Abstract group 82944.bd: $A_4^2:\POPlus(4,3)$

• Label: 82944.bd
• Order: \( 2^{10} \cdot 3^{4} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{14} \cdot 3^{4} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \)
• Perm deg.: $16$
• Trans deg.: $24$
• Rank: $3$

Group information

Description:	$A_4^2:\POPlus(4,3)$
Order:	\(82944\)\(\medspace = 2^{10} \cdot 3^{4} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$A_4^2\wr C_2.C_4.D_4$, of order \(1327104\)\(\medspace = 2^{14} \cdot 3^{4} \)
Composition factors:	$C_2$ x 10, $C_3$ x 4
Derived length:	$3$

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

Group statistics

Order	1	2	3	4	6	12
Elements	1	1839	6560	23760	36960	13824	82944
Conjugacy classes	1	12	24	15	50	4	106
Divisions	1	12	24	15	50	4	106
Autjugacy classes	1	7	6	7	12	1	34

Dimension	1	2	4	6	9	12	18	36	54	81	108
Irr. complex chars.	4	16	16	4	8	26	12	10	4	4	2	106
Irr. rational chars.	4	16	16	4	8	26	12	10	4	4	2	106

Minimal presentations

Permutation degree:	$16$
Transitive degree:	$24$
Rank:	$3$
Inequivalent generating triples:	$98317800$

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:	${\langle a, b, c, d, e, f, g, h, i, j, k \mid b^{6}=c^{3}=d^{6}=e^{6}=f^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $16$ $\langle(1,3,7,2,5,8)(4,6)(9,12,16,11,14,15)(10,13), (2,4,3)(5,7,6)(9,10)(12,15,13)(14,16), (1,2)(3,6)(4,5)(7,8)(9,11,10,12,14,13)(15,16)\rangle$
Transitive group:	24T16601	36T19063	36T19064		more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$A_4^2$ $\,\rtimes\,$ $\POPlus(4,3)$	$(C_2^4:A_4^2)$ $\,\rtimes\,$ $S_3^2$	$C_2^8$ $\,\rtimes\,$ $(C_3^2:S_3^2)$	$(C_2^6:A_4)$ $\,\rtimes\,$ $(C_3:S_3^2)$	all 6
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

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

Homology

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

Subgroups

There are 5123980 subgroups in 16051 conjugacy classes, 59 normal (5 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $A_4^2:\POPlus(4,3)$
Commutator:	$G' \simeq$ $C_2^8.C_3^4$	$G/G' \simeq$ $C_2^2$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $A_4^2:\POPlus(4,3)$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^8$	$G/\operatorname{Fit} \simeq$ $C_3^2:S_3^2$
Radical:	$R \simeq$ $A_4^2:\POPlus(4,3)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^8$	$G/\operatorname{soc} \simeq$ $C_3^2:S_3^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^7.C_2^3$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^4$

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

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Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 82944: $A_4^2:\POPlus(4,3)$

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

Order 27648: $C_2^8.C_3^2.D_6$ x 4, $C_2^8.C_3^2.D_6$ x 4

Order 20736: $C_2^8.C_3^4$

Order 13824: $C_2^8.C_3^3.C_2$ x 8, $C_2^2:A_4^2.S_4$ x 8, $C_2^8.C_3^3.C_2$ x 6, $C_2^8.C_3^3.C_2$ x 4, $C_2^8.C_3^3.C_2$ x 4, $C_2^8.C_3^3.C_2$ x 4, $C_2^8:(S_3\times C_3^2)$ x 4, $C_2^6.C_3^3.D_4$ x 2

Order 10368: $A_4^3:S_3$ x 2

Order 9216: $C_2^8.S_3^2$ x 8, $C_2^8.S_3^2$ x 4, $C_2^8.S_3^2$ x 2, $C_2^8.C_3.D_6$ x 2, $C_2^8:S_3^2$ x 2

Order 6912: $C_2^8.C_3^3$ x 8, $C_2^2.A_4^3$ x 8, $C_2^8.C_3^3$ x 6, $A_4^3:C_2^2$ x 4, $C_2^6.C_3^3.C_2^2$ x 2, $C_2^6.C_3^3.C_4$ x 2, $A_4^3:C_2^2$ x 2

Order 5184: $A_4^2.S_3^2$ x 2, $C_3\times A_4^3$ x 2

Order 4608: $C_2^8.C_3.S_3$ x 24, $C_2^8.C_3.C_6$ x 8, $C_2^6.C_3^2.D_4$ x 8, $C_2^8.C_3.C_6$ x 8, $C_2^8.C_3.S_3$ x 8, $C_2^6.C_3^2.D_4$ x 8, $C_2^8.C_3.S_3$ x 8, $C_2^6.(S_3\times A_4)$ x 8, $C_2^8.C_3.S_3$ x 4, $C_2^4:C_3.C_2^4.C_6$ x 4, $A_4^2.C_2^2\wr C_2$ x 4, $(C_2^2\times A_4):\GL(2,\mathbb{Z}/4)$ x 4, $C_2^4.\PSOPlus(4,3)$ x 4, $C_2^4:\PSOPlus(4,3)$ x 4, $(C_2^2\times A_4)\wr C_2$ x 4, $C_2^7.S_3^2$ x 4, $A_4^2.C_2^3.C_2^2$ x 2, $A_4^2.Q_8:C_2^2$ x 2, $C_2^5:A_4^2$ x 2

Order 3456: $C_2^6.C_3^3.C_2$ x 46, $A_4^2:S_4$ x 16, $C_2^6.C_3^3.C_2$ x 12, $C_2^2:A_4^2:S_3$ x 8, $C_2^4:C_3.A_4.C_6$ x 4, $C_2^6.C_3^3.C_2$ x 4, $A_4^2.D_6.C_2$ x 4, $(C_3\times A_4^2).D_4$ x 2, $C_2\times A_4^3$ x 2

