Abstract group 165888.bl: $A_4^2.(C_2\times A_4^2:C_4)$

• Label: 165888.bl
• Order: \( 2^{11} \cdot 3^{4} \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{15} \cdot 3^{4} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \)
• Perm deg.: $16$
• Trans deg.: $32$
• Rank: $2$

Group information

Description:	$A_4^2.(C_2\times A_4^2:C_4)$
Order:	\(165888\)\(\medspace = 2^{11} \cdot 3^{4} \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$C_2^8.S_3\wr D_4$, of order \(2654208\)\(\medspace = 2^{15} \cdot 3^{4} \)
Composition factors:	$C_2$ x 11, $C_3$ x 4
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	8	12
Elements	1	2703	6560	43632	23136	62208	27648	165888
Conjugacy classes	1	16	12	27	36	12	18	122
Divisions	1	16	12	25	36	6	18	114
Autjugacy classes	1	9	5	12	13	2	6	48

Dimension	1	2	4	6	8	9	12	18	24	36	54	72	81	108	162
Irr. complex chars.	8	0	8	8	8	16	20	8	8	16	8	2	8	4	0	122
Irr. rational chars.	4	2	8	8	8	8	20	12	8	16	8	2	4	4	2	114

Minimal presentations

Permutation degree:	$16$
Transitive degree:	$32$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	18	18	18
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h, i, j \mid c^{6}=d^{6}=e^{6}=f^{2}=g^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $16$ $\langle(1,2,3,4,6,8,5,7)(9,10,11,13,14,15,16,12), (2,4)(3,5,6)(7,8)(10,12,13,15)(14,16)\rangle$
Transitive group:	36T23377	36T23378	36T23379		more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$(A_4^2.A_4^2.C_2)$ . $C_4$	$A_4^2$ . $(C_2\times A_4^2:C_4)$	$(C_2^8.C_3^3.D_6)$ . $C_2$	$(A_4^2:(A_4^2:C_4))$ . $C_2$	all 11

Elements of the group are displayed as permutations of degree 16.

Homology

Abelianization:	$C_{2} \times C_{4} $
Schur multiplier:	$C_{2} \times C_{6}^{2}$
Commutator length:	$1$

Subgroups

There are 20 normal subgroups (6 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $A_4^2.(C_2\times A_4^2:C_4)$
Commutator:	$G' \simeq$ $C_2^8.C_3^4$	$G/G' \simeq$ $C_2\times C_4$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $A_4^2.(C_2\times A_4^2:C_4)$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^8$	$G/\operatorname{Fit} \simeq$ $C_3^4:(C_2\times C_4)$
Radical:	$R \simeq$ $A_4^2.(C_2\times A_4^2:C_4)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^8$	$G/\operatorname{soc} \simeq$ $C_3^4:(C_2\times C_4)$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^6.C_2^4.C_2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^4$

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 12 (order at least 13824) are shown, as well as all normal subgroups of any index.

Order 165888: $A_4^2.(C_2\times A_4^2:C_4)$

Order 82944: $A_4^2:(A_4^2:C_4)$ x 2, $C_2^8.C_3^3.D_6$

Order 41472: $A_4^2.A_4^2.C_2$ x 2, $C_2^8.C_3^4.C_2$

Order 27648: $C_2^8.C_3^2.D_6$ x 2, $C_2^6.C_3^3.C_2^3.C_2$ x 2

Order 20736: $A_4^3.D_6$ x 2, $C_2^8.C_3^4$

Order 18432: $A_4^2.C_2^5:C_4$ x 2

Order 13824: $C_2^8.C_3^3.C_2$ x 4, $C_2^2:A_4^2.S_4$ x 4, $C_2^6.C_3^3.D_4$ x 2, $C_2^6.C_3^3.D_4$ x 2, $A_4^2.C_2^4.C_6$ x 2, $C_2^6.C_3^2.D_6.C_2$ x 2, $A_4^2.A_4.C_2^3$ x 2, $C_2^4:C_3.A_4^2.C_2$ x 2, $A_4.C_2^5.C_6^2$ x 2, $C_2^6.C_3^3.D_4$ x 2, $A_4^3.C_2^3$ x 2, $C_2^2.A_4^2:S_4$ x 2

Order 10368: $(C_3^2\times A_4^2):(C_2\times C_4)$ x 2

Order 4608: $A_4^2.C_2^5$ x 2

Order 2304: $A_4^2.C_2^4$ x 2

Order 2048: $C_2^6.C_2^4.C_2$

Order 288: $\PSOPlus(4,3)$ x 2

Order 256: $C_2^8$

Order 144: $A_4^2$ x 2

Order 81: $C_3^4$

Order 16: $C_2^4$ x 2

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 12 (order at least 13824) are shown, as well as all normal subgroups of any index.

