Abstract group 1536.408640920: $\GL(2,\mathbb{Z}/4):C_2^4$

• Label: 1536.408640920
• Order: \( 2^{9} \cdot 3 \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{6} \)
• $\card{Z(G)}$: \( 2^{4} \)
• $\card{\Aut(G)}$: \( 2^{22} \cdot 3^{2} \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{17} \cdot 3 \cdot 7 \)
• Perm deg.: not computed
• Trans deg.: $96$
• Rank: $6$

Group information

Description:	$\GL(2,\mathbb{Z}/4):C_2^4$
Order:	\(1536\)\(\medspace = 2^{9} \cdot 3 \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$A_4.C_2^6.C_2^4.C_2^6.C_2.\PSL(2,7)$, of order \(264241152\)\(\medspace = 2^{22} \cdot 3^{2} \cdot 7 \)
Composition factors:	$C_2$ x 9, $C_3$
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	479	8	544	376	128	1536
Conjugacy classes	1	95	1	64	31	8	200
Divisions	1	95	1	64	31	8	200
Autjugacy classes	1	9	1	6	3	1	21

Dimension	1	2	3	4	6
Irr. complex chars.	64	48	64	8	16	200
Irr. rational chars.	64	48	64	8	16	200

Minimal presentations

Permutation degree:	not computed
Transitive degree:	$96$
Rank:	$6$
Inequivalent generating 6-tuples:	$14504891940$

Minimal degrees of faithful linear representations

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

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h \mid a^{2}=b^{6}=c^{4}=d^{2}=e^{2}=f^{2}= \!\cdots\! \rangle}$
Direct product:	$C_2$ ${}^3$ $\, \times\, $ $D_4$ $\, \times\, $ $S_4$
Semidirect product:	$C_2^7$ $\,\rtimes\,$ $D_6$	$(D_4\times C_2^5)$ $\,\rtimes\,$ $S_3$	$C_2^5$ $\,\rtimes\,$ $(S_3\times D_4)$	$(S_4\times C_2^5)$ $\,\rtimes\,$ $C_2$	all 61
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_2\times S_4)$ . $C_2^5$	$C_2$ . $(S_4\times C_2^5)$	$(C_2\times A_4)$ . $C_2^6$	$C_2^6$ . $(C_2\times D_6)$	all 16
Aut. group:	$\Aut(\SL(2,3).\SD_{16})$	$\Aut(\SL(2,3).\SD_{16})$	$\Aut(\SL(2,3).Q_{16})$	$\Aut(C_8:\GL(2,3))$	all 57

Elements of the group are displayed as words in the presentation generators from the presentation above.

Homology

Abelianization:	$C_{2}^{6} $
Schur multiplier:	$C_{2}^{16}$
Commutator length:	$1$

Subgroups

There are 540436 subgroups in 99595 conjugacy classes, 3982 normal (20 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^4$	$G/Z \simeq$ $C_2^2\times S_4$
Commutator:	$G' \simeq$ $C_2\times A_4$	$G/G' \simeq$ $C_2^6$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $S_4\times C_2^5$
Fitting:	$\operatorname{Fit} \simeq$ $D_4\times C_2^5$	$G/\operatorname{Fit} \simeq$ $S_3$
Radical:	$R \simeq$ $\GL(2,\mathbb{Z}/4):C_2^4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^6$	$G/\operatorname{soc} \simeq$ $C_2\times D_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^3\times D_4^2$
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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Subgroup information

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Classes of subgroups up to conjugation

Order 1536: $\GL(2,\mathbb{Z}/4):C_2^4$

Order 768: $\GL(2,\mathbb{Z}/4):C_2^3$ x 56, $S_4\times C_2^5$ x 2, $C_2^3\times \GL(2,\mathbb{Z}/4)$ x 2, $C_2^5:D_{12}$, $C_2^6.D_6$, $C_2^7:C_6$