Order 3072: $C_2^8.D_6$ x 4, $C_2^4.C_2^4:S_3.C_2$ x 4

Order 2592: $C_2^4.C_3^4.C_2$ x 4, $C_2^4.C_3^4.C_2$ x 2, $(A_4\times C_6^2):S_3$ x 2

Order 2304: $C_2^8.C_3^2$ x 28, $C_2^6.C_3^2.C_4$ x 12, $C_2^6.C_3^2.C_4$ x 8, $C_2^8.C_3^2$ x 8, $C_2^6.C_6^2$ x 8, $C_2^6.S_3^2$ x 8, $C_2^6.C_3:S_3.C_2$ x 8, $A_4^2.C_2^2:C_4$ x 8, $(C_2^3\times A_4):S_4$ x 8, $C_2^4:A_4^2$ x 8, $C_2^4:A_4^2$ x 6, $A_4^2.D_4.C_2$ x 4, $A_4^2.D_4.C_2$ x 4, $C_2^6.S_3^2$ x 4, $C_2^4.C_6^2.C_2^2$ x 4, $A_4^2.C_2^3.C_2$ x 4, $(C_2^4\times A_4).D_6$ x 4, $C_2^8.C_3^2$ x 4, $C_2^8.C_3^2$ x 4, $C_2\times A_4:\GL(2,\mathbb{Z}/4)$ x 4, $C_2^4:A_4^2$ x 4, $(C_2^3\times A_4):S_4$ x 4, $C_2\times A_4^2:D_4$ x 4, $C_2\times A_4^2.C_2^3$ x 2, $A_4^2.C_2^4$ x 2, $A_4^2.D_4:C_2$ x 2, $A_4^2.D_4.C_2$ x 2, $A_4^2.C_2^3.C_2$ x 2, $C_2\wr S_3^2$ x 2

Order 1728: $A_4^3$ x 62, $C_2^6:C_3^3$ x 12, $A_4^2:D_6$ x 12, $C_2^6:C_3^3$ x 8, $C_2^6:C_3^3$ x 6, $A_4^2:D_6$ x 6, $(C_3\times A_4^2):C_4$ x 6, $(C_6\times D_6):S_4$ x 4

Order 1536: $C_2^4:\GL(2,\mathbb{Z}/4)$ x 12, $C_2^6:S_4$ x 10, $C_2^4:\GL(2,\mathbb{Z}/4)$ x 8, $C_2^4.\GL(2,\mathbb{Z}/4)$ x 8, $C_2^7:D_6$ x 4, $C_2^4.\GL(2,\mathbb{Z}/4)$ x 4, $C_2^8:C_6$ x 4, $C_2^8:C_6$ x 4, $C_2^8.S_3$ x 2

Order 1296: $C_3^2\times A_4^2$ x 4

Order 1152: $(C_2^2\times A_4):S_4$ x 160, $C_2^2:\PSOPlus(4,3)$ x 100, $A_4:\GL(2,\mathbb{Z}/4)$ x 80, $C_2^3:A_4^2$ x 12, $(C_2\times C_6):\GL(2,\mathbb{Z}/4)$ x 12, $C_2^3\times A_4^2$ x 10, $C_2\times A_4^2:C_4$ x 10, $(C_2^3\times C_6):S_4$ x 8, $C_2^3:A_4^2$ x 8, $C_2^4:(S_3\times A_4)$ x 8, $C_2^4:(C_3\times S_4)$ x 8, $A_4^2:D_4$ x 8, $(C_2\times C_6):\GL(2,\mathbb{Z}/4)$ x 6, $A_4^2:C_2^3$ x 4, $A_4^2:C_2^3$ x 4, $A_4\times C_2^2:S_4$ x 4, $C_2\times A_4^2:C_4$ x 4, $A_4^2:D_4$ x 4, $C_2^5.S_3^2$ x 4, $C_2^3:A_4^2$ x 4, $(C_2^2\times S_4):A_4$ x 4, $A_4^2:C_2^3$ x 2, $A_4^2:Q_8$ x 2, $A_4^2:D_4$ x 2, $C_2^5.S_3^2$ x 2

Order 1024: $C_2^7.C_2^3$

Order 864: $A_4^2:S_3$ x 126, $A_4^2:S_3$ x 12, $A_4^2:C_6$ x 12, $C_6^2:S_4$ x 8, $C_6\times A_4^2$ x 6, $C_6^2:S_4$ x 6, $C_6^2:S_4$ x 4, $(C_6\times D_6):A_4$ x 4

Order 768: $C_2^3:\GL(2,\mathbb{Z}/4)$ x 24, $C_2^3.\GL(2,\mathbb{Z}/4)$ x 24, $C_2^3:\GL(2,\mathbb{Z}/4)$ x 14, $A_4.(C_2^5.C_2)$ x 14, $C_2^5:S_4$ x 12, $C_2^5:S_4$ x 8, $C_2^5.S_4$ x 8, $C_2^6.D_6$ x 8, $C_2\wr D_6$ x 8, $C_2^8.C_3$ x 8, $C_2^6:A_4$ x 8, $C_2^6:A_4$ x 6, $C_2^6.D_6$ x 4, $C_2^3:\GL(2,\mathbb{Z}/4)$ x 4, $C_2^3:\GL(2,\mathbb{Z}/4)$ x 4, $C_2^7:C_6$ x 4, $C_2^6:C_{12}$ x 4, $C_2^8.C_3$ x 2

