Abstract group 1536.408544248: $C_4^2.(C_2^3\times A_4)$

• Label: 1536.408544248
• Order: \( 2^{9} \cdot 3 \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 3 \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{14} \cdot 3^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \cdot 3 \)
• Perm deg.: $24$
• Trans deg.: $24$
• Rank: $3$

Group information

Description:	$C_4^2.(C_2^3\times A_4)$
Order:	\(1536\)\(\medspace = 2^{9} \cdot 3 \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_2^4.C_2^6.A_4.D_6$, of order \(147456\)\(\medspace = 2^{14} \cdot 3^{2} \)
Composition factors:	$C_2$ x 9, $C_3$
Derived length:	$3$

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

Group statistics

Order	1	2	3	4	6
Elements	1	223	128	288	896	1536
Conjugacy classes	1	25	2	30	14	72
Divisions	1	25	1	30	7	64
Autjugacy classes	1	7	1	4	2	15

Dimension	1	2	3	12
Irr. complex chars.	24	0	40	8	72
Irr. rational chars.	8	8	40	8	64

Minimal presentations

Permutation degree:	$24$
Transitive degree:	$24$
Rank:	$3$
Inequivalent generating triples:	$5460$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	12	12	12
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g \mid a^{6}=c^{2}=d^{4}=e^{2}=f^{4}=g^{2}=[b,g]= \!\cdots\! \rangle}$
Permutation group:	Degree $24$ $\langle(1,17,9)(2,18,10)(3,19,11)(4,20,12)(5,21,13)(6,22,14)(7,23,15)(8,24,16) \!\cdots\! \rangle$
Transitive group:	24T3393	24T3436	24T3613		more information
Direct product:	$C_2$ $\, \times\, $ $(C_4^2.(C_2^2\times A_4))$
Semidirect product:	$(C_2^5:A_4)$ $\,\rtimes\,$ $C_2^2$	$(C_4^2:A_4)$ $\,\rtimes\,$ $C_2^3$	$(C_4^2:C_2^3)$ $\,\rtimes\,$ $A_4$	$(C_4^2:C_2^3)$ $\,\rtimes\,$ $A_4$	all 13
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2^5$ . $(C_2^2:A_4)$	$C_2^5$ . $(C_2^2\times A_4)$	$C_2^4$ . $(C_2^3:A_4)$ (2)	$C_2^4$ . $(C_2^3\times A_4)$	all 13

Elements of the group are displayed as permutations of degree 24.

Homology

Abelianization:	$C_{2}^{2} \times C_{6} \simeq C_{2}^{3} \times C_{3}$
Schur multiplier:	$C_{2}^{7}$
Commutator length:	$1$

Subgroups

There are 72786 subgroups in 13760 conjugacy classes, 130 normal (11 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_4^2.(C_2^2\times A_4)$
Commutator:	$G' \simeq$ $C_4^2:C_2^2$	$G/G' \simeq$ $C_2^2\times C_6$
Frattini:	$\Phi \simeq$ $C_2^2$	$G/\Phi \simeq$ $C_2^5:A_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^4.C_2^5$	$G/\operatorname{Fit} \simeq$ $C_3$
Radical:	$R \simeq$ $C_4^2.(C_2^3\times A_4)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^3$	$G/\operatorname{soc} \simeq$ $C_2^2\wr C_3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^4.C_2^5$
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: $C_4^2.(C_2^3\times A_4)$

Order 768: $C_4^2.(C_2^2\times A_4)$ x 4, $(C_2^2\times C_4^2):A_4$ x 3

Order 512: $C_2^4.C_2^5$

Order 384: $(C_2\times C_4^2):A_4$ x 6, $C_2^5.A_4$ x 3, $C_2^5:A_4$ x 2, $C_2^5:A_4$

Order 256: $D_4^2:C_2^2$ x 24, $D_4^2:C_2^2$ x 11, $D_4^2:C_2^2$ x 9, $Q_8^2:C_2^2$ x 3

Order 192: $C_2^4.A_4$ x 18, $C_2^4:A_4$ x 8, $C_2^4:A_4$ x 6, $C_2^4.A_4$ x 3, $C_4^2:A_4$

Order 128: $D_4^2:C_2$ x 292, $D_4^2:C_2$ x 288, $Q_8^2:C_2$ x 96, $C_2\times D_4^2$ x 54, $C_4^2:C_2^3$ x 54, $C_4^2.C_2^3$ x 27, $C_4^2.C_2^3$ x 27, $C_4^2.C_2^3$ x 18, $C_4^2:C_2^3$ x 14, $C_4^2:C_2^3$ x 11, $C_4^2.C_2^3$ x 9, $D_4:C_2^4$ x 6, $C_2\times Q_8^2$

Order 96: $C_4^2:C_6$ x 12, $C_2^3:A_4$ x 12, $C_4^2:C_6$ x 9, $C_2^3:A_4$ x 2, $C_2^3\times A_4$

