Abstract group 6144.r: $C_4^3.(C_2^2\times S_4)$

• Label: 6144.r
• Order: \( 2^{11} \cdot 3 \)
• Exponent: \( 2^{4} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{17} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{8} \)
• Perm deg.: $50$
• Trans deg.: $96$
• Rank: $3$

Group information

Description:	$C_4^3.(C_2^2\times S_4)$
Order:	\(6144\)\(\medspace = 2^{11} \cdot 3 \)
Exponent:	\(48\)\(\medspace = 2^{4} \cdot 3 \)
Automorphism group:	$C_8^2.C_6.C_2^6.C_2^4$, of order \(393216\)\(\medspace = 2^{17} \cdot 3 \)
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	16	24
Elements	1	399	128	496	384	1664	512	1536	1024	6144
Conjugacy classes	1	11	1	28	3	104	4	16	8	176
Divisions	1	11	1	22	3	42	2	4	2	88
Autjugacy classes	1	6	1	15	2	25	2	1	1	54

Dimension	1	2	3	4	6	8	12	16	24	48
Irr. complex chars.	8	10	56	12	70	0	20	0	0	0	176
Irr. rational chars.	8	6	8	2	18	2	30	2	10	2	88

Minimal presentations

Permutation degree:	$50$
Transitive degree:	$96$
Rank:	$3$
Inequivalent generating triples:	$161280$

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:	${\langle a, b, c, d, e \mid a^{2}=c^{8}=d^{8}=e^{8}=[b,e]=[c,e]=[d,e]=1, b^{6}= \!\cdots\! \rangle}$
Permutation group:	Degree $50$ $\langle(1,6,7,29)(2,8,11,31)(3,17)(4,24)(5,28,27,48)(9,35)(10,39)(12,32,40,41) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 25 & 24 \\ 8 & 9 \end{array}\right), \left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 0 & 11 \\ 3 & 0 \end{array}\right), \left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 9 & 16 \\ 16 & 25 \end{array}\right), \left(\begin{array}{rr} 28 & 11 \\ 25 & 3 \end{array}\right), \left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 13 & 8 \\ 8 & 5 \end{array}\right), \left(\begin{array}{rr} 0 & 1 \\ 1 & 0 \end{array}\right), \left(\begin{array}{rr} 17 & 0 \\ 0 & 17 \end{array}\right), \left(\begin{array}{rr} 13 & 12 \\ 12 & 21 \end{array}\right), \left(\begin{array}{rr} 17 & 16 \\ 16 & 17 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/32\Z)$
Direct product:	$C_2$ $\, \times\, $ $((C_4^2\times C_8).S_4)$
Semidirect product:	$(C_8^2:C_8)$ $\,\rtimes\,$ $D_6$	$(C_4^3.S_4)$ $\,\rtimes\,$ $C_2^2$	$(C_2^4.C_4^3)$ $\,\rtimes\,$ $S_3$	$(C_2\times C_4^3.S_4)$ $\,\rtimes\,$ $C_2$	all 6
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_4^3$ . $(C_4:S_4)$	$(C_4^3.A_4)$ . $D_4$	$C_8$ . $(C_8^2:D_6)$	$(C_4.C_8^2)$ . $D_{12}$	all 35

Elements of the group are displayed as matrices in $\GL_{2}(\Z/{32}\Z)$.