Order 648: $C_3^3:S_4$ x 2

Order 576: $C_2^2:A_4^2$ x 156, $C_2^2:A_4^2$ x 100, $C_2^2\times A_4^2$ x 84, $A_4^2:C_2^2$ x 72, $A_4^2:C_4$ x 70, $\POPlus(4,3)$ x 22, $C_6:\GL(2,\mathbb{Z}/4)$ x 18, $C_6.\GL(2,\mathbb{Z}/4)$ x 18, $C_2^4:C_6^2$ x 12, $(C_2^2\times C_6):S_4$ x 12, $(C_2^2\times C_6).S_4$ x 12, $C_2^4:S_3^2$ x 12, $C_2^6:C_3^2$ x 8, $C_2^4:C_6^2$ x 6, $A_4^2:C_2^2$ x 4, $A_4^2:C_4$ x 4, $(C_2^2\times C_6):S_4$ x 4, $S_4^2$ x 2, $C_2^4:S_3^2$ x 2

Order 512: $C_2^6.D_4$ x 9, $C_2^7:C_2^2$ x 3, $C_2^4\wr C_2$ x 3

Order 432: $C_3\times A_4^2$ x 126, $C_6^2:A_4$ x 8, $A_4\times C_6^2$ x 6, $C_6^2:D_6$ x 6, $(C_3^2\times A_4):C_4$ x 6, $A_4:S_3^2$ x 4

Order 384: $C_2\wr S_3$ x 104, $C_2^4:S_4$ x 86, $C_2^2:\GL(2,\mathbb{Z}/4)$ x 80, $C_2^2.\GL(2,\mathbb{Z}/4)$ x 42, $C_2^4.S_4$ x 24, $C_2^5:A_4$ x 20, $A_4\times C_2^5$ x 14, $C_2^2\times \GL(2,\mathbb{Z}/4)$ x 14, $C_2^2\wr S_3$ x 12, $C_2^5:A_4$ x 12, $C_2\wr C_6$ x 8, $C_2^4:S_4$ x 8, $C_2^2.\GL(2,\mathbb{Z}/4)$ x 8, $C_2^5.D_6$ x 8, $C_2^5:A_4$ x 8, $C_2^5:C_{12}$ x 4, $C_2^2.\GL(2,\mathbb{Z}/4)$ x 4, $C_2^4.S_4$ x 4, $C_2^4:D_{12}$ x 4, $C_2^5.D_6$ x 4, $\GL(2,\mathbb{Z}/4):C_2^2$ x 4, $C_2^6:C_6$ x 4, $C_2^5:A_4$ x 4, $C_2^4:S_4$ x 4, $C_2^6:S_3$ x 2

Order 324: $A_4\times C_3^3$ x 2, $C_3^2:S_3^2$

Order 288: $\PSOPlus(4,3)$ x 380, $(C_2\times C_6):S_4$ x 116, $C_3:\GL(2,\mathbb{Z}/4)$ x 108, $C_2\times A_4^2$ x 72, $A_4\wr C_2$ x 44, $C_2\times C_6.S_4$ x 18, $C_2^3:C_6^2$ x 18, $(C_2\times D_6):A_4$ x 12, $C_2^5:C_3^2$ x 12, $C_3\times C_2^2:S_4$ x 12, $C_2\times C_6:S_4$ x 6, $C_6^2:D_4$ x 4, $C_6^2:D_4$ x 4, $C_3\times C_2^3:A_4$ x 4, $C_6^2.D_4$ x 4, $A_4\times S_4$ x 4, $C_2^3:S_3^2$ x 2, $C_6^2.D_4$ x 2

Order 256: $C_2^4.C_2^4$ x 18, $C_2^5:D_4$ x 18, $C_2^6:C_4$ x 15, $C_2^5:D_4$ x 15, $C_2^5.D_4$ x 9, $C_2^4.C_4^2$ x 6, $C_2^8$, $C_4^2:C_2^4$

Order 216: $C_3^2:S_4$ x 86, $C_6^2:C_6$ x 6, $C_6^2:C_6$ x 4, $C_3^2:S_4$ x 4, $C_6^2:S_3$ x 2

Order 192: $C_2\times \GL(2,\mathbb{Z}/4)$ x 148, $C_2.\GL(2,\mathbb{Z}/4)$ x 144, $C_2^2\wr C_3$ x 104, $A_4\times C_2^4$ x 96, $C_2^3:S_4$ x 84, $C_2^3.S_4$ x 80, $C_2^4:A_4$ x 74, $C_2^3:S_4$ x 28, $C_2^4.D_6$ x 14, $C_2^3.S_4$ x 14, $C_2^4:D_6$ x 14, $D_4:S_4$ x 8, $D_4\times S_4$ x 8, $C_2^4:C_{12}$ x 4, $C_2^4:D_6$ x 4, $C_2^4:C_{12}$ x 4, $C_2^5:C_6$ x 4, $\GL(2,\mathbb{Z}/4):C_2$ x 4, $C_2^4:A_4$ x 4, $C_2^5\times C_6$ x 2, $C_2^3\times S_4$ x 2

Order 162: $C_3^2\wr C_2$ x 2, $C_3^3:S_3$

Order 144: $A_4^2$ x 378, $C_2^2:C_6^2$ x 114, $C_2^4:C_3^2$ x 112, $C_6:S_4$ x 94, $C_6.S_4$ x 90, $S_3\times S_4$ x 12, $C_6^2:C_2^2$ x 10, $C_6^2:C_4$ x 10, $C_6^2:C_4$ x 4, $C_2^2\times C_6^2$ x 4, $C_6^2:C_2^2$ x 4, $D_6:D_6$ x 4, $D_6.D_6$ x 4, $D_6:D_6$ x 2, $D_6.D_6$ x 2

