Abstract group 6144.yu: $C_4^3.\GL(2,\mathbb{Z}/4)$

• Label: 6144.yu
• Order: \( 2^{11} \cdot 3 \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{16} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \)
• Perm deg.: $24$
• Trans deg.: $24$
• Rank: $3$

Group information

Description:	$C_4^3.\GL(2,\mathbb{Z}/4)$
Order:	\(6144\)\(\medspace = 2^{11} \cdot 3 \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$D_5:F_5^2$, of order \(196608\)\(\medspace = 2^{16} \cdot 3 \)
Composition factors:	$C_2$ x 11, $C_3$
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	8	12
Elements	1	527	128	1264	1408	2304	512	6144
Conjugacy classes	1	16	1	37	7	18	2	82
Divisions	1	16	1	37	5	18	2	80
Autjugacy classes	1	10	1	20	3	4	2	41

Dimension	1	2	3	4	6	8	12	16	24
Irr. complex chars.	8	10	24	0	14	4	16	2	4	82
Irr. rational chars.	8	6	24	2	14	4	16	2	4	80

Minimal presentations

Permutation degree:	$24$
Transitive degree:	$24$
Rank:	$3$
Inequivalent generating triples:	not computed

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:	${\langle a, b, c, d, e, f, g, h \mid b^{6}=d^{4}=e^{2}=f^{4}=g^{4}=h^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $24$ $\langle(1,2)(3,6)(4,5)(9,22,10,21)(11,17,12,18)(13,24,14,23)(15,19,16,20), (1,18,10,8,23,16) \!\cdots\! \rangle$
Transitive group:	24T9080				more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_4^3.A_4)$ $\,\rtimes\,$ $D_4$	$(C_4^3.A_4)$ $\,\rtimes\,$ $D_4$	$((C_2\times C_4^3).S_4)$ $\,\rtimes\,$ $C_2$	$((C_2\times C_4^3).S_4)$ $\,\rtimes\,$ $C_2$	all 11
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_2^3.C_2^5)$ . $S_4$	$(C_4^3:C_2^2)$ . $S_4$	$C_2^5$ . $(C_2^3:S_4)$	$(C_2^5.C_2^4)$ . $D_6$	all 30

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

Homology

Abelianization:	$C_{2}^{3} $
Schur multiplier:	$C_{2}^{4} \times C_{4}$
Commutator length:	$2$

Subgroups

There are 264601 subgroups in 15992 conjugacy classes, 65 normal (31 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_2^5.\GL(2,\mathbb{Z}/4)$
Commutator:	$G' \simeq$ $(C_2^2\times C_4^2).A_4$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_2^4$	$G/\Phi \simeq$ $C_2^2\wr S_3$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^5.C_2^5$	$G/\operatorname{Fit} \simeq$ $S_3$
Radical:	$R \simeq$ $C_4^3.\GL(2,\mathbb{Z}/4)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^3$	$G/\operatorname{soc} \simeq$ $C_2^3:\GL(2,\mathbb{Z}/4)$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^4.C_2^4.C_2^3$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$

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

All subgroups of index up to 32 (order at least 192) are shown, as well as all normal subgroups of any index.

Order 6144: $C_4^3.\GL(2,\mathbb{Z}/4)$

Order 3072: $(C_2\times C_4^2).\GL(2,\mathbb{Z}/4)$ x 4, $(C_2\times C_4^3).S_4$, $(C_2\times C_4^3).S_4$, $C_4^3.(C_2^2\times A_4)$

Order 2048: $C_2^4.C_2^4.C_2^3$

Order 1536: $(C_2^2\times C_4^2).S_4$ x 2, $(C_2^2\times C_4^2).S_4$ x 2, $(C_2^2\times C_4).\GL(2,\mathbb{Z}/4)$ x 2, $C_2^3.C_2^2\wr C_3$ x 2, $C_4^3.S_4$, $C_4^3.S_4$, $C_2^4.C_2^4.S_3$, $C_4^3.(C_2\times A_4)$, $C_4^3.(C_2\times A_4)$, $(C_2\times C_4^3).A_4$

Order 1024: $C_2^4.C_2^4.C_2^2$ x 8, $C_2^4.C_2^4.C_2^2$ x 4, $C_2^4.C_2^3.C_2^3$ x 4, $C_2^3.C_2^4.C_2^3$ x 2, $C_2^4.C_2^5.C_2$ x 2, $C_2^3.C_2^4.C_2^3$ x 2, $C_2^2.C_2^5.C_2^3$ x 2, $C_2^4.C_2^3.C_2^3$ x 2, $C_2^3.C_2^5.C_2^2$, $C_2^3.C_2^4.C_2^3$, $C_2^4.C_2^5.C_2$, $C_2^5.C_2^5$, $C_2^3.C_2^4.C_2^3$