Order 512: $C_2^3\times D_4^2$

Order 384: $\GL(2,\mathbb{Z}/4):C_2^2$ x 448, $C_2^4\times S_4$ x 91, $C_2^2\times \GL(2,\mathbb{Z}/4)$ x 56, $C_2^6:C_6$ x 28, $C_2^4:D_{12}$ x 28, $C_2^5.D_6$ x 28, $A_4\times C_2^5$ x 2, $C_{12}:C_2^5$, $C_2^5:C_{12}$, $C_2^4.S_4$

Order 256: $C_2^2\times D_4^2$ x 112, $D_4\times C_2^5$ x 4, $C_2^5:D_4$ x 4, $C_2^5:D_4$ x 4, $C_4^2:C_2^4$ x 2, $C_4^2:C_2^4$

Order 192: $C_2^3\times S_4$ x 746, $D_4\times S_4$ x 512, $C_2\times \GL(2,\mathbb{Z}/4)$ x 224, $C_2^5:C_6$ x 112, $C_2^3:D_{12}$ x 112, $C_2^4.D_6$ x 112, $C_{12}:C_2^4$ x 56, $A_4\times C_2^4$ x 45, $C_2^4:C_{12}$ x 14, $C_2^3.S_4$ x 14, $D_6\times C_2^4$ x 2, $D_6:C_2^4$ x 2, $C_{12}:C_2^4$, $C_2^3\times D_{12}$, $C_{12}:C_2^4$

Order 128: $C_2\times D_4^2$ x 1792, $D_4\times C_2^4$ x 336, $C_2^4:D_4$ x 224, $C_2^4:D_4$ x 224, $C_4^2:C_2^3$ x 112, $C_4^2:C_2^3$ x 56, $C_2^5:C_4$ x 8, $C_2^7$ x 4, $C_2^5\times C_4$ x 4, $C_2^4.D_4$ x 2, $C_2^3\times C_4^2$

Order 96: $C_2^2\times S_4$ x 1332, $D_4\times D_6$ x 448, $C_2^3\times A_4$ x 183, $\GL(2,\mathbb{Z}/4)$ x 128, $C_2^3\times D_6$ x 91, $D_4\times A_4$ x 64, $C_4:S_4$ x 64, $C_4\times S_4$ x 64, $C_2^3:D_6$ x 56, $C_{12}:C_2^3$ x 28, $C_2^2\times D_{12}$ x 28, $C_{12}:C_2^3$ x 28, $C_2^3:C_{12}$ x 28, $C_2^2.S_4$ x 28, $C_2^4\times C_6$ x 2, $C_2^3\times C_{12}$, $C_6.C_2^4$

Order 64: $D_4\times C_2^3$ x 5307, $D_4^2$ x 4096, $C_2^3:D_4$ x 1792, $C_2^3:D_4$ x 1792, $C_4^2:C_2^2$ x 896, $C_4^2:C_2^2$ x 448, $C_2^6$ x 232, $C_2^4:C_4$ x 224, $C_2^4\times C_4$ x 167, $C_2^3.D_4$ x 56, $C_2^2\times C_4^2$ x 28

Order 48: $C_2^2\times D_6$ x 746, $S_3\times D_4$ x 512, $C_2\times S_4$ x 504, $C_6:D_4$ x 224, $C_2^2\times A_4$ x 163, $C_6\times D_4$ x 112, $C_2\times D_{12}$ x 112, $C_4\times D_6$ x 112, $C_2^3\times C_6$ x 45, $C_2^2\times C_{12}$ x 14, $C_6.C_2^3$ x 14, $C_4\times A_4$ x 8, $A_4:C_4$ x 8

Order 32: $C_2^2\times D_4$ x 18708, $C_2^5$ x 3015, $C_4:D_4$ x 2048, $C_2^2\wr C_2$ x 2048, $C_2^3\times C_4$ x 1317, $C_4\times D_4$ x 1024, $C_2^3:C_4$ x 896, $C_4:D_4$ x 512, $C_2^2.D_4$ x 224, $C_2\times C_4^2$ x 112