Order 128: $C_2^3\wr C_2$ x 94, $C_2^5:C_4$ x 79, $C_2^4:D_4$ x 36, $C_2^4.D_4$ x 36, $C_2^4.D_4$ x 36, $C_2^4:D_4$ x 35, $C_2^4:Q_8$ x 12, $C_2^4:D_4$ x 12, $C_2^4.D_4$ x 9, $C_2^5:C_4$ x 9, $C_2^7$ x 9, $C_4^2:C_2^3$ x 9, $D_4^2:C_2$ x 9, $C_2^3.C_4^2$ x 6, $D_4^2:C_2$ x 6

Order 108: $C_3^2\times A_4$ x 86, $C_3:S_3^2$ x 8, $C_3\times C_6^2$ x 2, $C_3^2:D_6$ x 2, $C_3^3:C_4$ x 2

Order 96: $C_2^2:S_4$ x 250, $\GL(2,\mathbb{Z}/4)$ x 240, $C_2^3\times A_4$ x 168, $C_2^2.S_4$ x 106, $C_2^3:A_4$ x 80, $C_2^4:S_3$ x 68, $C_2^2\times S_4$ x 42, $C_2^3.D_6$ x 42, $C_2^3:A_4$ x 32, $C_2^4\times C_6$ x 14, $C_2^3:D_6$ x 14, $D_4\times A_4$ x 8, $C_4\times S_4$ x 8, $C_2^2.D_{12}$ x 8, $C_2^3.D_6$ x 4, $D_4:D_6$ x 4, $C_2^3:C_{12}$ x 4, $C_4:S_4$ x 4, $A_4:Q_8$ x 4, $C_2^4:C_6$ x 4

Order 81: $C_3^4$

Order 72: $C_3:S_4$ x 326, $C_6\times A_4$ x 94, $C_6^2:C_2$ x 40, $S_3\times A_4$ x 12, $C_3\times S_4$ x 12, $C_2\times C_6^2$ x 10, $C_6.D_6$ x 10, $C_6\wr C_2$ x 8, $C_6:D_6$ x 4, $C_6\times D_6$ x 4, $C_6:C_{12}$ x 4, $C_3:D_{12}$ x 4, $C_6.D_6$ x 4, $S_3\times D_6$ x 2, $C_3^2:Q_8$ x 2, $D_6:S_3$ x 2, $C_6.D_6$ x 2

Order 64: $C_2^3:D_4$ x 297, $C_2^4:C_4$ x 294, $C_2^6$ x 109, $C_2^4:C_4$ x 79, $C_2^4:C_4$ x 36, $C_2^2.C_4^2$ x 36, $D_4:D_4$ x 36, $C_2\wr C_2^2$ x 24, $C_4^2:C_2^2$ x 18, $D_4^2$ x 18, $C_4^2:C_2^2$ x 18, $C_2^3:D_4$ x 18, $D_4\times C_2^3$ x 15, $C_4^2:C_2^2$ x 9, $C_4^2:C_2^2$ x 9, $C_4^2:C_2^2$ x 9, $C_4^2:C_2^2$ x 6, $D_4:C_2^3$ x 3, $C_2^3:Q_8$ x 3, $C_4^2:C_2^2$ x 2

Order 54: $C_3^2:S_3$ x 24, $C_3^2:C_6$ x 8, $S_3\times C_3^2$ x 8, $C_3^2\times C_6$ x 2

Order 48: $C_2^2\times A_4$ x 268, $C_2^2:A_4$ x 224, $C_2\times S_4$ x 146, $A_4:C_4$ x 130, $C_6:D_4$ x 112, $C_6.D_4$ x 108, $C_2^3\times C_6$ x 84, $C_6.C_2^3$ x 14, $D_4:S_3$ x 8, $S_3\times D_4$ x 8, $C_6\times D_4$ x 4, $D_{12}:C_2$ x 4, $C_4\times A_4$ x 4, $C_2^2:C_{12}$ x 4, $C_2^2\times D_6$ x 2

Order 36: $C_3\times A_4$ x 322, $C_6^2$ x 44, $C_6:S_3$ x 32, $C_3^2:C_4$ x 30, $S_3^2$ x 16, $C_3:C_{12}$ x 4, $C_6\times S_3$ x 4

Order 32: $C_2^5$ x 486, $C_2^3:C_4$ x 439, $C_2^2\wr C_2$ x 436, $C_2^2\times D_4$ x 167, $C_4:D_4$ x 54, $C_2^3:C_4$ x 36, $C_2^2.D_4$ x 36, $C_4\times D_4$ x 36, $C_4^2:C_2$ x 18, $C_2^2:Q_8$ x 18, $C_2^3\times C_4$ x 15, $D_4:C_2^2$ x 12, $C_4^2:C_2$ x 12, $D_4:C_2^2$ x 9, $C_4^2:C_2$ x 9, $C_4:D_4$ x 6

Order 27: $C_3^3$ x 24

Order 24: $S_4$ x 160, $C_2\times A_4$ x 142, $C_2^2\times C_6$ x 132, $C_3:D_4$ x 120, $C_6:C_4$ x 70, $C_2\times D_6$ x 30, $C_4\times S_3$ x 8, $C_3\times D_4$ x 8, $C_2\times C_{12}$ x 4, $D_{12}$ x 4, $C_3:Q_8$ x 4

Order 18: $C_3:S_3$ x 76, $C_3\times C_6$ x 32, $C_3\times S_3$ x 32

Order 16: $C_2^4$ x 945, $C_2\times D_4$ x 321, $C_2^2:C_4$ x 312, $C_2^2\times C_4$ x 88, $C_4:C_4$ x 18, $D_4:C_2$ x 18, $C_4^2$ x 6, $C_2\times Q_8$ x 3

Order 12: $A_4$ x 144, $C_2\times C_6$ x 136, $D_6$ x 54, $C_3:C_4$ x 46, $C_{12}$ x 4