Order 768: $C_2^4:\GL(2,3)$ x 8, $C_2^3.\GL(2,\mathbb{Z}/4)$ x 4, $(C_2^2\times C_4^2).A_4$ x 3, $(C_2\times C_4^2).S_4$ x 2, $C_2^4.(C_2\times S_4)$ x 2, $C_2^4.(C_2\times S_4)$ x 2, $(D_4\times C_2^3).A_4$ x 2, $C_4^3:D_6$, $C_4\times C_2^3.S_4$, $C_4^3.A_4$, $C_4^3.A_4$, $C_2\times C_4^3:C_6$, $C_4^2:C_6\times D_4$

Order 512: $C_2^3.C_2^5.C_2$ x 24, $C_2^4.C_2^4.C_2$ x 16, $C_2^4.C_2^4.C_2$ x 16, $C_2^3.C_2^5.C_2$ x 16, $C_2^4.C_2^4.C_2$ x 8, $C_2^4.C_2^4.C_2$ x 8, $C_2^3.C_2^5.C_2$ x 8, $C_2^3.C_2^5.C_2$ x 8, $C_2^4.C_2^4.C_2$ x 8, $C_2^4.C_2^4.C_2$ x 8, $C_2^3.C_2^6$ x 7, $C_2^3.C_2^6$ x 6, $C_2^3.C_2^6$ x 5, $C_2^5.C_2^4$ x 4, $C_2^3.C_2^6$ x 4, $C_2^3.C_2^5.C_2$ x 4, $C_2^3.C_2^5.C_2$ x 4, $C_2^3.C_2^5.C_2$ x 4, $C_2^3.C_2^5.C_2$ x 4, $C_2^3.C_2^5.C_2$ x 4, $C_2^3.C_2^5.C_2$ x 4, $C_2^3.C_2^5.C_2$ x 4, $C_2^4.C_2^4.C_2$ x 4, $(C_2\times C_4).D_4^2$ x 4, $C_2.C_2^6.C_2^2$ x 4, $C_2^3.C_2^5.C_2$ x 4, $C_2^2.C_2^5.C_2^2$ x 4, $C_2^2.C_2^6.C_2$ x 2, $C_2^3.C_2^6$ x 2, $C_4^3.C_2^3$ x 2, $C_2\times C_2^2.C_2^5.C_2$ x 2, $C_4^2.C_2^5$ x 2, $C_2^3.C_2^5.C_2$ x 2, $C_2^3.C_2^5.C_2$ x 2, $C_2^3.C_2^5.C_2$ x 2, $C_2^3.C_2^5.C_2$ x 2, $C_2.C_2^6.C_2^2$ x 2, $C_4^2.C_2^3.C_2^2$ x 2, $C_2^3.C_2^6$, $C_2^4.C_2^5$, $(C_2^3\times C_4).C_2^4$, $C_2^3.C_2^5.C_2$, $(C_2^3\times C_4).C_2^4$, $C_2^2.C_2^5.C_2^2$, $(C_2^3\times C_4).C_2^4$, $(C_2^3\times C_4).C_2^4$, $C_2^3.C_2^5.C_2$, $(C_2^3\times C_4).C_2^4$, $(C_2\times C_4).C_2^6$

Order 384: $C_2^2.\GL(2,\mathbb{Z}/4)$ x 8, $C_2^4:\SL(2,3)$ x 4, $C_2^4.S_4$ x 4, $C_2^3:\GL(2,3)$ x 4, $C_2^5.A_4$ x 4, $C_4^3:C_6$ x 3, $C_4^2:(C_3\times D_4)$ x 2, $C_2^4.(C_2\times A_4)$ x 2, $C_2^4.(C_2\times A_4)$ x 2, $C_2^4.(C_2\times A_4)$ x 2, $C_2^4.S_4$ x 2, $(C_2^2\times C_4).S_4$ x 2, $C_4^2:D_{12}$ x 2, $Q_8.(C_2\times S_4)$ x 2, $C_2^4.S_4$ x 2, $(C_2\times C_4^2).A_4$, $C_4^3:C_6$, $C_4^3:C_6$, $C_4^3:C_6$, $C_4:\GL(2,\mathbb{Z}/4)$

