Abstract group 9216.ca: $A_4^2:C_2\wr C_4$

• Label: 9216.ca
• Order: \( 2^{10} \cdot 3^{2} \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{13} \cdot 3^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \)
• Perm deg.: $16$
• Trans deg.: $24$
• Rank: $2$

Group information

Description:	$A_4^2:C_2\wr C_4$
Order:	\(9216\)\(\medspace = 2^{10} \cdot 3^{2} \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$A_4^2.C_2^5.C_2^4$, of order \(73728\)\(\medspace = 2^{13} \cdot 3^{2} \)
Composition factors:	$C_2$ x 10, $C_3$ x 2
Derived length:	$3$

This group is nonabelian and monomial (hence solvable).

Group statistics

Order	1	2	3	4	6	8	12
Elements	1	559	80	3152	944	3456	1024	9216
Conjugacy classes	1	16	2	17	16	6	12	70
Divisions	1	16	2	16	16	3	6	60
Autjugacy classes	1	13	2	11	10	3	3	43

Dimension	1	2	4	6	8	9	12	18	24	36
Irr. complex chars.	8	2	19	8	4	8	14	2	2	3	70
Irr. rational chars.	4	4	11	8	8	4	10	4	4	3	60

Minimal presentations

Permutation degree:	$16$
Transitive degree:	$24$
Rank:	$2$
Inequivalent generating pairs:	$360$

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:	${\langle a, b, c, d, e, f, g, h \mid a^{4}=b^{12}=c^{6}=d^{2}=e^{2}=f^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $16$ $\langle(1,3,6,7)(2,4)(5,8)(9,11,10,12)(13,15,14,16), (1,2)(3,5,7)(6,8)(9,10)(11,12,13,14)(15,16)\rangle$
Transitive group:	24T9778	24T9823			more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$A_4^2$ $\,\rtimes\,$ $(C_2\wr C_4)$	$(A_4^2:C_2^4)$ $\,\rtimes\,$ $C_4$	$((C_2^3\times A_4^2):C_4)$ $\,\rtimes\,$ $C_2$	$((C_2^3\times A_4^2).C_4)$ $\,\rtimes\,$ $C_2$	all 7
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(A_4^2:C_2^3)$ . $D_4$	$C_2^6$ . $(C_6^2:C_4)$	$(C_2\times A_4^2:C_4)$ . $D_4$	$(A_4^2.C_2^4.C_2)$ . $C_2$	all 13

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

Homology

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

Subgroups

There are 246210 subgroups in 5450 conjugacy classes, 23 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $A_4^2:C_2^3:C_4$
Commutator:	$G' \simeq$ $C_2^3\times A_4^2$	$G/G' \simeq$ $C_2\times C_4$
Frattini:	$\Phi \simeq$ $C_2^3$	$G/\Phi \simeq$ $C_2\times A_4^2:C_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^6:C_4$	$G/\operatorname{Fit} \simeq$ $C_3^2:C_4$
Radical:	$R \simeq$ $A_4^2:C_2\wr C_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^5$	$G/\operatorname{soc} \simeq$ $(C_6\times C_{12}):C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^6.C_2^3.C_2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$

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 9216: $A_4^2:C_2\wr C_4$

Order 4608: $A_4^2.C_2^4.C_2$, $(C_2^3\times A_4^2).C_4$, $(C_2^3\times A_4^2):C_4$

Order 2304: $A_4^2.C_2^3.C_2$, $C_2^5.C_6^2.C_2$, $C_2\times A_4^2:D_4$, $A_4^2:C_2^4$, $C_2\times A_4:\GL(2,\mathbb{Z}/4)$, $A_4^2:\OD_{16}$, $(C_2^2\times A_4^2):C_4$