Order 9: $C_3^2$ x 74

Order 8: $C_2^3$ x 521, $D_4$ x 103, $C_2\times C_4$ x 88, $Q_8$ x 3

Order 6: $C_6$ x 50, $S_3$ x 32

Order 4: $C_2^2$ x 125, $C_4$ x 15

Order 3: $C_3$ x 24

Order 2: $C_2$ x 12

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 82944: $A_4^2:\POPlus(4,3)$

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

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

Order 20736: $C_2^8.C_3^4$

Order 13824: $C_2^8.C_3^3.C_2$ x 2, $C_2^8.C_3^3.C_2$ x 2, $C_2^6.C_3^3.D_4$, $C_2^8.C_3^3.C_2$, $C_2^8.C_3^3.C_2$, $C_2^8.C_3^3.C_2$, $C_2^2:A_4^2.S_4$, $C_2^8:(S_3\times C_3^2)$

Order 10368: $A_4^3:S_3$

Order 9216: $C_2^8.S_3^2$, $C_2^8.S_3^2$, $C_2^8.C_3.D_6$, $C_2^8.S_3^2$, $C_2^8:S_3^2$

Order 6912: $C_2^8.C_3^3$ x 2, $C_2^8.C_3^3$ x 2, $C_2^6.C_3^3.C_2^2$, $C_2^6.C_3^3.C_4$, $A_4^3:C_2^2$, $C_2^2.A_4^3$, $A_4^3:C_2^2$

Order 5184: $A_4^2.S_3^2$, $C_3\times A_4^3$

Order 4608: $C_2^8.C_3.S_3$ x 4, $C_2^8.C_3.S_3$ x 3, $A_4^2.C_2^2\wr C_2$ x 2, $C_2^8.C_3.C_6$, $C_2^6.C_3^2.D_4$, $C_2^8.C_3.C_6$, $A_4^2.C_2^3.C_2^2$, $C_2^8.C_3.S_3$, $C_2^6.C_3^2.D_4$, $C_2^4:C_3.C_2^4.C_6$, $C_2^8.C_3.S_3$, $A_4^2.Q_8:C_2^2$, $(C_2^2\times A_4):\GL(2,\mathbb{Z}/4)$, $C_2^4.\PSOPlus(4,3)$, $C_2^4:\PSOPlus(4,3)$, $C_2^5:A_4^2$, $C_2^6.(S_3\times A_4)$, $(C_2^2\times A_4)\wr C_2$, $C_2^7.S_3^2$

Order 3456: $C_2^6.C_3^3.C_2$ x 8, $C_2^6.C_3^3.C_2$ x 2, $C_2^4:C_3.A_4.C_6$, $C_2^6.C_3^3.C_2$, $A_4^2.D_6.C_2$, $(C_3\times A_4^2).D_4$, $C_2\times A_4^3$, $A_4^2:S_4$, $C_2^2:A_4^2:S_3$

Order 3072: $C_2^8.D_6$, $C_2^4.C_2^4:S_3.C_2$

Order 2592: $C_2^4.C_3^4.C_2$, $C_2^4.C_3^4.C_2$, $(A_4\times C_6^2):S_3$

Order 2304: $C_2^8.C_3^2$ x 4, $A_4^2.C_2^2:C_4$ x 3, $C_2^6.C_3^2.C_4$ x 2, $C_2^8.C_3^2$ x 2, $C_2\times A_4:\GL(2,\mathbb{Z}/4)$ x 2, $C_2^4:A_4^2$ x 2, $C_2\times A_4^2.C_2^3$, $A_4^2.C_2^4$, $A_4^2.D_4.C_2$, $A_4^2.D_4.C_2$, $A_4^2.D_4:C_2$, $C_2^6.C_3^2.C_4$, $C_2^6.S_3^2$, $C_2^4.C_6^2.C_2^2$, $A_4^2.D_4.C_2$, $A_4^2.C_2^3.C_2$, $(C_2^4\times A_4).D_6$, $C_2^8.C_3^2$, $C_2^8.C_3^2$, $C_2^6.C_6^2$, $C_2^6.S_3^2$, $C_2^6.C_3:S_3.C_2$, $A_4^2.C_2^3.C_2$, $C_2^4:A_4^2$, $(C_2^3\times A_4):S_4$, $C_2^4:A_4^2$, $(C_2^3\times A_4):S_4$, $C_2\times A_4^2:D_4$, $C_2\wr S_3^2$

Order 1728: $A_4^3$ x 9, $C_2^6:C_3^3$ x 2, $C_2^6:C_3^3$ x 2, $A_4^2:D_6$ x 2, $A_4^2:D_6$ x 2, $(C_3\times A_4^2):C_4$ x 2, $C_2^6:C_3^3$, $(C_6\times D_6):S_4$

Order 1536: $C_2^6:S_4$ x 3, $C_2^4:\GL(2,\mathbb{Z}/4)$ x 3, $C_2^4.\GL(2,\mathbb{Z}/4)$ x 2, $C_2^4:\GL(2,\mathbb{Z}/4)$, $C_2^8.S_3$, $C_2^7:D_6$, $C_2^4.\GL(2,\mathbb{Z}/4)$, $C_2^8:C_6$, $C_2^8:C_6$

