Abstract group 663552.jo: $C_2^5.A_4^3:A_4$

• Label: 663552.jo
• Order: \( 2^{13} \cdot 3^{4} \)
• Exponent: \( 2^{2} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2 \cdot 3 \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{20} \cdot 3^{6} \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{8} \cdot 3^{2} \cdot 5 \)
• Perm deg.: not computed
• Trans deg.: $36$
• Rank: $4$

Group information

Description:	$C_2^5.A_4^3:A_4$
Order:	\(663552\)\(\medspace = 2^{13} \cdot 3^{4} \)
Exponent:	\(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
Automorphism group:	Group of order \(3822059520\)\(\medspace = 2^{20} \cdot 3^{6} \cdot 5 \)
Composition factors:	$C_2$ x 13, $C_3$ x 4
Derived length:	$4$

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

Group statistics

Order	1	2	3	4	6	9	12	18
Elements	1	15871	19160	41472	209192	147456	82944	147456	663552
Conjugacy classes	1	151	6	192	250	4	96	4	704
Divisions	1	151	4	192	236	2	96	2	684
Autjugacy classes	1	19	4	8	26	1	4	1	64

Minimal presentations

Permutation degree:	not computed
Transitive degree:	$36$
Rank:	$4$
Inequivalent generating quadruples:	not computed

Minimal degrees of faithful linear representations

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

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h, i, j, k, l, m \mid d^{6}=e^{6}=f^{2}=g^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $36$ $\langle(1,5,2,6)(3,8,4,7)(9,12)(10,11)(17,18)(19,20)(25,33,26,34)(27,36,28,35) \!\cdots\! \rangle$
Transitive group:	36T33629	36T33721			more information
Direct product:	$C_2$ $\, \times\, $ $(C_2^4.A_4^3:A_4)$
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$A_4^3$ . $(C_2^5:A_4)$	$C_2^5$ . $(A_4^3:A_4)$	$C_2^8$ . $(C_6^3:A_4)$	$C_2^{11}$ . $(C_3^3:A_4)$	all 22

Elements of the group are displayed as permutations of degree 36.

Homology

Abelianization:	$C_{6} \simeq C_{2} \times C_{3}$
Schur multiplier:	$C_{2}^{10}$
Commutator length:	$2$

Subgroups

There are 118 normal subgroups (16 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_2^4.A_4^3:A_4$
Commutator:	$G' \simeq$ $C_2^6.C_3^3.C_2^6$	$G/G' \simeq$ $C_6$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_2^5.A_4^3:A_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^{11}$	$G/\operatorname{Fit} \simeq$ $C_3^3:A_4$
Radical:	$R \simeq$ $C_2^5.A_4^3:A_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^{11}$	$G/\operatorname{soc} \simeq$ $C_3^3:A_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^9.C_2^4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3\wr C_3$

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 9 (order at least 73728) are shown, as well as all normal subgroups of any index.