Order 1536: $C_2^6:D_{12}$, $C_2^5.C_2^3.S_3$

Order 1152: $A_4^2:C_2^3$ x 5, $A_4^2:D_4$ x 2, $C_2^3\times A_4^2$, $C_2\times A_4^2:C_4$, $C_2^5.C_6^2$, $A_4:\GL(2,\mathbb{Z}/4)$, $C_2\times A_4^2:C_4$, $A_4^2:C_8$, $(C_2^2\times A_4):D_{12}$

Order 1024: $C_2^6.C_2^3.C_2$

Order 768: $C_2^5:D_{12}$ x 2, $C_2^3.\GL(2,\mathbb{Z}/4)$ x 2, $C_2^5:D_{12}$ x 2, $(C_2^3\times S_4).C_2^2$ x 2, $A_4.(C_2.C_4:D_4)$ x 2, $A_4.(C_2^3.D_4)$ x 2, $C_2^5:S_4$, $C_2^3:\GL(2,\mathbb{Z}/4)$, $C_2^3\times \GL(2,\mathbb{Z}/4)$, $C_2^3:\GL(2,\mathbb{Z}/4)$, $C_2^5:D_{12}$, $A_4.(C_2^3.D_4)$, $(C_2^3\times A_4).C_2^3$, $A_4.C_2^5.C_2$, $C_2^6:C_{12}$, $C_2^7.C_6$

Order 576: $A_4^2:C_2^2$ x 10, $C_2^2\times A_4^2$ x 3, $C_2\times C_{12}:S_4$ x 2, $C_6.\GL(2,\mathbb{Z}/4)$ x 2, $C_2^2\times C_6:S_4$, $A_4^2:C_4$, $C_6:\GL(2,\mathbb{Z}/4)$, $C_4\times A_4^2$, $A_4^2:C_4$, $C_2^4.C_6^2$, $C_3^2:C_2\wr C_4$

Order 512: $C_2^6.C_2^3$, $C_2^3.(C_2^3\times C_4).C_2$, $(C_2^3\times D_4).C_2^3$, $C_2^3.C_2^5.C_2$, $C_2^7.C_4$, $C_2^5.C_2^3.C_2$, $C_2^7:C_4$

Order 384: $C_2^2\times \GL(2,\mathbb{Z}/4)$ x 19, $C_2^2\wr S_3$ x 9, $C_4:\GL(2,\mathbb{Z}/4)$ x 8, $C_2^2.\GL(2,\mathbb{Z}/4)$ x 8, $C_2^4:D_{12}$ x 7, $C_2^2.\GL(2,\mathbb{Z}/4)$ x 4, $C_2^4:D_{12}$ x 4, $C_2^5:C_{12}$ x 4, $C_2^2:\GL(2,\mathbb{Z}/4)$ x 4, $C_2^2.\GL(2,\mathbb{Z}/4)$ x 3, $C_2^5:C_{12}$ x 2, $C_2^5:C_{12}$ x 2, $C_2\wr S_3$ x 2, $C_2^4:D_{12}$ x 2, $C_2^4.D_{12}$ x 2, $A_4\times C_2^5$, $C_2^5:A_4$, $C_2^4\times S_4$, $C_2^4.S_4$, $C_2^4.S_4$, $C_2^4:D_{12}$

Order 288: $C_2\times C_6:S_4$ x 9, $C_{12}:S_4$ x 4, $C_2\times A_4^2$ x 3, $\PSOPlus(4,3)$ x 3, $C_2^3.C_6^2$ x 2, $C_3:\GL(2,\mathbb{Z}/4)$ x 2, $C_2\times C_6.S_4$, $C_6^2:D_4$, $(C_2\times C_6^2).C_4$, $(C_2\times C_6^2):C_4$, $C_2^3:C_6^2$