Order 1296: $C_3^2\times A_4^2$ x 2

Order 1152: $(C_2^2\times A_4):S_4$ x 20, $A_4:\GL(2,\mathbb{Z}/4)$ x 13, $C_2^2:\PSOPlus(4,3)$ x 10, $C_2^3\times A_4^2$ x 4, $C_2\times A_4^2:C_4$ x 4, $C_2^3:A_4^2$ x 2, $A_4^2:C_2^3$ x 2, $(C_2\times C_6):\GL(2,\mathbb{Z}/4)$ x 2, $A_4^2:D_4$ x 2, $(C_2\times C_6):\GL(2,\mathbb{Z}/4)$ x 2, $(C_2^3\times C_6):S_4$, $C_2^3:A_4^2$, $C_2^4:(S_3\times A_4)$, $C_2^4:(C_3\times S_4)$, $A_4^2:C_2^3$, $A_4\times C_2^2:S_4$, $A_4^2:C_2^3$, $C_2\times A_4^2:C_4$, $A_4^2:Q_8$, $A_4^2:D_4$, $A_4^2:D_4$, $C_2^5.S_3^2$, $C_2^5.S_3^2$, $C_2^3:A_4^2$, $(C_2^2\times S_4):A_4$

Order 1024: $C_2^7.C_2^3$

Order 864: $A_4^2:S_3$ x 17, $C_6^2:S_4$ x 2, $C_6\times A_4^2$ x 2, $A_4^2:S_3$ x 2, $A_4^2:C_6$ x 2, $C_6^2:S_4$ x 2, $C_6^2:S_4$, $(C_6\times D_6):A_4$

Order 768: $C_2^3:\GL(2,\mathbb{Z}/4)$ x 5, $C_2^3.\GL(2,\mathbb{Z}/4)$ x 5, $C_2^3:\GL(2,\mathbb{Z}/4)$ x 5, $A_4.(C_2^5.C_2)$ x 5, $C_2^5:S_4$ x 2, $C_2^6.D_6$ x 2, $C_2\wr D_6$ x 2, $C_2^8.C_3$ x 2, $C_2^6:A_4$ x 2, $C_2^5:S_4$, $C_2^5.S_4$, $C_2^6.D_6$, $C_2^3:\GL(2,\mathbb{Z}/4)$, $C_2^3:\GL(2,\mathbb{Z}/4)$, $C_2^8.C_3$, $C_2^6:A_4$, $C_2^7:C_6$, $C_2^6:C_{12}$

Order 648: $C_3^3:S_4$

Order 576: $C_2^2:A_4^2$ x 19, $C_2^2\times A_4^2$ x 15, $A_4^2:C_2^2$ x 12, $A_4^2:C_4$ x 11, $C_2^2:A_4^2$ x 10, $C_6:\GL(2,\mathbb{Z}/4)$ x 5, $C_6.\GL(2,\mathbb{Z}/4)$ x 5, $\POPlus(4,3)$ x 4, $C_2^4:C_6^2$ x 2, $C_2^4:C_6^2$ x 2, $(C_2^2\times C_6):S_4$ x 2, $(C_2^2\times C_6).S_4$ x 2, $C_2^4:S_3^2$ x 2, $C_2^6:C_3^2$, $A_4^2:C_2^2$, $S_4^2$, $A_4^2:C_4$, $(C_2^2\times C_6):S_4$, $C_2^4:S_3^2$

Order 512: $C_2^6.D_4$ x 4, $C_2^7:C_2^2$ x 2, $C_2^4\wr C_2$ x 2

Order 432: $C_3\times A_4^2$ x 17, $C_6^2:A_4$ x 2, $A_4\times C_6^2$ x 2, $C_6^2:D_6$ x 2, $(C_3^2\times A_4):C_4$ x 2, $A_4:S_3^2$

Order 384: $C_2^2:\GL(2,\mathbb{Z}/4)$ x 17, $C_2\wr S_3$ x 15, $C_2^4:S_4$ x 13, $C_2^2.\GL(2,\mathbb{Z}/4)$ x 13, $A_4\times C_2^5$ x 5, $C_2^4.S_4$ x 5, $C_2^2\times \GL(2,\mathbb{Z}/4)$ x 5, $C_2^5:A_4$ x 4, $C_2^2\wr S_3$ x 3, $C_2\wr C_6$ x 2, $C_2^4:S_4$ x 2, $C_2^2.\GL(2,\mathbb{Z}/4)$ x 2, $C_2^5.D_6$ x 2, $C_2^5:A_4$ x 2, $C_2^5:C_{12}$, $C_2^2.\GL(2,\mathbb{Z}/4)$, $C_2^4.S_4$, $C_2^4:D_{12}$, $C_2^5.D_6$, $C_2^5:A_4$, $\GL(2,\mathbb{Z}/4):C_2^2$, $C_2^6:C_6$, $C_2^5:A_4$, $C_2^4:S_4$, $C_2^6:S_3$

Order 324: $A_4\times C_3^3$, $C_3^2:S_3^2$

Order 288: $\PSOPlus(4,3)$ x 46, $C_3:\GL(2,\mathbb{Z}/4)$ x 17, $(C_2\times C_6):S_4$ x 16, $C_2\times A_4^2$ x 12, $C_2\times C_6.S_4$ x 5, $C_2^3:C_6^2$ x 5, $A_4\wr C_2$ x 5, $C_6^2:D_4$ x 2, $(C_2\times D_6):A_4$ x 2, $C_2^5:C_3^2$ x 2, $C_3\times C_2^2:S_4$ x 2, $C_2\times C_6:S_4$ x 2, $C_2^3:S_3^2$, $C_6^2:D_4$, $C_3\times C_2^3:A_4$, $C_6^2.D_4$, $C_6^2.D_4$, $A_4\times S_4$

Order 256: $C_2^6:C_4$ x 7, $C_2^5:D_4$ x 7, $C_2^4.C_2^4$ x 6, $C_2^5:D_4$ x 6, $C_2^5.D_4$ x 4, $C_2^4.C_4^2$ x 2, $C_2^8$, $C_4^2:C_2^4$

Order 216: $C_3^2:S_4$ x 13, $C_6^2:C_6$ x 2, $C_6^2:C_6$, $C_3^2:S_4$, $C_6^2:S_3$