Order 24: $C_2\times D_6$ x 1332, $C_2^2\times C_6$ x 183, $C_3:D_4$ x 128, $D_{12}$ x 64, $C_4\times S_3$ x 64, $C_3\times D_4$ x 64, $S_4$ x 32, $C_2\times A_4$ x 31, $C_2\times C_{12}$ x 28, $C_6:C_4$ x 28

Order 16: $C_2\times D_4$ x 14224, $C_2^4$ x 10214, $C_2^2\times C_4$ x 2342, $C_2^2:C_4$ x 512, $C_4:C_4$ x 128, $C_4^2$ x 64

Order 12: $D_6$ x 504, $C_2\times C_6$ x 163, $C_{12}$ x 8, $C_3:C_4$ x 8, $A_4$

Order 8: $C_2^3$ x 8892, $D_4$ x 1856, $C_2\times C_4$ x 916

Order 6: $S_3$ x 32, $C_6$ x 31

Order 4: $C_2^2$ x 1902, $C_4$ x 64

Order 3: $C_3$

Order 2: $C_2$ x 95

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1536: $\GL(2,\mathbb{Z}/4):C_2^4$

Order 768: $S_4\times C_2^5$, $\GL(2,\mathbb{Z}/4):C_2^3$, $C_2^5:D_{12}$, $C_2^3\times \GL(2,\mathbb{Z}/4)$, $C_2^6.D_6$, $C_2^7:C_6$

Order 512: $C_2^3\times D_4^2$

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

Order 256: $D_4\times C_2^5$ x 3, $C_2^5:D_4$ x 3, $C_2^2\times D_4^2$ x 2, $C_2^5:D_4$ x 2, $C_4^2:C_2^4$ x 2, $C_4^2:C_2^4$

Order 192: $C_2^3\times S_4$ x 6, $A_4\times C_2^4$ x 3, $D_6\times C_2^4$, $C_{12}:C_2^4$, $D_6:C_2^4$, $C_{12}:C_2^4$, $C_2^3\times D_{12}$, $C_{12}:C_2^4$, $C_2^5:C_6$, $C_2^4:C_{12}$, $C_2\times \GL(2,\mathbb{Z}/4)$, $C_2^3.S_4$, $D_4\times S_4$, $C_2^3:D_{12}$, $C_2^4.D_6$

Order 128: $D_4\times C_2^4$ x 21, $C_2^5:C_4$ x 5, $C_2^4:D_4$ x 4, $C_2^5\times C_4$ x 3, $C_2^4:D_4$ x 3, $C_4^2:C_2^3$ x 3, $C_2^7$ x 2, $C_2\times D_4^2$ x 2, $C_2^4.D_4$ x 2, $C_4^2:C_2^3$, $C_2^3\times C_4^2$

Order 96: $C_2^2\times S_4$ x 6, $C_2^3\times D_6$ x 4, $C_2^3\times A_4$ x 4, $C_2^4\times C_6$, $C_{12}:C_2^3$, $C_2^3\times C_{12}$, $C_2^3:D_6$, $C_6.C_2^4$, $D_4\times D_6$, $C_2^2\times D_{12}$, $C_{12}:C_2^3$, $D_4\times A_4$, $C_2^3:C_{12}$, $\GL(2,\mathbb{Z}/4)$, $C_2^2.S_4$, $C_4:S_4$, $C_4\times S_4$

Order 64: $D_4\times C_2^3$ x 45, $C_2^6$ x 15, $C_2^4\times C_4$ x 14, $C_2^4:C_4$ x 6, $C_2^3:D_4$ x 4, $C_2^3:D_4$ x 3, $C_4^2:C_2^2$ x 3, $D_4^2$ x 2, $C_2^3.D_4$ x 2, $C_4^2:C_2^2$, $C_2^2\times C_4^2$

Order 48: $C_2^2\times D_6$ x 6, $C_2\times S_4$ x 5, $C_2^2\times A_4$ x 4, $C_2^3\times C_6$ x 3, $C_6\times D_4$, $C_2^2\times C_{12}$, $C_6:D_4$, $C_6.C_2^3$, $S_3\times D_4$, $C_2\times D_{12}$, $C_4\times D_6$, $C_4\times A_4$, $A_4:C_4$