Order 256: $C_2^2.D_4^2$ x 3, $C_2^5:D_4$ x 3, $C_2^6:C_4$ x 3, $C_2^4:\OD_{16}$ x 3, $C_2^6.C_4$ x 3, $C_2^2.D_4^2$ x 2, $C_2^2.D_4^2$ x 2, $C_2^2.D_4^2$ x 2, $C_2^4.C_4^2$ x 2, $C_2^4.C_4^2$ x 2, $C_2^5:D_4$ x 2, $C_2^5.D_4$, $C_2^5.D_4$, $C_2^5:D_4$, $C_2^6:C_4$, $(C_2^4\times C_4).C_4$, $C_2^5:D_4$, $C_2^5:D_4$, $C_2^6:C_4$

Order 192: $C_2\times \GL(2,\mathbb{Z}/4)$ x 79, $C_2.\GL(2,\mathbb{Z}/4)$ x 20, $C_2^3:S_4$ x 19, $C_2^3\times S_4$ x 18, $C_2^3:D_{12}$ x 16, $A_4\times C_2^4$ x 11, $C_2^3.S_4$ x 11, $C_2^4:C_{12}$ x 8, $C_2^4:C_{12}$ x 8, $C_2.\GL(2,\mathbb{Z}/4)$ x 6, $C_2^2\wr C_3$ x 5, $C_2^3.D_{12}$ x 4, $C_2^3.D_{12}$ x 2, $C_2^3.D_{12}$ x 2, $C_2^3:D_{12}$ x 2, $C_2^4:C_{12}$ x 2, $C_2^3:D_{12}$ x 2, $C_2^3.D_{12}$, $C_2^3.D_{12}$, $C_2^3.D_{12}$, $D_6:C_2^4$, $C_2^3.S_4$, $C_2^4:C_{12}$, $C_2^4:D_6$, $C_2^3:D_{12}$

Order 144: $C_6:S_4$ x 19, $C_2^2:C_6^2$ x 5, $C_{12}\times A_4$ x 2, $C_6.D_{12}$, $C_6^2:C_2^2$, $A_4^2$, $C_6^2:C_2^2$, $C_6:D_{12}$, $C_6^2:C_4$, $C_3^2:\OD_{16}$, $C_6.S_4$, $C_6^2:C_4$

Order 128: $C_2^4:D_4$ x 22, $C_2^4.D_4$ x 14, $C_2^4:D_4$ x 12, $C_2^4.D_4$ x 10, $C_2^5:C_4$ x 9, $C_2^3.C_4^2$ x 9, $C_2^4:D_4$ x 8, $C_2^5:C_4$ x 7, $C_2^5:C_4$ x 6, $C_2^4:C_8$ x 6, $C_2^3:\OD_{16}$ x 6, $C_2^4.D_4$ x 6, $C_2^4.D_4$ x 4, $C_2^3\wr C_2$ x 4, $C_2^5.C_4$ x 3, $C_2^5:C_4$ x 3, $C_2^4:D_4$ x 3, $C_2^4.D_4$ x 3, $C_2^4:D_4$ x 3, $C_2^4.D_4$ x 3, $C_2^3.\OD_{16}$ x 2, $D_4\times C_2^4$ x 2, $C_2^3.C_4^2$ x 2, $C_2^4.D_4$ x 2, $C_2^7$, $C_2^5\times C_4$

Order 96: $C_2^2\times S_4$ x 82, $\GL(2,\mathbb{Z}/4)$ x 50, $C_2^3\times A_4$ x 37, $C_2^2.S_4$ x 28, $C_4:S_4$ x 20, $C_2^3:D_6$ x 19, $C_2^3:C_{12}$ x 18, $C_{12}:D_4$ x 8, $C_2^2.D_{12}$ x 8, $C_2^2:D_{12}$ x 5, $C_2^3:A_4$ x 5, $C_2^2:S_4$ x 5, $C_6.C_4^2$ x 4, $C_2^3:C_{12}$ x 4, $C_2^4:S_3$ x 4, $C_2^3.D_6$ x 3, $C_2^3\times C_{12}$ x 2, $C_2^2\times D_{12}$ x 2, $C_2^2.D_{12}$ x 2, $C_2^4\times C_6$, $C_2^3\times D_6$, $C_6.C_2^4$