Order 64: $D_4^2$ x 864, $C_4^2:C_2^2$ x 864, $C_4^2.C_2^2$ x 432, $C_4^2.C_2^2$ x 432, $Q_8:D_4$ x 288, $C_4^2:C_2^2$ x 201, $C_4^2:C_2^2$ x 198, $D_4:C_2^3$ x 144, $C_4^2.C_2^2$ x 144, $C_2^3:D_4$ x 108, $C_2^3:D_4$ x 76, $D_4\times C_2^3$ x 54, $C_4^2:C_2^2$ x 54, $C_4^2:C_2^2$ x 54, $D_4:C_2^3$ x 27, $C_4^2:C_2^2$ x 27, $Q_8^2$ x 16, $C_2^2\times C_4^2$ x 11, $C_4^2.C_2^2$ x 9, $C_4^2.C_2^2$ x 6

Order 48: $C_2^2\times A_4$ x 7, $C_4^2:C_3$ x 3, $C_2^2:A_4$ x 2

Order 32: $C_4:D_4$ x 864, $C_2^2\times D_4$ x 729, $C_2^2\wr C_2$ x 584, $C_4^2:C_2$ x 432, $C_4\times D_4$ x 432, $D_4:C_2^2$ x 384, $C_4:D_4$ x 368, $D_4:C_2^2$ x 360, $C_4^2:C_2$ x 216, $C_2\times C_4^2$ x 96, $C_4:Q_8$ x 72, $C_2^3:C_4$ x 54, $C_4\times Q_8$ x 48, $C_2^3\times C_4$ x 27, $C_2^2.D_4$ x 18, $C_2^5$ x 14, $C_2^2\times Q_8$ x 7

Order 24: $C_2\times A_4$ x 9, $C_2^2\times C_6$

Order 16: $C_2\times D_4$ x 1404, $D_4:C_2$ x 576, $C_2^2\times C_4$ x 228, $C_2^2:C_4$ x 216, $C_2^4$ x 135, $C_4^2$ x 74, $C_4:C_4$ x 72, $C_2\times Q_8$ x 48

Order 12: $C_2\times C_6$ x 7, $A_4$ x 3

Order 8: $D_4$ x 432, $C_2^3$ x 271, $C_2\times C_4$ x 270, $Q_8$ x 40

Order 6: $C_6$ x 7

Order 4: $C_2^2$ x 147, $C_4$ x 30

Order 3: $C_3$

Order 2: $C_2$ x 25

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1536: $C_4^2.(C_2^3\times A_4)$

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

Order 512: $C_2^4.C_2^5$

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

Order 256: $D_4^2:C_2^2$ x 3, $D_4^2:C_2^2$ x 3, $D_4^2:C_2^2$ x 2, $Q_8^2:C_2^2$

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

Order 128: $D_4^2:C_2$ x 14, $D_4^2:C_2$ x 11, $C_4^2:C_2^3$ x 8, $C_2\times D_4^2$ x 6, $C_4^2:C_2^3$ x 5, $Q_8^2:C_2$ x 5, $C_4^2.C_2^3$ x 4, $C_4^2.C_2^3$ x 4, $D_4:C_2^4$ x 3, $C_4^2:C_2^3$ x 3, $C_4^2.C_2^3$ x 2, $C_4^2.C_2^3$ x 2, $C_2\times Q_8^2$

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

Order 64: $C_4^2:C_2^2$ x 32, $D_4^2$ x 24, $C_4^2.C_2^2$ x 17, $C_4^2.C_2^2$ x 16, $C_4^2:C_2^2$ x 16, $C_4^2:C_2^2$ x 15, $D_4:C_2^3$ x 13, $C_2^3:D_4$ x 10, $C_2^3:D_4$ x 9, $D_4\times C_2^3$ x 8, $C_4^2:C_2^2$ x 8, $Q_8:D_4$ x 7, $D_4:C_2^3$ x 6, $C_4^2.C_2^2$ x 6, $C_4^2:C_2^2$ x 5, $C_4^2:C_2^2$ x 4, $C_2^2\times C_4^2$ x 3, $Q_8^2$ x 2, $C_4^2.C_2^2$ x 2, $C_4^2.C_2^2$

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

Order 32: $C_2^2\times D_4$ x 46, $C_2^2\wr C_2$ x 27, $D_4:C_2^2$ x 26, $C_4^2:C_2$ x 24, $C_4:D_4$ x 24, $C_4:D_4$ x 18, $C_4\times D_4$ x 12, $C_2\times C_4^2$ x 11, $C_4^2:C_2$ x 10, $D_4:C_2^2$ x 7, $C_2^3:C_4$ x 6, $C_2^5$ x 5, $C_4:Q_8$ x 5, $C_2^2\times Q_8$ x 4, $C_2^3\times C_4$ x 4, $C_4\times Q_8$ x 2, $C_2^2.D_4$ x 2