Order 32: $C_2^2\times D_4$ x 47, $C_2^5$ x 41, $C_2^3\times C_4$ x 24, $C_2^3:C_4$ x 6, $C_4:D_4$ x 4, $C_2^2\wr C_2$ x 3, $C_4\times D_4$ x 3, $C_2^2.D_4$ x 2, $C_4:D_4$, $C_2\times C_4^2$

Order 24: $C_2\times D_6$ x 6, $C_2^2\times C_6$ x 4, $C_2\times A_4$ x 3, $S_4$ x 2, $C_2\times C_{12}$, $C_3:D_4$, $C_6:C_4$, $D_{12}$, $C_4\times S_3$, $C_3\times D_4$

Order 16: $C_2^4$ x 57, $C_2\times D_4$ x 40, $C_2^2\times C_4$ x 24, $C_2^2:C_4$ x 6, $C_4:C_4$ x 2, $C_4^2$

Order 12: $D_6$ x 5, $C_2\times C_6$ x 4, $A_4$, $C_{12}$, $C_3:C_4$

Order 8: $C_2^3$ x 55, $C_2\times C_4$ x 18, $D_4$ x 17

Order 6: $C_6$ x 3, $S_3$ x 2

Order 4: $C_2^2$ x 34, $C_4$ x 6

Order 3: $C_3$

Order 2: $C_2$ x 9

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1536: $\GL(2,\mathbb{Z}/4):C_2^4$ ($C_1$)

Order 768: $\GL(2,\mathbb{Z}/4):C_2^3$ ($C_2$) x 56, $S_4\times C_2^5$ ($C_2$) x 2, $C_2^3\times \GL(2,\mathbb{Z}/4)$ ($C_2$) x 2, $C_2^5:D_{12}$ ($C_2$), $C_2^6.D_6$ ($C_2$), $C_2^7:C_6$ ($C_2$)

Order 384: $\GL(2,\mathbb{Z}/4):C_2^2$ ($C_2^2$) x 448, $C_2^4\times S_4$ ($C_2^2$) x 59, $C_2^2\times \GL(2,\mathbb{Z}/4)$ ($C_2^2$) x 56, $C_2^6:C_6$ ($C_2^2$) x 28, $C_2^4:D_{12}$ ($C_2^2$) x 28, $C_2^5.D_6$ ($C_2^2$) x 28, $A_4\times C_2^5$ ($C_2^2$) x 2, $C_2^5:C_{12}$ ($C_2^2$), $C_2^4.S_4$ ($C_2^2$)

Order 256: $D_4\times C_2^5$ ($S_3$)

Order 192: $D_4\times S_4$ ($C_2^3$) x 512, $C_2^3\times S_4$ ($C_2^3$) x 266, $C_2\times \GL(2,\mathbb{Z}/4)$ ($C_2^3$) x 224, $C_2^5:C_6$ ($C_2^3$) x 112, $C_2^3:D_{12}$ ($C_2^3$) x 112, $C_2^4.D_6$ ($C_2^3$) x 112, $A_4\times C_2^4$ ($C_2^3$) x 29, $C_2^3\times S_4$ ($D_4$) x 16, $C_2^4:C_{12}$ ($C_2^3$) x 14, $C_2^3.S_4$ ($C_2^3$) x 14

Order 128: $D_4\times C_2^4$ ($D_6$) x 28, $C_2^7$ ($D_6$) x 2, $C_2^5\times C_4$ ($D_6$)

Order 96: $C_2^2\times S_4$ ($C_2^4$) x 212, $\GL(2,\mathbb{Z}/4)$ ($C_2^4$) x 128, $C_2^2\times S_4$ ($C_2\times D_4$) x 112, $D_4\times A_4$ ($C_2^4$) x 64, $C_4:S_4$ ($C_2^4$) x 64, $C_4\times S_4$ ($C_2^4$) x 64, $C_2^3\times A_4$ ($C_2^4$) x 63, $C_2^3:C_{12}$ ($C_2^4$) x 28, $C_2^2.S_4$ ($C_2^4$) x 28, $C_2^3\times A_4$ ($C_2\times D_4$) x 8