Order 72: $C_6:D_6$ x 5, $C_6\times A_4$ x 5, $C_3:S_4$ x 5, $C_3:D_{12}$ x 2, $C_2\times C_6^2$, $C_2\times C_3^2:C_4$, $C_6\times C_{12}$, $C_6^2:C_2$, $C_6.D_6$, $C_3^2:C_8$

Order 64: $C_2^3:D_4$ x 161, $C_2^4:C_4$ x 78, $C_2^3.D_4$ x 49, $D_4\times C_2^3$ x 48, $C_2^3:D_4$ x 30, $C_2^4:C_4$ x 24, $C_2^2.C_4^2$ x 18, $C_2^6$ x 14, $C_2^3.D_4$ x 12, $C_2^3:D_4$ x 12, $C_2^3.D_4$ x 12, $C_2^3:C_8$ x 12, $C_2\wr C_4$ x 12, $C_2^4\times C_4$ x 12, $C_2^3.D_4$ x 7, $\OD_{16}:C_2^2$ x 6, $C_2^4:C_4$ x 6, $C_2^2:\OD_{16}$ x 3, $C_2^3.D_4$ x 3

Order 48: $C_2\times S_4$ x 100, $C_6:D_4$ x 77, $C_2^2\times A_4$ x 49, $C_2^2\times D_6$ x 16, $D_6:C_4$ x 16, $C_2\times D_{12}$ x 12, $C_2^3\times C_6$ x 11, $C_6.C_2^3$ x 11, $C_4\times A_4$ x 10, $A_4:C_4$ x 10, $C_2^2\times C_{12}$ x 8, $C_2^2:C_{12}$ x 6, $C_6.D_4$ x 6, $C_{12}:C_4$ x 4, $C_2^2:A_4$

Order 36: $C_6:S_3$ x 10, $C_6^2$ x 3, $C_3^2:C_4$, $C_3\times C_{12}$, $C_3^2:C_4$, $C_3\times A_4$

Order 32: $C_2^2\times D_4$ x 333, $C_2^3:C_4$ x 245, $C_2^2\wr C_2$ x 198, $C_2^5$ x 114, $C_2^3\times C_4$ x 73, $C_4:D_4$ x 48, $C_2.C_4^2$ x 30, $C_2^2.D_4$ x 15, $\OD_{16}:C_2$ x 10, $C_2^3:C_4$ x 10, $C_2^2:C_8$ x 6, $C_2\times \OD_{16}$ x 3

Order 24: $C_2\times D_6$ x 64, $C_3:D_4$ x 46, $C_2^2\times C_6$ x 35, $C_6:C_4$ x 26, $C_2\times A_4$ x 26, $S_4$ x 20, $C_2\times C_{12}$ x 14, $D_{12}$ x 12

Order 18: $C_3\times C_6$ x 3, $C_3:S_3$ x 3

Order 16: $C_2\times D_4$ x 522, $C_2^4$ x 392, $C_2^2\times C_4$ x 170, $C_2^2:C_4$ x 147, $C_4:C_4$ x 12, $\OD_{16}$ x 6, $C_2\times C_8$ x 3

Order 12: $D_6$ x 62, $C_2\times C_6$ x 39, $C_3:C_4$ x 8, $C_{12}$ x 6, $A_4$ x 4

Order 9: $C_3^2$

Order 8: $C_2^3$ x 440, $D_4$ x 132, $C_2\times C_4$ x 118, $C_8$ x 3

Order 6: $C_6$ x 16, $S_3$ x 10

Order 4: $C_2^2$ x 146, $C_4$ x 16

Order 3: $C_3$ x 2

Order 2: $C_2$ x 16

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 9216: $A_4^2:C_2\wr C_4$

Order 4608: $A_4^2.C_2^4.C_2$, $(C_2^3\times A_4^2).C_4$, $(C_2^3\times A_4^2):C_4$