Order 663552: $C_2^5.A_4^3:A_4$

Order 331776: $C_2^6.C_3^3.C_2^6.C_3$

Order 221184: $C_2^6.C_3^3.C_2^6.C_2$

Order 165888: $A_4^3.C_2^3:A_4$ x 20, $C_2^5.A_4\wr C_3$

Order 110592: $C_2^6.C_3^3.C_2^6$ x 42, $A_4^2.A_4.C_2^6$

Order 82944: $A_4:S_4^2.A_4$ x 20, $C_2^{10}:C_3\wr C_3$

Order 73728: $A_4^2.C_2^6.C_2^3$

Order 55296: $C_2^6.C_3^3.C_2^5$ x 20, $C_2^6.C_6^3.C_2^2$

Order 27648: $C_2^6.C_3^3.C_2^4$ x 20, $C_2^6.C_6^3.C_2$

Order 24576: $C_2^5.C_2^6.C_6.C_2$

Order 13824: $A_4^3.C_2^3$ x 16, $C_2^6.C_6^3$ x 5

Order 10368: $(C_2^2\times C_6^3).A_4$

Order 8192: $C_2^9.C_2^4$

Order 6912: $A_4:S_4^2$ x 16, $C_2^6.C_3^3.C_2^2$ x 5

Order 3456: $C_2\times A_4^3$

Order 2048: $C_2^{11}$

Order 1728: $A_4^3$

Order 1024: $C_2^{10}$

Order 512: $C_2^9$ x 5

Order 256: $C_2^8$ x 5

Order 128: $C_2^7$

Order 81: $C_3\wr C_3$

Order 64: $C_2^6$

Order 32: $C_2^5$

Order 16: $C_2^4$

Order 8: $C_2^3$ x 5

Order 4: $C_2^2$ x 5

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 9 (order at least 73728) are shown, as well as all normal subgroups of any index.

Order 663552: $C_2^5.A_4^3:A_4$

Order 331776: $C_2^6.C_3^3.C_2^6.C_3$

Order 221184: $C_2^6.C_3^3.C_2^6.C_2$

Order 165888: $A_4^3.C_2^3:A_4$, $C_2^5.A_4\wr C_3$

Order 110592: $C_2^6.C_3^3.C_2^6$ x 4, $A_4^2.A_4.C_2^6$

Order 82944: $A_4:S_4^2.A_4$, $C_2^{10}:C_3\wr C_3$

Order 73728: $A_4^2.C_2^6.C_2^3$

Order 55296: $C_2^6.C_6^3.C_2^2$, $C_2^6.C_3^3.C_2^5$

Order 27648: $C_2^6.C_3^3.C_2^4$, $C_2^6.C_6^3.C_2$

Order 24576: $C_2^5.C_2^6.C_6.C_2$

Order 13824: $C_2^6.C_6^3$, $A_4^3.C_2^3$

Order 10368: $(C_2^2\times C_6^3).A_4$

Order 8192: $C_2^9.C_2^4$

Order 6912: $C_2^6.C_3^3.C_2^2$, $A_4:S_4^2$

Order 3456: $C_2\times A_4^3$

Order 2048: $C_2^{11}$

Order 1728: $A_4^3$

Order 1024: $C_2^{10}$

Order 512: $C_2^9$

Order 256: $C_2^8$

Order 128: $C_2^7$

Order 81: $C_3\wr C_3$

Order 64: $C_2^6$

Order 32: $C_2^5$

Order 16: $C_2^4$

Order 8: $C_2^3$

Order 4: $C_2^2$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 663552: $C_2^5.A_4^3:A_4$ ($C_1$)

Order 331776: $C_2^6.C_3^3.C_2^6.C_3$ ($C_2$)

Order 221184: $C_2^6.C_3^3.C_2^6.C_2$ ($C_3$)

Order 110592: $C_2^6.C_3^3.C_2^6$ ($C_6$)

Order 55296: $C_2^6.C_3^3.C_2^5$ ($A_4$) x 20, $C_2^6.C_6^3.C_2^2$ ($A_4$)

Order 27648: $C_2^6.C_3^3.C_2^4$ ($C_2\times A_4$) x 20, $C_2^6.C_6^3.C_2$ ($C_2\times A_4$)

Order 13824: $A_4^3.C_2^3$ ($C_2^2:A_4$) x 16, $C_2^6.C_6^3$ ($C_2^2:A_4$) x 5

Order 6912: $A_4:S_4^2$ ($C_2^3:A_4$) x 16, $C_2^6.C_3^3.C_2^2$ ($C_2^3:A_4$) x 5

Order 3456: $C_2\times A_4^3$ ($C_2^4:A_4$)

Order 2048: $C_2^{11}$ ($C_3^3:A_4$)

Order 1728: $A_4^3$ ($C_2^5:A_4$)

Order 1024: $C_2^{10}$ ($C_2\times C_3^3:A_4$)