Order 24: $C_2\times A_4$ x 3, $C_2^2\times C_6$

Order 16: $C_2\times D_4$ x 62, $C_2^4$ x 26, $C_2^2\times C_4$ x 18, $D_4:C_2$ x 14, $C_2\times Q_8$ x 14, $C_2^2:C_4$ x 11, $C_4^2$ x 8, $C_4:C_4$ x 3

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

Order 8: $C_2^3$ x 37, $D_4$ x 20, $C_2\times C_4$ x 17, $Q_8$ x 6

Order 6: $C_6$ x 2

Order 4: $C_2^2$ x 22, $C_4$ x 4

Order 3: $C_3$

Order 2: $C_2$ x 7

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1536: $C_4^2.(C_2^3\times A_4)$ ($C_1$)

Order 768: $C_4^2.(C_2^2\times A_4)$ ($C_2$) x 4, $(C_2^2\times C_4^2):A_4$ ($C_2$) x 3

Order 512: $C_2^4.C_2^5$ ($C_3$)

Order 384: $(C_2\times C_4^2):A_4$ ($C_2^2$) x 6, $C_2^5:A_4$ ($C_2^2$)

Order 256: $D_4^2:C_2^2$ ($C_6$) x 4, $D_4^2:C_2^2$ ($C_6$) x 3

Order 192: $C_4^2:A_4$ ($C_2^3$)

Order 128: $D_4^2:C_2$ ($C_2\times C_6$) x 6, $C_4^2:C_2^3$ ($A_4$) x 3, $C_4^2:C_2^3$ ($A_4$) x 2, $C_4^2:C_2^3$ ($C_2\times C_6$)

Order 64: $C_4^2:C_2^2$ ($C_2\times A_4$) x 18, $C_4^2:C_2^2$ ($C_2\times A_4$) x 8, $C_2^3:D_4$ ($C_2\times A_4$) x 6, $C_2^2\times C_4^2$ ($C_2\times A_4$) x 3, $C_4^2:C_2^2$ ($C_2^2\times C_6$)

Order 32: $C_4:D_4$ ($C_2^2\times A_4$) x 12, $C_2^2\wr C_2$ ($C_2^2\times A_4$) x 12, $C_2\times C_4^2$ ($C_2^2\times A_4$) x 9, $C_2^5$ ($C_2^2\times A_4$) x 2, $C_2^5$ ($C_2^2:A_4$)

Order 16: $C_2^4$ ($C_2^3:A_4$) x 7, $C_4^2$ ($C_2^3\times A_4$) x 3, $C_2^4$ ($C_2^3\times A_4$) x 2

Order 8: $C_2^3$ ($C_2^2\wr C_3$) x 7

Order 4: $C_2^2$ ($C_2^5:A_4$)

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

Order 1: $C_1$ ($C_4^2.(C_2^3\times A_4)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1536: $C_4^2.(C_2^3\times A_4)$ ($C_1$)

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

Order 512: $C_2^4.C_2^5$ ($C_3$)

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

Order 256: $D_4^2:C_2^2$ ($C_6$), $D_4^2:C_2^2$ ($C_6$)

Order 192: $C_4^2:A_4$ ($C_2^3$)

Order 128: $D_4^2:C_2$ ($C_2\times C_6$), $C_4^2:C_2^3$ ($C_2\times C_6$), $C_4^2:C_2^3$ ($A_4$), $C_4^2:C_2^3$ ($A_4$)

Order 64: $C_4^2:C_2^2$ ($C_2\times A_4$) x 2, $C_4^2:C_2^2$ ($C_2^2\times C_6$), $C_4^2:C_2^2$ ($C_2\times A_4$), $C_2^3:D_4$ ($C_2\times A_4$), $C_2^2\times C_4^2$ ($C_2\times A_4$)

Order 32: $C_2\times C_4^2$ ($C_2^2\times A_4$) x 2, $C_2^5$ ($C_2^2:A_4$), $C_2^5$ ($C_2^2\times A_4$), $C_4:D_4$ ($C_2^2\times A_4$), $C_2^2\wr C_2$ ($C_2^2\times A_4$)

Order 16: $C_2^4$ ($C_2^3:A_4$) x 2, $C_4^2$ ($C_2^3\times A_4$), $C_2^4$ ($C_2^3\times A_4$)

Order 8: $C_2^3$ ($C_2^2\wr C_3$) x 2

Order 4: $C_2^2$ ($C_2^5:A_4$)

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

Order 1: $C_1$ ($C_4^2.(C_2^3\times A_4)$)

Series

Derived series

$C_4^2.(C_2^3\times A_4)$

$\rhd$

$C_4^2:C_2^2$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Chief series

$C_4^2.(C_2^3\times A_4)$

$\rhd$

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

$\rhd$

$C_2^5:A_4$

$\rhd$

$C_4^2:A_4$

$\rhd$

$C_4^2:C_2^2$

$\rhd$

$C_2^4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Lower central series

$C_4^2.(C_2^3\times A_4)$

$\rhd$

$C_4^2:C_2^2$

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 4 larger groups in the database.

Character theory

Complex character table

See the $72 \times 72$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $64 \times 64$ rational character table.