Order 2304: $A_4^2.C_2^3.C_2$, $C_2^5.C_6^2.C_2$, $C_2\times A_4^2:D_4$, $A_4^2:C_2^4$, $C_2\times A_4:\GL(2,\mathbb{Z}/4)$, $A_4^2:\OD_{16}$, $(C_2^2\times A_4^2):C_4$

Order 1536: $C_2^6:D_{12}$, $C_2^5.C_2^3.S_3$

Order 1152: $A_4^2:C_2^3$ x 4, $C_2^3\times A_4^2$, $C_2\times A_4^2:C_4$, $C_2^5.C_6^2$, $A_4:\GL(2,\mathbb{Z}/4)$, $C_2\times A_4^2:C_4$, $A_4^2:D_4$, $A_4^2:C_8$, $(C_2^2\times A_4):D_{12}$

Order 1024: $C_2^6.C_2^3.C_2$

Order 768: $C_2^5:S_4$, $C_2^5:D_{12}$, $C_2^3:\GL(2,\mathbb{Z}/4)$, $C_2^3\times \GL(2,\mathbb{Z}/4)$, $C_2^3.\GL(2,\mathbb{Z}/4)$, $C_2^5:D_{12}$, $C_2^3:\GL(2,\mathbb{Z}/4)$, $C_2^5:D_{12}$, $(C_2^3\times S_4).C_2^2$, $A_4.(C_2.C_4:D_4)$, $A_4.(C_2^3.D_4)$, $A_4.(C_2^3.D_4)$, $(C_2^3\times A_4).C_2^3$, $A_4.C_2^5.C_2$, $C_2^6:C_{12}$, $C_2^7.C_6$

Order 576: $A_4^2:C_2^2$ x 7, $C_2^2\times A_4^2$ x 3, $C_2^2\times C_6:S_4$, $A_4^2:C_4$, $C_6:\GL(2,\mathbb{Z}/4)$, $C_2\times C_{12}:S_4$, $C_4\times A_4^2$, $A_4^2:C_4$, $C_2^4.C_6^2$, $C_6.\GL(2,\mathbb{Z}/4)$, $C_3^2:C_2\wr C_4$

Order 512: $C_2^6.C_2^3$, $C_2^3.(C_2^3\times C_4).C_2$, $(C_2^3\times D_4).C_2^3$, $C_2^3.C_2^5.C_2$, $C_2^7.C_4$, $C_2^5.C_2^3.C_2$, $C_2^7:C_4$

Order 384: $C_2^2\times \GL(2,\mathbb{Z}/4)$ x 9, $C_2^4:D_{12}$ x 5, $C_2^2\wr S_3$ x 4, $C_2^2.\GL(2,\mathbb{Z}/4)$ x 4, $C_2^5:C_{12}$ x 3, $C_2^2.\GL(2,\mathbb{Z}/4)$ x 3, $C_2^2.\GL(2,\mathbb{Z}/4)$ x 2, $C_2^2:\GL(2,\mathbb{Z}/4)$ x 2, $C_4:\GL(2,\mathbb{Z}/4)$ x 2, $A_4\times C_2^5$, $C_2^5:A_4$, $C_2^4\times S_4$, $C_2^5:C_{12}$, $C_2^5:C_{12}$, $C_2\wr S_3$, $C_2^4.S_4$, $C_2^4:D_{12}$, $C_2^4.S_4$, $C_2^4:D_{12}$, $C_2^4.D_{12}$, $C_2^4:D_{12}$

Order 288: $C_2\times C_6:S_4$ x 4, $C_2\times A_4^2$ x 3, $\PSOPlus(4,3)$ x 2, $C_2^3.C_6^2$, $C_3:\GL(2,\mathbb{Z}/4)$, $C_2\times C_6.S_4$, $C_{12}:S_4$, $C_6^2:D_4$, $(C_2\times C_6^2).C_4$, $(C_2\times C_6^2):C_4$, $C_2^3:C_6^2$