Homology

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

Subgroups

There are 49801 subgroups in 3257 conjugacy classes, 60 normal (34 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $C_8^2:D_{12}$
Commutator:	$G' \simeq$ $C_4^3.A_4$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_4^3$	$G/\Phi \simeq$ $C_2^2\times S_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^4.C_4^3$	$G/\operatorname{Fit} \simeq$ $S_3$
Radical:	$R \simeq$ $C_4^3.(C_2^2\times S_4)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^4$	$G/\operatorname{soc} \simeq$ $C_4^2:D_{12}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times C_8^2.D_8$
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

Order 6144: $C_4^3.(C_2^2\times S_4)$

Order 3072: $(C_4^2\times C_8).S_4$ x 4, $C_2\times C_4^3.S_4$ x 2, $C_4^3.(C_4\times A_4)$

Order 2048: $C_2\times C_8^2.D_8$

Order 1536: $C_4^3.S_4$ x 6, $C_2\times C_4^2.\GL(2,3)$ x 2, $(C_4^2\times C_8).A_4$ x 2, $C_4^3.D_{12}$, $C_2\times C_4^3.A_4$

Order 1024: $C_8^2.D_8$ x 8, $C_2\times C_4^3.D_4$ x 4, $C_4^3.C_2^4$, $C_2\times C_8^2.C_8$, $C_2^4.C_4^3$

Order 768: $C_4^2:D_{24}$ x 4, $C_4^2.\GL(2,3)$ x 4, $C_4^3:D_6$ x 2, $C_4^3.A_4$ x 2, $C_4^3.C_{12}$, $C_2\times C_4^2.\SL(2,3)$

Order 512: $C_4^3.D_4$ x 24, $C_8^2.D_4$ x 12, $C_2\times C_4^2.D_8$ x 8, $C_8^2.C_8$ x 6, $C_8^2:C_8$ x 4, $C_4^3.C_2^3$ x 2, $C_4^3.C_2^3$ x 2, $C_8^2.C_2^3$ x 2, $C_2\times C_4^3.C_4$ x 2, $(C_2\times C_8^2):C_2^2$ x 2, $C_4^3.D_4$ x 2, $C_2\times C_8^2.C_4$, $C_2\times C_4\times C_8^2$, $C_8^2.C_2^3$, $C_8^2.C_2^3$, $C_8^2.C_2^3$

Order 384: $C_4^2:D_{12}$ x 6, $C_4^2:C_{24}$ x 2, $C_2^4.S_4$ x 2, $C_4^3:C_6$, $C_4^2.\SL(2,3)$, $C_2^3:D_{24}$

Order 256: $C_4^2.D_8$ x 32, $C_8^2.C_2^2$ x 24, $C_4^3.C_2^2$ x 20, $C_4^2:D_8$ x 16, $(C_2\times C_8^2):C_2$ x 12, $C_8^2.C_2^2$ x 12, $C_4^3.C_4$ x 10, $C_4.C_8^2$ x 10, $C_4\times C_8^2$ x 8, $C_8:\OD_{32}$ x 6, $C_8:\OD_{32}$ x 6, $C_8^2.C_4$ x 6, $C_2^2.C_8^2$ x 5, $C_2^4.\OD_{16}$ x 4, $C_4^2.C_2^4$ x 4, $(C_2^2\times D_8):C_4$ x 4, $C_8^2:C_2^2$ x 4, $C_8^2:C_2^2$ x 4, $C_4^2.C_2^4$ x 4, $C_2^2\times C_8^2$ x 4, $C_2\times C_4^2\times C_8$ 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, $C_8^2.C_4$ x 2, $C_8^2.C_4$ x 2, $C_2\times C_4:\OD_{32}$ x 2, $C_4^3:C_2^2$ x 2, $C_2\times C_4^2.D_4$ x 2, $C_8^2:C_2^2$ x 2, $C_8^2:C_2^2$ x 2, $C_4^2.C_2^4$ x 2, $C_8^2.C_2^2$, $C_8^2:C_2^2$

Order 192: $C_4^2:D_6$ x 8, $C_8:S_4$ x 4, $C_4\wr C_3$ x 2, $C_2^3:D_{12}$ x 2, $C_2^4.A_4$, $C_2^3:C_{24}$