Order 256: $C_2^3.C_2^5$ x 72, $C_4.D_4^2$ x 52, $C_2^3.C_2^5$ x 32, $C_2^5.D_4$ x 32, $C_2^3.C_2^5$ x 32, $C_4^2.(C_2\times D_4)$ x 32, $C_4^3.C_2^2$ x 26, $C_4^3:C_2^2$ x 24, $C_4^2.(C_2\times D_4)$ x 24, $(C_4\times D_4):D_4$ x 24, $C_2^3.C_2^5$ x 17, $C_4:D_4^2$ x 16, $(C_2\times C_4^2).D_4$ x 16, $(C_2\times C_4^2).D_4$ x 16, $C_2^4:\SD_{16}$ x 16, $C_2^4:D_8$ x 16, $(C_2\times C_4^2).D_4$ x 16, $(C_2\times C_4^2).D_4$ x 16, $C_4^3:C_2^2$ x 16, $C_4.D_4^2$ x 16, $C_4^3:C_2^2$ x 16, $C_2^3.C_2^5$ x 16, $C_4^3.C_2^2$ x 16, $C_4^3.C_2^2$ x 16, $C_4^2.(C_2\times D_4)$ x 16, $C_4.D_4^2$ x 16, $C_4^2.C_2^4$ x 16, $(C_2^2\times D_4):D_4$ x 16, $C_2^3.C_2^5$ x 12, $C_2^3.C_2^5$ x 12, $C_2^3.C_2^5$ x 12, $C_4^2.(C_2\times D_4)$ x 12, $C_4^3:C_2^2$ x 11, $C_2^3.C_2^5$ x 10, $C_4:D_4^2$ x 10, $C_4^3.C_2^2$ x 10, $C_4^2.C_2^4$ x 8, $C_4^2:\SD_{16}$ x 8, $C_4^2:D_8$ x 8, $(C_2\times Q_8):\SD_{16}$ x 8, $(C_2\times Q_8):D_8$ x 8, $(C_2\times C_4^2).D_4$ x 8, $(C_2\times C_4^2).D_4$ x 8, $(C_2\times C_4^2).D_4$ x 8, $C_2^4.\OD_{16}$ x 8, $C_2^2.D_4^2$ x 8, $C_2^5.D_4$ x 8, $C_4^2.C_2^4$ x 8, $C_2^3.C_2^5$ x 6, $C_4.D_4^2$ x 6, $D_4^2:C_2^2$ x 5, $C_4^3.C_2^2$ x 5, $C_4^3.C_2^2$ x 5, $D_4^2:C_2^2$ x 4, $C_4^2.C_2^4$ x 4, $C_4^2.C_2^4$ x 4, $C_4^2:C_2^4$ x 4, $(C_2\times Q_8).Q_{16}$ x 4, $(C_2\times Q_8).D_8$ x 4, $C_2^4.(C_2\times D_4)$ x 4, $C_2^4.(C_2\times D_4)$ x 4, $C_4^2.\OD_{16}$ x 4, $(C_2\times C_4^2).D_4$ x 4, $(C_2^2\times C_8).D_4$ x 4, $(C_2\times C_4^2).D_4$ x 4, $(C_2\times \OD_{16}):D_4$ x 4, $C_2^4.C_4^2$ x 4, $C_4.D_4^2$ x 4, $C_2^3.C_2^5$ x 4, $C_4^3.C_2^2$ x 4, $C_4.D_4^2$ x 4, $C_2^4.C_2^4$ x 4, $C_2\times \OD_{16}:D_4$ x 4, $C_2^5.D_4$ x 4, $C_2\times C_4^2.D_4$ x 4, $C_4^2.C_2^4$ x 4, $C_4^2.C_2^4$ x 4, $(C_2\times Q_8):\SD_{16}$ x 4, $(C_2\times Q_8):D_8$ x 4, $(C_2\times D_4):\SD_{16}$ x 4, $(C_2\times D_4):D_8$ x 4, $C_4^2.C_2^4$ x 4, $C_4^2.C_4^2$ x 4, $C_4^2.(C_2\times D_4)$ x 4, $(C_2\times D_4):\OD_{16}$ x 4, $C_4^2.C_2^4$ x 3, $C_4^2.C_2^4$ x 3, $C_4^3.C_2^2$ x 3, $D_8:C_2^4$ x 2, $C_4^2.C_2^4$ x 2, $C_4^2.C_2^4$ x 2, $C_2^3.C_2^5$ x 2, $C_4.Q_8^2$ x 2, $C_4^2:\OD_{16}$ x 2, $C_4^2:\OD_{16}$ x 2, $C_4^3.C_4$ x 2, $C_4^3.C_4$ x 2, $(C_2\times D_4).C_4^2$ x 2, $C_4^2.C_4^2$ x 2, $C_4^3.C_2^2$ x 2, $C_4^3:C_2^2$ x 2, $C_4^3:C_2^2$ x 2, $C_4.D_4^2$ x 2, $C_4^3:C_2^2$ x 2, $C_4.D_4^2$ x 2, $C_4^3:C_2^2$ x 2, $(C_2\times C_4^2):D_4$ x 2, $(C_2\times \OD_{16}):D_4$ x 2, $C_2\times \OD_{16}:D_4$ x 2, $(D_4\times C_2^3):C_4$ x 2, $C_4^2.C_2^4$ x 2, $C_4^2.C_2^4$ x 2, $D_4^2:C_2^2$, $C_4^2.C_2^4$, $C_4^2.C_2^4$, $C_4^2.C_2^4$, $(C_4\times \OD_{16}):C_4$, $(C_2\times D_4).C_4^2$, $C_4^3:C_4$, $C_4^3:C_2^2$, $C_4^2.(C_2\times D_4)$, $C_2\times \OD_{16}:D_4$, $C_2^4.C_2^4$, $C_4^3:C_2^2$, $C_4^3:C_2^2$, $C_2\times \OD_{16}:D_4$, $C_4^2.(C_2\times D_4)$, $(C_2^2\times C_4^2):C_2^2$, $C_4^3:C_2^2$, $C_2^4.C_2^4$