Order 256: $C_2^6:C_4$ x 3, $C_2^4:\OD_{16}$ x 3, $C_2^6.C_4$ x 3, $C_2^2.D_4^2$ x 2, $C_2^5:D_4$ x 2, $C_2^2.D_4^2$ x 2, $C_2^2.D_4^2$ x 2, $C_2^2.D_4^2$ x 2, $C_2^5:D_4$ x 2, $C_2^5.D_4$, $C_2^5.D_4$, $C_2^5:D_4$, $C_2^6:C_4$, $(C_2^4\times C_4).C_4$, $C_2^4.C_4^2$, $C_2^5:D_4$, $C_2^5:D_4$, $C_2^4.C_4^2$, $C_2^6:C_4$

Order 192: $C_2\times \GL(2,\mathbb{Z}/4)$ x 27, $C_2^3\times S_4$ x 10, $C_2.\GL(2,\mathbb{Z}/4)$ x 7, $A_4\times C_2^4$ x 7, $C_2^3:S_4$ x 7, $C_2^4:C_{12}$ x 6, $C_2^3.S_4$ x 6, $C_2^3:D_{12}$ x 6, $C_2.\GL(2,\mathbb{Z}/4)$ x 4, $C_2^4:C_{12}$ x 4, $C_2^2\wr C_3$ x 3, $C_2^3.D_{12}$, $C_2^3.D_{12}$, $C_2^3.D_{12}$, $C_2^3:D_{12}$, $C_2^3.D_{12}$, $C_2^3.D_{12}$, $C_2^3.D_{12}$, $D_6:C_2^4$, $C_2^4:C_{12}$, $C_2^3.S_4$, $C_2^4:C_{12}$, $C_2^4:D_6$, $C_2^3:D_{12}$, $C_2^3:D_{12}$

Order 144: $C_6:S_4$ x 7, $C_2^2:C_6^2$ x 3, $C_6.D_{12}$, $C_6^2:C_2^2$, $A_4^2$, $C_6^2:C_2^2$, $C_6:D_{12}$, $C_{12}\times A_4$, $C_6^2:C_4$, $C_3^2:\OD_{16}$, $C_6.S_4$, $C_6^2:C_4$

Order 128: $C_2^4:D_4$ x 15, $C_2^4.D_4$ x 10, $C_2^4:D_4$ x 9, $C_2^5:C_4$ x 8, $C_2^3.C_4^2$ x 7, $C_2^5:C_4$ x 7, $C_2^3:\OD_{16}$ x 6, $C_2^5:C_4$ x 5, $C_2^4.D_4$ x 5, $C_2^4:C_8$ x 4, $C_2^5.C_4$ x 3, $C_2^5:C_4$ x 3, $C_2^4.D_4$ x 3, $C_2^4:D_4$ x 3, $C_2^3\wr C_2$ x 3, $C_2^3.\OD_{16}$ x 2, $D_4\times C_2^4$ x 2, $C_2^4:D_4$ x 2, $C_2^4.D_4$ x 2, $C_2^4.D_4$ x 2, $C_2^4:D_4$ x 2, $C_2^4.D_4$ x 2, $C_2^4.D_4$ x 2, $C_2^7$, $C_2^5\times C_4$, $C_2^3.C_4^2$

Order 96: $C_2^2\times S_4$ x 29, $C_2^3\times A_4$ x 21, $\GL(2,\mathbb{Z}/4)$ x 14, $C_2^2.S_4$ x 14, $C_2^3:D_6$ x 9, $C_2^3:C_{12}$ x 8, $C_2^2:D_{12}$ x 4, $C_4:S_4$ x 4, $C_2^2.D_{12}$ x 4, $C_2^3:A_4$ x 3, $C_2^3:C_{12}$ x 3, $C_2^3.D_6$ x 3, $C_6.C_4^2$ x 2, $C_2^2:S_4$ x 2, $C_2^4:S_3$ x 2, $C_{12}:D_4$ x 2, $C_2^4\times C_6$, $C_2^3\times D_6$, $C_2^3\times C_{12}$, $C_6.C_2^4$, $C_2^2\times D_{12}$, $C_2^2.D_{12}$