Order 165888: $A_4^2.(C_2\times A_4^2:C_4)$

Order 82944: $C_2^8.C_3^3.D_6$, $A_4^2:(A_4^2:C_4)$

Order 41472: $A_4^2.A_4^2.C_2$, $C_2^8.C_3^4.C_2$

Order 27648: $C_2^8.C_3^2.D_6$, $C_2^6.C_3^3.C_2^3.C_2$

Order 20736: $C_2^8.C_3^4$, $A_4^3.D_6$

Order 18432: $A_4^2.C_2^5:C_4$

Order 13824: $C_2^8.C_3^3.C_2$ x 2, $C_2^6.C_3^3.D_4$, $C_2^6.C_3^3.D_4$, $A_4^2.C_2^4.C_6$, $C_2^6.C_3^2.D_6.C_2$, $A_4^2.A_4.C_2^3$, $C_2^4:C_3.A_4^2.C_2$, $A_4.C_2^5.C_6^2$, $C_2^6.C_3^3.D_4$, $A_4^3.C_2^3$, $C_2^2.A_4^2:S_4$, $C_2^2:A_4^2.S_4$

Order 10368: $(C_3^2\times A_4^2):(C_2\times C_4)$

Order 4608: $A_4^2.C_2^5$

Order 2304: $A_4^2.C_2^4$

Order 2048: $C_2^6.C_2^4.C_2$

Order 288: $\PSOPlus(4,3)$

Order 256: $C_2^8$

Order 144: $A_4^2$

Order 81: $C_3^4$

Order 16: $C_2^4$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 165888: $A_4^2.(C_2\times A_4^2:C_4)$ ($C_1$)

Order 82944: $A_4^2:(A_4^2:C_4)$ ($C_2$) x 2, $C_2^8.C_3^3.D_6$ ($C_2$)

Order 41472: $A_4^2.A_4^2.C_2$ ($C_4$) x 2, $C_2^8.C_3^4.C_2$ ($C_2^2$)

Order 20736: $C_2^8.C_3^4$ ($C_2\times C_4$)

Order 4608: $A_4^2.C_2^5$ ($C_3^2:C_4$) x 2

Order 2304: $A_4^2.C_2^4$ ($C_2\times C_3^2:C_4$) x 2

Order 288: $\PSOPlus(4,3)$ ($A_4^2:C_4$) x 2

Order 256: $C_2^8$ ($C_3^4:(C_2\times C_4)$)

Order 144: $A_4^2$ ($C_2\times A_4^2:C_4$) x 2

Order 16: $C_2^4$ ($(C_3^2\times A_4^2):(C_2\times C_4)$) x 2

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

Normal subgroups up to automorphism (quotient in parentheses)

Order 165888: $A_4^2.(C_2\times A_4^2:C_4)$ ($C_1$)

Order 82944: $C_2^8.C_3^3.D_6$ ($C_2$), $A_4^2:(A_4^2:C_4)$ ($C_2$)

Order 41472: $A_4^2.A_4^2.C_2$ ($C_4$), $C_2^8.C_3^4.C_2$ ($C_2^2$)

Order 20736: $C_2^8.C_3^4$ ($C_2\times C_4$)

Order 4608: $A_4^2.C_2^5$ ($C_3^2:C_4$)

Order 2304: $A_4^2.C_2^4$ ($C_2\times C_3^2:C_4$)

Order 288: $\PSOPlus(4,3)$ ($A_4^2:C_4$)

Order 256: $C_2^8$ ($C_3^4:(C_2\times C_4)$)

Order 144: $A_4^2$ ($C_2\times A_4^2:C_4$)

Order 16: $C_2^4$ ($(C_3^2\times A_4^2):(C_2\times C_4)$)

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

Series

Derived series

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

$\rhd$

$C_2^8.C_3^4$

$\rhd$

$C_2^8$

$\rhd$

$C_1$

Chief series

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

$\rhd$

$C_2^8.C_3^3.D_6$

$\rhd$

$C_2^8.C_3^4.C_2$

$\rhd$

$C_2^8.C_3^4$

$\rhd$

$A_4^2.C_2^4$

$\rhd$

$C_2^8$

$\rhd$

$C_2^4$

$\rhd$

$C_1$

Lower central series

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

$\rhd$

$C_2^8.C_3^4$

Upper central series

$C_1$

Supergroups

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

This group is a maximal quotient of 2 larger groups in the database.

Character theory

Complex character table

See the $122 \times 122$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $114 \times 114$ rational character table.