Order 128: $C_2\times C_8^2$ x 44, $C_4^2.C_2^3$ x 40, $C_8:D_8$ x 32, $C_8\times D_8$ x 32, $C_4^2.D_4$ x 24, $C_4^2.D_4$ x 24, $C_4^2.D_4$ x 24, $C_4^2.D_4$ x 20, $C_4^2.C_2^3$ x 20, $C_4^2\times C_8$ x 18, $C_8:D_8$ x 16, $C_2.C_8^2$ x 15, $C_4^3:C_2$ x 12, $C_8^2:C_2$ x 12, $C_8:\OD_{16}$ x 12, $C_2^2\times C_4\times C_8$ x 12, $\OD_{32}:C_4$ x 12, $C_4:\OD_{32}$ x 10, $C_4^2.D_4$ x 10, $C_8:C_{16}$ x 8, $C_4^2.D_4$ x 8, $C_8:C_{16}$ x 8, $C_8:Q_{16}$ x 6, $C_8:D_8$ x 6, $C_4^2.C_8$ x 4, $C_4^2.D_4$ x 4, $D_8:C_2^3$ x 4, $C_4^2.D_4$ x 4, $C_4^2.D_4$ x 4, $C_4^2.D_4$ x 4, $C_2^3:D_8$ x 4, $C_4^2.D_4$ x 4, $C_4^2.C_2^3$ x 4, $C_4^2.C_2^3$ x 4, $C_2^4.Q_8$ x 4, $C_4^2:C_2^3$ x 4, $C_8.C_2^4$ x 2, $C_4^2.C_2^3$ x 2, $C_2^2\times \OD_{32}$ x 2, $C_4^2.D_4$ x 2, $C_4^2.D_4$ x 2, $C_4^2.D_4$ x 2, $C_4^2.C_2^3$ x 2, $(C_2^3\times C_8):C_2$ x 2, $C_2\times C_4^3$

Order 96: $C_4:S_4$ x 6, $C_4^2:S_3$ x 4, $C_4^2:C_6$ x 3, $C_8\times A_4$ x 2, $C_2^2\times S_4$ x 2, $C_2^3:C_{12}$, $C_2\times D_{24}$

Order 64: $C_2\times C_4\times C_8$ x 110, $D_8:C_4$ x 64, $C_8^2$ x 56, $D_8:C_2^2$ x 40, $D_4:C_8$ x 32, $C_8:D_4$ x 32, $C_4\times D_8$ x 32, $C_4^2:C_2^2$ x 32, $C_8.C_2^3$ x 28, $C_8.D_4$ x 24, $C_4.D_8$ x 24, $C_8:D_4$ x 24, $C_8\times D_4$ x 24, $C_4:D_8$ x 22, $C_4^2.C_2^2$ x 20, $C_2^3.Q_8$ x 20, $C_8:C_8$ x 16, $C_2\times \OD_{32}$ x 14, $(C_2^2\times C_8):C_2$ x 12, $C_4:C_{16}$ x 12, $C_8:Q_8$ x 12, $C_8:D_4$ x 12, $C_2^3\times C_8$ x 11, $C_4:Q_{16}$ x 10, $C_4:\OD_{16}$ x 10, $C_2^2.D_8$ x 8, $C_2^2\times D_8$ x 8, $C_2^3:C_8$ x 4, $C_4^3$ x 4, $C_2^2\times Q_{16}$ x 4, $C_2^2\times \SD_{16}$ x 4, $C_2^2\times \OD_{16}$ x 4, $C_2^3:D_4$ x 4, $C_4^2:C_2^2$ x 4, $C_2^2.D_8$ x 4, $C_4^2.C_4$ x 4, $C_2^2\times C_4^2$ x 3, $D_4:C_2^3$ x 2, $C_4^2.C_2^2$ x 2, $C_4^2:C_2^2$ x 2, $C_4^2:C_2^2$ x 2, $C_2^2\times C_{16}$ x 2

Order 48: $C_2\times S_4$ x 8, $D_{24}$ x 4, $C_2\times D_{12}$ x 2, $C_4\times A_4$ x 2, $C_2^2\times A_4$, $C_4^2:C_3$, $C_2\times C_{24}$