Order 72: $C_6:D_6$ x 4, $C_6\times A_4$ x 3, $C_3:S_4$ x 2, $C_2\times C_6^2$, $C_2\times C_3^2:C_4$, $C_6\times C_{12}$, $C_6^2:C_2$, $C_6.D_6$, $C_3:D_{12}$, $C_3^2:C_8$

Order 64: $C_2^3:D_4$ x 67, $C_2^4:C_4$ x 54, $D_4\times C_2^3$ x 30, $C_2^3.D_4$ x 22, $C_2^4:C_4$ x 17, $C_2^3:D_4$ x 13, $C_2^6$ x 12, $C_2^2.C_4^2$ x 10, $C_2^3:C_8$ x 10, $C_2^4\times C_4$ x 9, $C_2\wr C_4$ x 7, $C_2^3.D_4$ x 6, $C_2^3:D_4$ x 6, $C_2^3.D_4$ x 6, $\OD_{16}:C_2^2$ x 5, $C_2^4:C_4$ x 5, $C_2^3.D_4$ x 4, $C_2^2:\OD_{16}$ x 3, $C_2^3.D_4$ x 2

Order 48: $C_2\times S_4$ x 30, $C_6:D_4$ x 26, $C_2^2\times A_4$ x 25, $C_2^2\times D_6$ x 9, $C_2^3\times C_6$ x 7, $C_6.C_2^3$ x 6, $A_4:C_4$ x 6, $D_6:C_4$ x 6, $C_2\times D_{12}$ x 5, $C_2^2:C_{12}$ x 5, $C_2^2\times C_{12}$ x 4, $C_4\times A_4$ x 4, $C_6.D_4$ x 4, $C_2^2:A_4$, $C_{12}:C_4$

Order 36: $C_6:S_3$ x 7, $C_6^2$ x 3, $C_3^2:C_4$, $C_3\times C_{12}$, $C_3^2:C_4$, $C_3\times A_4$

Order 32: $C_2^2\times D_4$ x 132, $C_2^3:C_4$ x 127, $C_2^5$ x 73, $C_2^2\wr C_2$ x 56, $C_2^3\times C_4$ x 45, $C_2.C_4^2$ x 13, $C_4:D_4$ x 12, $C_2^2.D_4$ x 8, $\OD_{16}:C_2$ x 7, $C_2^3:C_4$ x 7, $C_2^2:C_8$ x 6, $C_2\times \OD_{16}$ x 3

Order 24: $C_2\times D_6$ x 25, $C_2^2\times C_6$ x 20, $C_3:D_4$ x 13, $C_6:C_4$ x 13, $C_2\times A_4$ x 13, $C_2\times C_{12}$ x 7, $S_4$ x 6, $D_{12}$ x 3

Order 18: $C_3\times C_6$ x 3, $C_3:S_3$ x 2

Order 16: $C_2^4$ x 201, $C_2\times D_4$ x 163, $C_2^2\times C_4$ x 93, $C_2^2:C_4$ x 67, $\OD_{16}$ x 5, $C_4:C_4$ x 4, $C_2\times C_8$ x 3

Order 12: $D_6$ x 23, $C_2\times C_6$ x 22, $C_3:C_4$ x 5, $A_4$ x 3, $C_{12}$ x 3

Order 9: $C_3^2$

Order 8: $C_2^3$ x 217, $C_2\times C_4$ x 68, $D_4$ x 36, $C_8$ x 3

Order 6: $C_6$ x 10, $S_3$ x 4

Order 4: $C_2^2$ x 86, $C_4$ x 11

Order 3: $C_3$ x 2

Order 2: $C_2$ x 13

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 9216: $A_4^2:C_2\wr C_4$ ($C_1$)