Order 192: $C_2^4.A_4$ x 13, $C_4^2:D_6$ x 6, $C_2^3:\SL(2,3)$ x 6, $C_2^3.S_4$ x 4, $Q_8:S_4$ x 4, $C_2\times \GL(2,\mathbb{Z}/4)$ x 4, $C_2^4.A_4$ x 3, $C_4\wr C_3$ x 3, $C_2^3.(C_2\times A_4)$ x 2, $Q_8.(C_2\times A_4)$ x 2, $A_4:C_4^2$, $C_4^2:C_{12}$, $C_2^5:C_6$, $C_2^3:D_{12}$

Order 128: $C_2^4:Q_8$ x 4, $C_4^2:C_2^3$ x 2, $C_2^3.C_2^4$ x 2, $C_2\times C_4^3$, $C_2.Q_8^2$

Order 64: $C_2^3.C_2^3$ x 2, $C_2^3:Q_8$ x 2, $C_4^3$ x 2, $D_4:C_2^3$ x 2, $D_4\times C_2^3$, $C_2^2\times C_4^2$

Order 32: $C_2^5$ x 2, $C_2^2\times Q_8$ x 2, $C_2^3\times C_4$, $C_2\times C_4^2$

Order 24: $C_3:D_4$ x 8

Order 16: $C_2^2\times C_4$ x 2, $C_2^4$

Order 8: $D_4$ x 16, $C_2^3$ x 3

Order 6: $S_3$ x 2

Order 4: $C_2^2$ x 13, $C_4$

Order 3: $C_3$

Order 2: $C_2$ x 11

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 32 (order at least 192) are shown, as well as all normal subgroups of any index.

Order 6144: $C_4^3.\GL(2,\mathbb{Z}/4)$

Order 3072: $(C_2\times C_4^3).S_4$, $(C_2\times C_4^3).S_4$, $(C_2\times C_4^2).\GL(2,\mathbb{Z}/4)$, $C_4^3.(C_2^2\times A_4)$

Order 2048: $C_2^4.C_2^4.C_2^3$

Order 1536: $(C_2^2\times C_4^2).S_4$, $C_4^3.S_4$, $C_4^3.S_4$, $(C_2^2\times C_4^2).S_4$, $(C_2^2\times C_4).\GL(2,\mathbb{Z}/4)$, $C_2^4.C_2^4.S_3$, $C_2^3.C_2^2\wr C_3$, $C_4^3.(C_2\times A_4)$, $C_4^3.(C_2\times A_4)$, $(C_2\times C_4^3).A_4$

Order 1024: $C_2^3.C_2^4.C_2^3$ x 2, $C_2^3.C_2^5.C_2^2$, $C_2^3.C_2^4.C_2^3$, $C_2^3.C_2^4.C_2^3$, $C_2^4.C_2^5.C_2$, $C_2^4.C_2^4.C_2^2$, $C_2^4.C_2^3.C_2^3$, $C_2^2.C_2^5.C_2^3$, $C_2^4.C_2^5.C_2$, $C_2^5.C_2^5$, $C_2^4.C_2^4.C_2^2$, $C_2^4.C_2^3.C_2^3$, $C_2^3.C_2^4.C_2^3$