Order 32: $C_4\times C_8$ x 124, $C_2^2\times C_8$ x 100, $D_8:C_2$ x 64, $C_2\times D_8$ x 52, $\OD_{16}:C_2$ x 48, $D_4:C_4$ x 32, $C_2\times \SD_{16}$ x 32, $C_4\wr C_2$ x 32, $C_2\times \OD_{16}$ x 30, $D_4:C_2^2$ x 28, $C_2\times C_4^2$ x 27, $C_4:D_4$ x 24, $C_4\times D_4$ x 24, $C_2\times Q_{16}$ x 20, $C_2^2:C_8$ x 16, $C_8.C_4$ x 16, $C_8:C_4$ x 16, $C_4^2:C_2$ x 12, $\OD_{32}$ x 12, $C_4:C_8$ x 12, $C_4:Q_8$ x 10, $C_4:D_4$ x 10, $C_2^2\times D_4$ x 8, $C_2\times C_{16}$ x 8, $C_2^3\times C_4$ x 5, $C_2^2.D_4$ x 4, $C_2^3:C_4$ x 4, $C_2^2\times Q_8$ x 2

Order 24: $D_{12}$ x 6, $S_4$ x 4, $C_2\times A_4$ x 3, $C_{24}$ x 2, $C_2\times D_6$ x 2, $C_2\times C_{12}$

Order 16: $C_2\times C_8$ x 155, $C_2\times D_4$ x 58, $D_8$ x 48, $D_4:C_2$ x 48, $C_2^2\times C_4$ x 48, $\SD_{16}$ x 32, $C_4^2$ x 31, $\OD_{16}$ x 28, $Q_{16}$ x 16, $C_2^2:C_4$ x 16, $C_2\times Q_8$ x 14, $C_4:C_4$ x 12, $C_{16}$ x 4, $C_2^4$ x 3

Order 12: $D_6$ x 8, $C_{12}$ x 2, $C_2\times C_6$, $A_4$

Order 8: $C_2\times C_4$ x 79, $D_4$ x 52, $C_8$ x 42, $C_2^3$ x 23, $Q_8$ x 12

Order 6: $S_3$ x 4, $C_6$ x 3

Order 4: $C_2^2$ x 32, $C_4$ x 22

Order 3: $C_3$

Order 2: $C_2$ x 11

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 6144: $C_4^3.(C_2^2\times S_4)$

Order 3072: $(C_4^2\times C_8).S_4$, $C_4^3.(C_4\times A_4)$, $C_2\times C_4^3.S_4$

Order 2048: $C_2\times C_8^2.D_8$

Order 1536: $C_4^3.S_4$ x 2, $C_4^3.D_{12}$, $C_2\times C_4^2.\GL(2,3)$, $C_2\times C_4^3.A_4$, $(C_4^2\times C_8).A_4$

Order 1024: $C_2\times C_4^3.D_4$ x 2, $C_8^2.D_8$ x 2, $C_4^3.C_2^4$, $C_2\times C_8^2.C_8$, $C_2^4.C_4^3$

Order 768: $C_4^3.A_4$ x 2, $C_4^3:D_6$, $C_4^2:D_{24}$, $C_4^2.\GL(2,3)$, $C_4^3.C_{12}$, $C_2\times C_4^2.\SL(2,3)$

Order 512: $C_4^3.D_4$ x 6, $C_8^2.D_4$ x 4, $C_2\times C_4^2.D_8$ x 3, $C_4^3.C_2^3$ x 2, $C_8^2.C_2^3$ x 2, $C_8^2.C_8$ x 2, $C_8^2:C_8$ x 2, $C_4^3.D_4$ x 2, $C_2\times C_8^2.C_4$, $C_2\times C_4\times C_8^2$, $C_4^3.C_2^3$, $C_8^2.C_2^3$, $C_2\times C_4^3.C_4$, $C_8^2.C_2^3$, $C_8^2.C_2^3$, $(C_2\times C_8^2):C_2^2$