Order 4608: $A_4^2.C_2^4.C_2$ ($C_2$), $(C_2^3\times A_4^2).C_4$ ($C_2$), $(C_2^3\times A_4^2):C_4$ ($C_2$)

Order 2304: $C_2^5.C_6^2.C_2$ ($C_4$), $A_4^2:C_2^4$ ($C_4$), $C_2\times A_4:\GL(2,\mathbb{Z}/4)$ ($C_2^2$)

Order 1152: $C_2^3\times A_4^2$ ($C_2\times C_4$), $A_4^2:C_2^3$ ($D_4$), $C_2\times A_4^2:C_4$ ($D_4$)

Order 576: $C_2^2\times A_4^2$ ($C_2^2:C_4$)

Order 288: $C_2\times A_4^2$ ($C_2^3:C_4$)

Order 256: $C_2^6:C_4$ ($C_3^2:C_4$)

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

Order 128: $C_2^7$ ($C_2\times C_3^2:C_4$)

Order 64: $C_2^6$ ($C_6^2:C_4$)

Order 32: $C_2^5$ ($(C_6\times C_{12}):C_4$)

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

Order 8: $C_2^3$ ($C_2\times A_4^2:C_4$)

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

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

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

Normal subgroups up to automorphism (quotient in parentheses)

Order 9216: $A_4^2:C_2\wr C_4$ ($C_1$)

Order 4608: $A_4^2.C_2^4.C_2$ ($C_2$), $(C_2^3\times A_4^2).C_4$ ($C_2$), $(C_2^3\times A_4^2):C_4$ ($C_2$)

Order 2304: $C_2^5.C_6^2.C_2$ ($C_4$), $A_4^2:C_2^4$ ($C_4$), $C_2\times A_4:\GL(2,\mathbb{Z}/4)$ ($C_2^2$)

Order 1152: $C_2^3\times A_4^2$ ($C_2\times C_4$), $A_4^2:C_2^3$ ($D_4$), $C_2\times A_4^2:C_4$ ($D_4$)

Order 576: $C_2^2\times A_4^2$ ($C_2^2:C_4$)

Order 288: $C_2\times A_4^2$ ($C_2^3:C_4$)

Order 256: $C_2^6:C_4$ ($C_3^2:C_4$)

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

Order 128: $C_2^7$ ($C_2\times C_3^2:C_4$)

Order 64: $C_2^6$ ($C_6^2:C_4$)

Order 32: $C_2^5$ ($(C_6\times C_{12}):C_4$)

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

Order 8: $C_2^3$ ($C_2\times A_4^2:C_4$)

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

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

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

Series

Derived series

$A_4^2:C_2\wr C_4$

$\rhd$

$C_2^3\times A_4^2$

$\rhd$

$C_2^4$

$\rhd$

$C_1$

Chief series

$A_4^2:C_2\wr C_4$

$\rhd$

$A_4^2.C_2^4.C_2$

$\rhd$

$C_2\times A_4:\GL(2,\mathbb{Z}/4)$

$\rhd$

$C_2^3\times A_4^2$

$\rhd$

$C_2^2\times A_4^2$

$\rhd$

$C_2\times A_4^2$

$\rhd$

$A_4^2$

$\rhd$

$C_2^4$

$\rhd$

$C_1$

Lower central series

$A_4^2:C_2\wr C_4$

$\rhd$

$C_2^3\times A_4^2$

$\rhd$

$C_2^2\times A_4^2$

$\rhd$

$C_2\times A_4^2$

$\rhd$

$A_4^2$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2^2$

$\lhd$

$C_2^3$

$\lhd$

$C_2^2:C_4$

Supergroups

This group is a maximal subgroup of 6 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 $70 \times 70$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $60 \times 60$ rational character table.