Order 768: $(C_2^2\times C_4^2).A_4$ x 2, $(C_2\times C_4^2).S_4$, $C_2^3.\GL(2,\mathbb{Z}/4)$, $C_4^3:D_6$, $C_2^4.(C_2\times S_4)$, $C_2^4:\GL(2,3)$, $C_2^4.(C_2\times S_4)$, $C_4\times C_2^3.S_4$, $C_4^3.A_4$, $(D_4\times C_2^3).A_4$, $C_4^3.A_4$, $C_2\times C_4^3:C_6$, $C_4^2:C_6\times D_4$

Order 512: $C_2^3.C_2^6$ x 4, $C_2^3.C_2^5.C_2$ x 4, $C_2^3.C_2^6$ x 3, $C_2^3.C_2^6$ x 3, $C_2^3.C_2^6$ x 3, $C_2^4.C_2^4.C_2$ x 3, $C_2^4.C_2^4.C_2$ x 3, $C_2^3.C_2^5.C_2$ x 3, $C_2^3.C_2^5.C_2$ x 3, $C_2^4.C_2^4.C_2$ x 3, $C_2^5.C_2^4$ x 2, $C_2^3.C_2^6$ x 2, $C_4^3.C_2^3$ x 2, $C_2^3.C_2^5.C_2$ x 2, $C_2^3.C_2^5.C_2$ x 2, $C_2^3.C_2^5.C_2$ x 2, $C_2^3.C_2^5.C_2$ x 2, $C_2^3.C_2^5.C_2$ x 2, $C_2^4.C_2^4.C_2$ x 2, $C_2^3.C_2^5.C_2$ x 2, $C_2^4.C_2^4.C_2$ x 2, $C_2^3.C_2^5.C_2$ x 2, $C_2^4.C_2^4.C_2$ x 2, $C_2^3.C_2^5.C_2$ x 2, $C_2^4.C_2^4.C_2$ x 2, $C_2^3.C_2^5.C_2$ x 2, $C_2^3.C_2^5.C_2$ x 2, $C_2^2.C_2^5.C_2^2$ x 2, $C_2^2.C_2^6.C_2$, $C_2^3.C_2^6$, $C_2^4.C_2^5$, $(C_2^3\times C_4).C_2^4$, $C_2^3.C_2^5.C_2$, $C_2\times C_2^2.C_2^5.C_2$, $C_4^2.C_2^5$, $C_2^3.C_2^5.C_2$, $C_2^3.C_2^5.C_2$, $C_2^3.C_2^5.C_2$, $(C_2\times C_4).D_4^2$, $C_2.C_2^6.C_2^2$, $(C_2^3\times C_4).C_2^4$, $C_2^2.C_2^5.C_2^2$, $C_2.C_2^6.C_2^2$, $(C_2^3\times C_4).C_2^4$, $(C_2^3\times C_4).C_2^4$, $C_2^3.C_2^5.C_2$, $(C_2^3\times C_4).C_2^4$, $C_4^2.C_2^3.C_2^2$, $(C_2\times C_4).C_2^6$

Order 384: $C_4^3:C_6$ x 3, $C_4^2:D_{12}$ x 2, $C_2^5.A_4$ x 2, $(C_2\times C_4^2).A_4$, $C_4^2:(C_3\times D_4)$, $C_2^4:\SL(2,3)$, $C_2^4.(C_2\times A_4)$, $C_2^4.S_4$, $C_2^4.(C_2\times A_4)$, $C_2^4.(C_2\times A_4)$, $C_4^3:C_6$, $C_4^3:C_6$, $C_4^3:C_6$, $C_2^2.\GL(2,\mathbb{Z}/4)$, $C_2^4.S_4$, $(C_2^2\times C_4).S_4$, $C_2^3:\GL(2,3)$, $Q_8.(C_2\times S_4)$, $C_4:\GL(2,\mathbb{Z}/4)$, $C_2^4.S_4$