Order 64: $D_4\times C_2^3$ ($C_2\times D_6$) x 112, $C_2^6$ ($C_2\times D_6$) x 29, $C_2^4\times C_4$ ($C_2\times D_6$) x 14, $D_4\times C_2^3$ ($S_4$)

Order 48: $C_2\times S_4$ ($C_2^2\times D_4$) x 112, $C_2^2\times A_4$ ($C_2^2\times D_4$) x 28, $C_2\times S_4$ ($C_2^5$) x 24, $C_2^2\times A_4$ ($C_2^5$) x 23, $C_4\times A_4$ ($C_2^5$) x 8, $A_4:C_4$ ($C_2^5$) x 8

Order 32: $C_2^2\times D_4$ ($C_2^2\times D_6$) x 64, $C_2^5$ ($C_2^2\times D_6$) x 63, $C_2^2\times D_4$ ($C_2\times S_4$) x 28, $C_2^3\times C_4$ ($C_2^2\times D_6$) x 28, $C_2^5$ ($S_3\times D_4$) x 8, $C_2^5$ ($C_2\times S_4$) x 2, $C_2^3\times C_4$ ($C_2\times S_4$)

Order 24: $S_4$ ($D_4\times C_2^3$) x 16, $C_2\times A_4$ ($D_4\times C_2^3$) x 14, $C_2\times A_4$ ($C_2^6$)

Order 16: $C_2\times D_4$ ($C_2^2\times S_4$) x 112, $C_2^4$ ($C_2^2\times S_4$) x 29, $C_2^4$ ($D_4\times D_6$) x 28, $C_2^4$ ($C_2^3\times D_6$) x 23, $C_2^2\times C_4$ ($C_2^2\times S_4$) x 14, $C_2^2\times C_4$ ($C_2^3\times D_6$) x 8

Order 12: $A_4$ ($D_4\times C_2^4$)

Order 8: $D_4$ ($C_2^3\times S_4$) x 64, $C_2^3$ ($C_2^3\times S_4$) x 63, $C_2\times C_4$ ($C_2^3\times S_4$) x 28, $C_2^3$ ($C_{12}:C_2^4$) x 14, $C_2^3$ ($D_4\times S_4$) x 8, $C_2^3$ ($D_6\times C_2^4$)

Order 4: $C_2^2$ ($\GL(2,\mathbb{Z}/4):C_2^2$) x 28, $C_2^2$ ($C_2^4\times S_4$) x 23, $C_4$ ($C_2^4\times S_4$) x 8, $C_2^2$ ($C_{12}:C_2^5$)

Order 2: $C_2$ ($\GL(2,\mathbb{Z}/4):C_2^3$) x 14, $C_2$ ($S_4\times C_2^5$)

Order 1: $C_1$ ($\GL(2,\mathbb{Z}/4):C_2^4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1536: $\GL(2,\mathbb{Z}/4):C_2^4$ ($C_1$)

Order 768: $S_4\times C_2^5$ ($C_2$), $\GL(2,\mathbb{Z}/4):C_2^3$ ($C_2$), $C_2^5:D_{12}$ ($C_2$), $C_2^3\times \GL(2,\mathbb{Z}/4)$ ($C_2$), $C_2^6.D_6$ ($C_2$), $C_2^7:C_6$ ($C_2$)

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

Order 256: $D_4\times C_2^5$ ($S_3$)

Order 192: $C_2^3\times S_4$ ($C_2^3$) x 3, $A_4\times C_2^4$ ($C_2^3$) x 2, $C_2^3\times S_4$ ($D_4$), $C_2^5:C_6$ ($C_2^3$), $C_2^4:C_{12}$ ($C_2^3$), $C_2\times \GL(2,\mathbb{Z}/4)$ ($C_2^3$), $C_2^3.S_4$ ($C_2^3$), $D_4\times S_4$ ($C_2^3$), $C_2^3:D_{12}$ ($C_2^3$), $C_2^4.D_6$ ($C_2^3$)