Order 192: $C_2\times \GL(2,\mathbb{Z}/4)$ x 31, $C_2.\GL(2,\mathbb{Z}/4)$ x 30, $A_4\times C_2^4$ x 23, $C_2^2\wr C_3$ x 15, $C_2^4:A_4$ x 11, $C_2^3:S_4$ x 11, $C_2^3.S_4$ x 10, $C_2^4.D_6$ x 5, $C_2^3.S_4$ x 5, $C_2^4:D_6$ x 5, $C_2^3:S_4$ x 4, $D_4:S_4$ x 2, $D_4\times S_4$ x 2, $C_2^4:C_{12}$, $C_2^4:D_6$, $C_2^4:C_{12}$, $C_2^5\times C_6$, $C_2^3\times S_4$, $C_2^5:C_6$, $\GL(2,\mathbb{Z}/4):C_2$, $C_2^4:A_4$

Order 162: $C_3^3:S_3$, $C_3^2\wr C_2$

Order 144: $A_4^2$ x 45, $C_2^2:C_6^2$ x 19, $C_2^4:C_3^2$ x 15, $C_6:S_4$ x 15, $C_6.S_4$ x 14, $C_6^2:C_2^2$ x 4, $C_6^2:C_4$ x 4, $C_2^2\times C_6^2$ x 2, $S_3\times S_4$ x 2, $C_6^2:C_4$, $C_6^2:C_2^2$, $D_6:D_6$, $D_6:D_6$, $D_6.D_6$, $D_6.D_6$

Order 128: $C_2^5:C_4$ x 29, $C_2^3\wr C_2$ x 23, $C_2^4:D_4$ x 14, $C_2^4:D_4$ x 9, $C_2^4.D_4$ x 9, $C_2^4.D_4$ x 9, $C_2^7$ x 5, $C_2^4:Q_8$ x 4, $C_2^4.D_4$ x 4, $C_2^5:C_4$ x 4, $C_4^2:C_2^3$ x 4, $D_4^2:C_2$ x 4, $C_2^4:D_4$ x 4, $C_2^3.C_4^2$ x 2, $D_4^2:C_2$ x 2

Order 108: $C_3^2\times A_4$ x 13, $C_3:S_3^2$ x 2, $C_3\times C_6^2$, $C_3^2:D_6$, $C_3^3:C_4$

Order 96: $\GL(2,\mathbb{Z}/4)$ x 39, $C_2^3\times A_4$ x 36, $C_2^2:S_4$ x 32, $C_2^2.S_4$ x 24, $C_2^4:S_3$ x 15, $C_2^3.D_6$ x 13, $C_2^2\times S_4$ x 11, $C_2^3:A_4$ x 10, $C_2^3:A_4$ x 5, $C_2^4\times C_6$ x 5, $C_2^3:D_6$ x 5, $D_4\times A_4$ x 2, $C_4\times S_4$ x 2, $C_2^2.D_{12}$ x 2, $C_2^3.D_6$, $D_4:D_6$, $C_2^3:C_{12}$, $C_4:S_4$, $A_4:Q_8$, $C_2^4:C_6$

Order 81: $C_3^4$

Order 72: $C_3:S_4$ x 39, $C_6\times A_4$ x 15, $C_6^2:C_2$ x 9, $C_2\times C_6^2$ x 4, $C_6.D_6$ x 4, $C_6:D_6$ x 2, $S_3\times A_4$ x 2, $C_3\times S_4$ x 2, $C_6\wr C_2$ x 2, $C_6\times D_6$, $S_3\times D_6$, $C_6:C_{12}$, $C_3^2:Q_8$, $C_3:D_{12}$, $D_6:S_3$, $C_6.D_6$, $C_6.D_6$

Order 64: $C_2^3:D_4$ x 70, $C_2^4:C_4$ x 68, $C_2^6$ x 33, $C_2^4:C_4$ x 29, $C_2^4:C_4$ x 9, $C_2^2.C_4^2$ x 9, $D_4:D_4$ x 9, $C_2^3:D_4$ x 8, $D_4\times C_2^3$ x 7, $C_4^2:C_2^2$ x 6, $C_2\wr C_2^2$ x 6, $C_4^2:C_2^2$ x 5, $D_4^2$ x 5, $C_4^2:C_2^2$ x 4, $C_4^2:C_2^2$ x 4, $C_4^2:C_2^2$ x 4, $D_4:C_2^3$ x 2, $C_4^2:C_2^2$ x 2, $C_2^3:Q_8$ x 2, $C_4^2:C_2^2$

Order 54: $C_3^2:S_3$ x 6, $C_3^2:C_6$ x 2, $S_3\times C_3^2$ x 2, $C_3^2\times C_6$

Order 48: $C_2^2\times A_4$ x 47, $C_2^2:A_4$ x 29, $C_6:D_4$ x 26, $C_6.D_4$ x 25, $C_2\times S_4$ x 23, $C_2^3\times C_6$ x 21, $A_4:C_4$ x 20, $C_6.C_2^3$ x 5, $D_4:S_3$ x 2, $S_3\times D_4$ x 2, $C_2^2\times D_6$, $C_6\times D_4$, $D_{12}:C_2$, $C_4\times A_4$, $C_2^2:C_{12}$

Order 36: $C_3\times A_4$ x 38, $C_6^2$ x 11, $C_6:S_3$ x 8, $C_3^2:C_4$ x 7, $S_3^2$ x 4, $C_3:C_{12}$, $C_6\times S_3$