Order 256: $C_2^3.C_2^5$ x 22, $C_4.D_4^2$ x 17, $C_4^3.C_2^2$ x 12, $C_4^3:C_2^2$ x 12, $C_2^3.C_2^5$ x 10, $C_2^3.C_2^5$ x 10, $C_2^3.C_2^5$ x 9, $C_2^3.C_2^5$ x 7, $C_4^3:C_2^2$ x 7, $C_4^3:C_2^2$ x 7, $C_4^3.C_2^2$ x 7, $C_4:D_4^2$ x 7, $C_4:D_4^2$ x 6, $C_4.D_4^2$ x 6, $C_2^3.C_2^5$ x 6, $C_4^3.C_2^2$ x 6, $C_4^3:C_2^2$ x 6, $C_4^2.(C_2\times D_4)$ x 6, $(C_4\times D_4):D_4$ x 6, $(C_2^2\times D_4):D_4$ x 6, $C_2^3.C_2^5$ x 5, $C_4^3.C_2^2$ x 5, $C_2^5.D_4$ x 4, $C_2^3.C_2^5$ x 4, $C_4^3.C_2^2$ x 4, $C_4^2.(C_2\times D_4)$ x 4, $D_4^2:C_2^2$ x 3, $D_4^2:C_2^2$ x 3, $C_4^3.C_2^2$ x 3, $C_2^3.C_2^5$ x 3, $(C_2\times C_4^2).D_4$ x 3, $(C_2\times C_4^2).D_4$ x 3, $(C_2\times Q_8):\SD_{16}$ x 3, $(C_2\times Q_8):D_8$ x 3, $(C_2\times C_4^2).D_4$ x 3, $(C_2\times C_4^2).D_4$ x 3, $(C_2\times C_4^2).D_4$ x 3, $(C_2\times C_4^2).D_4$ x 3, $(C_2\times C_4^2).D_4$ x 3, $C_2^3.C_2^5$ x 3, $C_4.D_4^2$ x 3, $C_2^3.C_2^5$ x 3, $C_4.D_4^2$ x 3, $C_4^3.C_2^2$ x 3, $C_4^3.C_2^2$ x 3, $C_2^2.D_4^2$ x 3, $C_4^2.C_2^4$ x 2, $C_4^2.C_2^4$ x 2, $C_4^2.C_2^4$ x 2, $C_4^2.C_2^4$ x 2, $C_4.Q_8^2$ x 2, $C_4^2:\SD_{16}$ x 2, $C_4^2:D_8$ x 2, $C_2^4:\SD_{16}$ x 2, $C_2^4:D_8$ x 2, $(C_2\times Q_8).Q_{16}$ x 2, $(C_2\times Q_8).D_8$ x 2, $C_2^4.(C_2\times D_4)$ x 2, $C_2^4.(C_2\times D_4)$ x 2, $C_4^2:\OD_{16}$ x 2, $C_4^2:\OD_{16}$ x 2, $C_4^2.\OD_{16}$ x 2, $(C_2\times C_4^2).D_4$ x 2, $C_4^3.C_4$ x 2, $C_4^3.C_4$ x 2, $C_4^3.C_2^2$ x 2, $C_4^2.(C_2\times D_4)$ x 2, $C_4.D_4^2$ x 2, $C_4^3:C_2^2$ x 2, $C_4.D_4^2$ x 2, $C_4^3:C_2^2$ x 2, $C_2^5.D_4$ x 2, $C_4^2.(C_2\times D_4)$ x 2, $C_4^2.C_2^4$ x 2, $(C_2\times Q_8):\SD_{16}$ x 2, $(C_2\times Q_8):D_8$ x 2, $(C_2\times D_4):\SD_{16}$ x 2, $(C_2\times D_4):D_8$ x 2, $C_4^2.C_2^4$ x 2, $C_4^2.C_4^2$ x 2, $C_4^2.(C_2\times D_4)$ x 2, $(C_2\times D_4):\OD_{16}$ x 2, $D_8:C_2^4$, $D_4^2:C_2^2$, $C_4^2.C_2^4$, $C_4^2.C_2^4$, $C_4^2.C_2^4$, $C_4^2.C_2^4$, $C_2^3.C_2^5$, $C_4^2:C_2^4$, $C_4^2.C_2^4$, $C_4^2.C_2^4$, $C_2^4.\OD_{16}$, $(C_2^2\times C_8).D_4$, $(C_2\times C_4^2).D_4$, $(C_2\times \OD_{16}):D_4$, $(C_2\times D_4).C_4^2$, $C_2^4.C_4^2$, $C_4^2.C_4^2$, $(C_4\times \OD_{16}):C_4$, $(C_2\times D_4).C_4^2$, $C_4^3:C_4$, $C_4^3:C_2^2$, $C_4.D_4^2$, $C_4^3:C_2^2$, $C_4^3:C_2^2$, $C_4.D_4^2$, $(C_2\times C_4^2):D_4$, $C_2^4.C_2^4$, $C_2\times \OD_{16}:D_4$, $(C_2\times \OD_{16}):D_4$, $C_2^5.D_4$, $C_4^2.(C_2\times D_4)$, $C_2\times \OD_{16}:D_4$, $C_2^4.C_2^4$, $C_4^3:C_2^2$, $C_4^3:C_2^2$, $C_2\times \OD_{16}:D_4$, $C_2\times C_4^2.D_4$, $C_2\times \OD_{16}:D_4$, $C_4^2.(C_2\times D_4)$, $(D_4\times C_2^3):C_4$, $(C_2^2\times C_4^2):C_2^2$, $C_4^3:C_2^2$, $C_2^4.C_2^4$, $C_4^2.C_2^4$, $C_4^2.C_2^4$, $C_4^2.C_2^4$, $C_4^2.C_2^4$, $C_4^2.C_2^4$

