhd$

$C_4^2:C_2^2$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Chief series

$(C_2^2\times C_4^2).S_4$

$\rhd$

$(C_2\times C_4^2).S_4$

$\rhd$

$(C_2\times C_4^2):A_4$

$\rhd$

$C_4^2:A_4$

$\rhd$

$C_4^2:C_2^2$

$\rhd$

$C_2^4$

$\rhd$

$C_4^2$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Lower central series

$(C_2^2\times C_4^2).S_4$

$\rhd$

$C_4^2:A_4$

Upper central series

$C_1$

$\lhd$

$C_2$

Supergroups

This group is a maximal subgroup of 6 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 $48 \times 48$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $38 \times 38$ rational character table.