Order 32: $C_2^3:C_4$ x 113, $C_2^5$ x 103, $C_2^2\wr C_2$ x 88, $C_2^2\times D_4$ x 49, $C_4:D_4$ x 15, $C_2^2.D_4$ x 11, $C_2^3:C_4$ x 9, $C_4\times D_4$ x 9, $C_2^3\times C_4$ x 7, $C_2^2:Q_8$ x 6, $C_4^2:C_2$ x 5, $D_4:C_2^2$ x 4, $D_4:C_2^2$ x 4, $C_4^2:C_2$ x 4, $C_4^2:C_2$ x 3, $C_4:D_4$ x 2

Order 27: $C_3^3$ x 6

Order 24: $C_2^2\times C_6$ x 31, $C_3:D_4$ x 25, $S_4$ x 23, $C_2\times A_4$ x 22, $C_6:C_4$ x 19, $C_2\times D_6$ x 9, $C_4\times S_3$ x 2, $C_3\times D_4$ x 2, $C_2\times C_{12}$, $D_{12}$, $C_3:Q_8$

Order 18: $C_3:S_3$ x 16, $C_3\times C_6$ x 8, $C_3\times S_3$ x 5

Order 16: $C_2^4$ x 181, $C_2\times D_4$ x 78, $C_2^2:C_4$ x 73, $C_2^2\times C_4$ x 33, $C_4:C_4$ x 6, $D_4:C_2$ x 6, $C_4^2$ x 2, $C_2\times Q_8$ x 2

Order 12: $C_2\times C_6$ x 31, $A_4$ x 20, $D_6$ x 13, $C_3:C_4$ x 11, $C_{12}$

Order 9: $C_3^2$ x 15

Order 8: $C_2^3$ x 117, $C_2\times C_4$ x 33, $D_4$ x 27, $Q_8$ x 2

Order 6: $C_6$ x 12, $S_3$ x 8

Order 4: $C_2^2$ x 41, $C_4$ x 7

Order 3: $C_3$ x 6

Order 2: $C_2$ x 7

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 82944: $A_4^2:\POPlus(4,3)$ ($C_1$)

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

Order 20736: $C_2^8.C_3^4$ ($C_2^2$)

Order 13824: $C_2^8.C_3^3.C_2$ ($S_3$) x 4, $C_2^8.C_3^3.C_2$ ($S_3$) x 4

Order 6912: $C_2^8.C_3^3$ ($D_6$) x 4, $C_2^8.C_3^3$ ($D_6$) x 4

Order 4608: $C_2^8.C_3.S_3$ ($C_3:S_3$) x 2

Order 2304: $C_2^8.C_3^2$ ($S_3^2$) x 8, $C_2^4:A_4^2$ ($S_3^2$) x 4, $A_4^2.C_2^4$ ($S_3^2$) x 2, $C_2^8.C_3^2$ ($C_6:S_3$) x 2, $C_2^8.C_3^2$ ($S_3^2$) x 2

Order 768: $C_2^8.C_3$ ($C_3:S_3^2$) x 4, $C_2^6:A_4$ ($C_3:S_3^2$) x 4

Order 256: $C_2^8$ ($C_3^2:S_3^2$)

Order 144: $A_4^2$ ($\POPlus(4,3)$) x 2

Order 48: $C_2^2:A_4$ ($A_4^2:D_6$) x 4

Order 16: $C_2^4$ ($A_4^2.S_3^2$) x 2

Order 1: $C_1$ ($A_4^2:\POPlus(4,3)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 82944: $A_4^2:\POPlus(4,3)$ ($C_1$)

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

Order 20736: $C_2^8.C_3^4$ ($C_2^2$)

Order 13824: $C_2^8.C_3^3.C_2$ ($S_3$), $C_2^8.C_3^3.C_2$ ($S_3$)

Order 6912: $C_2^8.C_3^3$ ($D_6$), $C_2^8.C_3^3$ ($D_6$)

Order 4608: $C_2^8.C_3.S_3$ ($C_3:S_3$)

Order 2304: $A_4^2.C_2^4$ ($S_3^2$), $C_2^8.C_3^2$ ($S_3^2$), $C_2^8.C_3^2$ ($C_6:S_3$), $C_2^8.C_3^2$ ($S_3^2$), $C_2^4:A_4^2$ ($S_3^2$)

Order 768: $C_2^8.C_3$ ($C_3:S_3^2$), $C_2^6:A_4$ ($C_3:S_3^2$)

Order 256: $C_2^8$ ($C_3^2:S_3^2$)

Order 144: $A_4^2$ ($\POPlus(4,3)$)

Order 48: $C_2^2:A_4$ ($A_4^2:D_6$)

Order 16: $C_2^4$ ($A_4^2.S_3^2$)

Order 1: $C_1$ ($A_4^2:\POPlus(4,3)$)

Series

Derived series

$A_4^2:\POPlus(4,3)$

$\rhd$

$C_2^8.C_3^4$

$\rhd$

$C_2^8$

$\rhd$

$C_1$

Chief series

$A_4^2:\POPlus(4,3)$

$\rhd$

$C_2^8.C_3^4.C_2$

$\rhd$

$C_2^8.C_3^4$

$\rhd$

$C_2^8.C_3^3$

$\rhd$

$C_2^8.C_3^3$

$\rhd$

$A_4^2.C_2^4$

$\rhd$

$C_2^8.C_3^2$

$\rhd$

$C_2^8.C_3$

$\rhd$

$C_2^8$

$\rhd$

$C_2^4$

$\rhd$

$C_1$

Lower central series

$A_4^2:\POPlus(4,3)$

$\rhd$

$C_2^8.C_3^4$

Upper central series

$C_1$

Supergroups

This group is a maximal subgroup of 7 larger groups in the database.

This group is a maximal quotient of 1 larger groups in the database.

Character theory

Complex character table

Every character has rational values, so the complex character table is the same as the rational character table below.

Rational character table

See the $106 \times 106$ rational character table. Alternatively, you may search for characters of this group with desired properties.