Order 192: $C_2^4.A_4$ x 5, $C_4\wr C_3$ x 3, $C_2^4.A_4$ x 2, $C_4^2:D_6$ x 2, $C_2^3:\SL(2,3)$ x 2, $A_4:C_4^2$, $C_2^3.(C_2\times A_4)$, $C_4^2:C_{12}$, $C_2^3.S_4$, $Q_8.(C_2\times A_4)$, $C_2^5:C_6$, $Q_8:S_4$, $C_2\times \GL(2,\mathbb{Z}/4)$, $C_2^3:D_{12}$

Order 128: $C_2\times C_4^3$, $C_4^2:C_2^3$, $C_2^4:Q_8$, $C_2.Q_8^2$, $C_2^3.C_2^4$

Order 64: $C_4^3$ x 2, $C_2^3.C_2^3$, $C_2^3:Q_8$, $D_4:C_2^3$, $D_4\times C_2^3$, $C_2^2\times C_4^2$

Order 32: $C_2^5$, $C_2^2\times Q_8$, $C_2^3\times C_4$, $C_2\times C_4^2$

Order 24: $C_3:D_4$

Order 16: $C_2^2\times C_4$ x 2, $C_2^4$

Order 8: $C_2^3$ x 2, $D_4$ x 2

Order 6: $S_3$

Order 4: $C_2^2$ x 3, $C_4$

Order 3: $C_3$

Order 2: $C_2$ x 6

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 6144: $C_4^3.\GL(2,\mathbb{Z}/4)$ ($C_1$)

Order 3072: $(C_2\times C_4^2).\GL(2,\mathbb{Z}/4)$ ($C_2$) x 4, $(C_2\times C_4^3).S_4$ ($C_2$), $(C_2\times C_4^3).S_4$ ($C_2$), $C_4^3.(C_2^2\times A_4)$ ($C_2$)

Order 1536: $(C_2^2\times C_4^2).S_4$ ($C_2^2$) x 2, $(C_2^2\times C_4^2).S_4$ ($C_2^2$) x 2, $C_2^3.C_2^2\wr C_3$ ($C_2^2$) x 2, $(C_2\times C_4^3).A_4$ ($C_2^2$)

Order 1024: $C_2^5.C_2^5$ ($S_3$)

Order 768: $(C_2^2\times C_4^2).A_4$ ($C_2^3$), $C_4^3.A_4$ ($D_4$), $C_4^3.A_4$ ($D_4$)

Order 512: $C_2^5.C_2^4$ ($D_6$) x 2, $C_4^3.C_2^3$ ($D_6$)

Order 384: $(C_2\times C_4^2).A_4$ ($C_2\times D_4$)

Order 256: $C_2^3.C_2^5$ ($S_4$) x 2, $C_2^3.C_2^5$ ($C_2\times D_6$), $C_4.Q_8^2$ ($C_3:D_4$), $C_4^3.C_2^2$ ($C_3:D_4$), $C_4^3:C_2^2$ ($S_4$)

Order 128: $C_2^4:Q_8$ ($C_2\times S_4$) x 4, $C_4^2:C_2^3$ ($C_2\times S_4$) x 2, $C_2^3.C_2^4$ ($C_2\times S_4$) x 2, $C_2\times C_4^3$ ($C_2\times S_4$), $C_2.Q_8^2$ ($C_6:D_4$)

Order 64: $C_2^3.C_2^3$ ($\GL(2,\mathbb{Z}/4)$) x 2, $C_2^3:Q_8$ ($C_2^2\times S_4$) x 2, $C_4^3$ ($\GL(2,\mathbb{Z}/4)$) x 2, $D_4:C_2^3$ ($\GL(2,\mathbb{Z}/4)$) x 2, $D_4\times C_2^3$ ($C_2^2:S_4$), $C_2^2\times C_4^2$ ($C_2^2\times S_4$)

Order 32: $C_2^5$ ($C_2^3:S_4$) x 2, $C_2^2\times Q_8$ ($C_2\times \GL(2,\mathbb{Z}/4)$) x 2, $C_2^3\times C_4$ ($C_2^3:S_4$), $C_2\times C_4^2$ ($C_2\times \GL(2,\mathbb{Z}/4)$)

Order 16: $C_2^2\times C_4$ ($C_2\wr S_3$) x 2, $C_2^4$ ($C_2^2\wr S_3$)