Order 512: $C_2^9$ ($(C_3\times C_6^2):A_4$) x 5

Order 256: $C_2^8$ ($C_6^3:A_4$) x 5

Order 128: $C_2^7$ ($(C_2\times C_6^3).A_4$)

Order 64: $C_2^6$ ($(C_2^2\times C_6^3).A_4$)

Order 32: $C_2^5$ ($A_4^3:A_4$)

Order 16: $C_2^4$ ($C_2\times A_4^3:A_4$)

Order 8: $C_2^3$ ($A_4:S_4^2.A_4$) x 5

Order 4: $C_2^2$ ($A_4^3.C_2^3:A_4$) x 5

Order 2: $C_2$ ($C_2^4.A_4^3:A_4$)

Order 1: $C_1$ ($C_2^5.A_4^3:A_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 663552: $C_2^5.A_4^3:A_4$ ($C_1$)

Order 331776: $C_2^6.C_3^3.C_2^6.C_3$ ($C_2$)

Order 221184: $C_2^6.C_3^3.C_2^6.C_2$ ($C_3$)

Order 110592: $C_2^6.C_3^3.C_2^6$ ($C_6$)

Order 55296: $C_2^6.C_6^3.C_2^2$ ($A_4$), $C_2^6.C_3^3.C_2^5$ ($A_4$)

Order 27648: $C_2^6.C_3^3.C_2^4$ ($C_2\times A_4$), $C_2^6.C_6^3.C_2$ ($C_2\times A_4$)

Order 13824: $C_2^6.C_6^3$ ($C_2^2:A_4$), $A_4^3.C_2^3$ ($C_2^2:A_4$)

Order 6912: $C_2^6.C_3^3.C_2^2$ ($C_2^3:A_4$), $A_4:S_4^2$ ($C_2^3:A_4$)

Order 3456: $C_2\times A_4^3$ ($C_2^4:A_4$)

Order 2048: $C_2^{11}$ ($C_3^3:A_4$)

Order 1728: $A_4^3$ ($C_2^5:A_4$)

Order 1024: $C_2^{10}$ ($C_2\times C_3^3:A_4$)

Order 512: $C_2^9$ ($(C_3\times C_6^2):A_4$)

Order 256: $C_2^8$ ($C_6^3:A_4$)

Order 128: $C_2^7$ ($(C_2\times C_6^3).A_4$)

Order 64: $C_2^6$ ($(C_2^2\times C_6^3).A_4$)

Order 32: $C_2^5$ ($A_4^3:A_4$)

Order 16: $C_2^4$ ($C_2\times A_4^3:A_4$)

Order 8: $C_2^3$ ($A_4:S_4^2.A_4$)

Order 4: $C_2^2$ ($A_4^3.C_2^3:A_4$)

Order 2: $C_2$ ($C_2^4.A_4^3:A_4$)

Order 1: $C_1$ ($C_2^5.A_4^3:A_4$)

Series

Derived series

$C_2^5.A_4^3:A_4$

$\rhd$

$C_2^6.C_3^3.C_2^6$

$\rhd$

$A_4^3$

$\rhd$

$C_2^6$

$\rhd$

$C_1$

Chief series

$C_2^5.A_4^3:A_4$

$\rhd$

$C_2^6.C_3^3.C_2^6.C_3$

$\rhd$

$C_2^6.C_3^3.C_2^6$

$\rhd$

$C_2^6.C_3^3.C_2^4$

$\rhd$

$C_2^6.C_3^3.C_2^2$

$\rhd$

$A_4^3$

$\rhd$

$C_2^6$

$\rhd$

$C_1$

Lower central series

$C_2^5.A_4^3:A_4$

$\rhd$

$C_2^6.C_3^3.C_2^6$

Upper central series

$C_1$

$\lhd$

$C_2$

Supergroups

This group is a maximal subgroup of 12 larger groups in the database.

This group is a maximal quotient of 6 larger groups in the database.

Character theory

Complex character table

The $704 \times 704$ character table is not available for this group.

Rational character table

The $684 \times 684$ rational character table is not available for this group.