Order 128: $C_2^7$ ($D_6$), $D_4\times C_2^4$ ($D_6$), $C_2^5\times C_4$ ($D_6$)

Order 96: $C_2^2\times S_4$ ($C_2^4$) x 3, $C_2^3\times A_4$ ($C_2^4$) x 2, $C_2^3\times A_4$ ($C_2\times D_4$), $C_2^2\times S_4$ ($C_2\times D_4$), $D_4\times A_4$ ($C_2^4$), $C_2^3:C_{12}$ ($C_2^4$), $\GL(2,\mathbb{Z}/4)$ ($C_2^4$), $C_2^2.S_4$ ($C_2^4$), $C_4:S_4$ ($C_2^4$), $C_4\times S_4$ ($C_2^4$)

Order 64: $C_2^6$ ($C_2\times D_6$) x 2, $D_4\times C_2^3$ ($C_2\times D_6$), $D_4\times C_2^3$ ($S_4$), $C_2^4\times C_4$ ($C_2\times D_6$)

Order 48: $C_2^2\times A_4$ ($C_2^5$) x 2, $C_2\times S_4$ ($C_2^5$) x 2, $C_2^2\times A_4$ ($C_2^2\times D_4$), $C_2\times S_4$ ($C_2^2\times D_4$), $C_4\times A_4$ ($C_2^5$), $A_4:C_4$ ($C_2^5$)

Order 32: $C_2^5$ ($C_2^2\times D_6$) x 2, $C_2^5$ ($C_2\times S_4$), $C_2^5$ ($S_3\times D_4$), $C_2^2\times D_4$ ($C_2^2\times D_6$), $C_2^2\times D_4$ ($C_2\times S_4$), $C_2^3\times C_4$ ($C_2^2\times D_6$), $C_2^3\times C_4$ ($C_2\times S_4$)

Order 24: $C_2\times A_4$ ($C_2^6$), $C_2\times A_4$ ($D_4\times C_2^3$), $S_4$ ($D_4\times C_2^3$)

Order 16: $C_2^4$ ($C_2^3\times D_6$) x 2, $C_2^4$ ($C_2^2\times S_4$) x 2, $C_2^4$ ($D_4\times D_6$), $C_2\times D_4$ ($C_2^2\times S_4$), $C_2^2\times C_4$ ($C_2^3\times D_6$), $C_2^2\times C_4$ ($C_2^2\times S_4$)

Order 12: $A_4$ ($D_4\times C_2^4$)

Order 8: $C_2^3$ ($C_2^3\times S_4$) x 2, $C_2^3$ ($D_6\times C_2^4$), $C_2^3$ ($C_{12}:C_2^4$), $C_2^3$ ($D_4\times S_4$), $D_4$ ($C_2^3\times S_4$), $C_2\times C_4$ ($C_2^3\times S_4$)

Order 4: $C_2^2$ ($C_2^4\times S_4$) x 2, $C_2^2$ ($C_{12}:C_2^5$), $C_2^2$ ($\GL(2,\mathbb{Z}/4):C_2^2$), $C_4$ ($C_2^4\times S_4$)

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

Order 1: $C_1$ ($\GL(2,\mathbb{Z}/4):C_2^4$)

Series

Derived series

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

$\rhd$

$C_2\times A_4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Chief series

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

$\rhd$

$C_2^6.D_6$

$\rhd$

$C_2^5:C_{12}$

$\rhd$

$C_2^5\times C_4$

$\rhd$

$C_2^6$

$\rhd$

$C_2^5$

$\rhd$

$C_2^4$

$\rhd$

$C_2^3$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

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

$\rhd$

$C_2\times A_4$

$\rhd$

$A_4$

Upper central series

$C_1$

$\lhd$

$C_2^4$

$\lhd$

$D_4\times C_2^3$

Supergroups

This group is a maximal subgroup of 10 larger groups in the database.

This group is a maximal quotient of 4 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 $200 \times 200$ rational character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.