Order 8: $C_2^3$ ($C_2^4.(C_2\times S_4)$) x 2, $C_2^3$ ($C_2^3:\GL(2,\mathbb{Z}/4)$)

Order 4: $C_2^2$ ($C_2^2.C_2^6.S_3$), $C_4$ ($C_4^2.\GL(2,\mathbb{Z}/4)$)

Order 2: $C_2$ ($C_2^5.\GL(2,\mathbb{Z}/4)$)

Order 1: $C_1$ ($C_4^3.\GL(2,\mathbb{Z}/4)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 6144: $C_4^3.\GL(2,\mathbb{Z}/4)$ ($C_1$)

Order 3072: $(C_2\times C_4^3).S_4$ ($C_2$), $(C_2\times C_4^3).S_4$ ($C_2$), $(C_2\times C_4^2).\GL(2,\mathbb{Z}/4)$ ($C_2$), $C_4^3.(C_2^2\times A_4)$ ($C_2$)

Order 1536: $(C_2^2\times C_4^2).S_4$ ($C_2^2$), $(C_2^2\times C_4^2).S_4$ ($C_2^2$), $C_2^3.C_2^2\wr C_3$ ($C_2^2$), $(C_2\times C_4^3).A_4$ ($C_2^2$)

Order 1024: $C_2^5.C_2^5$ ($S_3$)

Order 768: $(C_2^2\times C_4^2).A_4$ ($C_2^3$), $C_4^3.A_4$ ($D_4$), $C_4^3.A_4$ ($D_4$)

Order 512: $C_2^5.C_2^4$ ($D_6$), $C_4^3.C_2^3$ ($D_6$)

Order 384: $(C_2\times C_4^2).A_4$ ($C_2\times D_4$)

Order 256: $C_2^3.C_2^5$ ($C_2\times D_6$), $C_4.Q_8^2$ ($C_3:D_4$), $C_2^3.C_2^5$ ($S_4$), $C_4^3.C_2^2$ ($C_3:D_4$), $C_4^3:C_2^2$ ($S_4$)

Order 128: $C_2\times C_4^3$ ($C_2\times S_4$), $C_4^2:C_2^3$ ($C_2\times S_4$), $C_2^4:Q_8$ ($C_2\times S_4$), $C_2.Q_8^2$ ($C_6:D_4$), $C_2^3.C_2^4$ ($C_2\times S_4$)

Order 64: $C_4^3$ ($\GL(2,\mathbb{Z}/4)$) x 2, $C_2^3.C_2^3$ ($\GL(2,\mathbb{Z}/4)$), $C_2^3:Q_8$ ($C_2^2\times S_4$), $D_4:C_2^3$ ($\GL(2,\mathbb{Z}/4)$), $D_4\times C_2^3$ ($C_2^2:S_4$), $C_2^2\times C_4^2$ ($C_2^2\times S_4$)

Order 32: $C_2^5$ ($C_2^3:S_4$), $C_2^2\times Q_8$ ($C_2\times \GL(2,\mathbb{Z}/4)$), $C_2^3\times C_4$ ($C_2^3:S_4$), $C_2\times C_4^2$ ($C_2\times \GL(2,\mathbb{Z}/4)$)

Order 16: $C_2^2\times C_4$ ($C_2\wr S_3$) x 2, $C_2^4$ ($C_2^2\wr S_3$)

Order 8: $C_2^3$ ($C_2^4.(C_2\times S_4)$), $C_2^3$ ($C_2^3:\GL(2,\mathbb{Z}/4)$)

Order 4: $C_2^2$ ($C_2^2.C_2^6.S_3$), $C_4$ ($C_4^2.\GL(2,\mathbb{Z}/4)$)

Order 2: $C_2$ ($C_2^5.\GL(2,\mathbb{Z}/4)$)

Order 1: $C_1$ ($C_4^3.\GL(2,\mathbb{Z}/4)$)

Series

Derived series

$C_4^3.\GL(2,\mathbb{Z}/4)$

$\rhd$

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

$\rhd$

$C_2.Q_8^2$

$\rhd$

$C_2^3$

$\rhd$

$C_1$

Chief series

$C_4^3.\GL(2,\mathbb{Z}/4)$

$\rhd$

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

$\rhd$

$C_2^3.C_2^2\wr C_3$

$\rhd$

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

$\rhd$

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

$\rhd$

$C_2.Q_8^2$

$\rhd$

$C_2\times C_4^2$

$\rhd$

$C_2^3$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_4^3.\GL(2,\mathbb{Z}/4)$

$\rhd$

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

$\rhd$

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

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_4$

Supergroups

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

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

Character theory

Complex character table

See the $82 \times 82$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $80 \times 80$ rational character table.