Order 384: $C_4^2:D_{12}$ x 2, $C_4^2:C_{24}$, $C_4^3:C_6$, $C_4^2.\SL(2,3)$, $C_2^3:D_{24}$, $C_2^4.S_4$

Order 256: $C_4^3.C_2^2$ x 8, $C_4.C_8^2$ x 8, $C_8^2.C_2^2$ x 8, $C_4^2.D_8$ x 5, $C_2^2.C_8^2$ x 4, $C_4\times C_8^2$ x 4, $C_4^3.C_4$ x 4, $C_4^2:D_8$ x 4, $C_8^2.C_2^2$ x 4, $C_2^2\times C_8^2$ x 4, $C_2\times C_4^2\times C_8$ x 3, $C_8^2:C_2^2$ x 3, $C_8^2:C_2^2$ x 3, $C_2^4.\OD_{16}$ x 2, $C_4^2.C_2^4$ x 2, $C_4^2.C_2^4$ x 2, $C_8:\OD_{32}$ x 2, $C_8^2.C_4$ x 2, $C_8:\OD_{32}$ x 2, $(C_2\times C_8^2):C_2$ x 2, $C_8^2.C_4$ x 2, $(C_2^2\times D_8):C_4$ x 2, $C_8^2:C_2^2$ x 2, $C_4^2.C_2^4$ x 2, $C_4^2.C_2^4$ x 2, $C_4^2.C_2^4$, $C_4^2.C_2^4$, $C_8^2.C_4$, $C_2\times C_4:\OD_{32}$, $C_4^3:C_2^2$, $C_2\times C_4^2.D_4$, $C_8^2.C_2^2$, $C_8^2:C_2^2$, $C_8^2:C_2^2$

Order 192: $C_4^2:D_6$ x 2, $C_4\wr C_3$ x 2, $C_2^4.A_4$, $C_8:S_4$, $C_2^3:D_{12}$, $C_2^3:C_{24}$

Order 128: $C_2\times C_8^2$ x 28, $C_4^2\times C_8$ x 13, $C_4^2.C_2^3$ x 11, $C_2^2\times C_4\times C_8$ x 11, $C_2.C_8^2$ x 9, $C_4^2.D_4$ x 8, $C_4^2.D_4$ x 8, $C_4^2.D_4$ x 8, $C_4^2.C_2^3$ x 8, $C_8:D_8$ x 6, $C_8\times D_8$ x 6, $C_4:\OD_{32}$ x 4, $C_4^3:C_2$ x 4, $C_4^2.D_4$ x 4, $C_8:D_8$ x 4, $C_8:\OD_{16}$ x 4, $C_4^2.D_4$ x 4, $\OD_{32}:C_4$ x 4, $C_4^2.D_4$ x 3, $C_4^2.D_4$ x 3, $C_4^2.D_4$ x 3, $C_2^3:D_8$ x 3, $C_4^2.D_4$ x 3, $C_4^2.C_8$ x 2, $C_8:Q_{16}$ x 2, $C_8:D_8$ x 2, $C_8^2:C_2$ x 2, $D_8:C_2^3$ x 2, $C_4^2.D_4$ x 2, $C_4^2.D_4$ x 2, $C_4^2.D_4$ x 2, $C_4^2.C_2^3$ x 2, $C_4^2.C_2^3$ x 2, $C_2^4.Q_8$ x 2, $C_4^2:C_2^3$ x 2, $C_8:C_{16}$ x 2, $C_2\times C_4^3$, $C_8:C_{16}$, $C_8.C_2^4$, $C_4^2.C_2^3$, $C_2^2\times \OD_{32}$, $C_4^2.D_4$, $C_4^2.D_4$, $C_4^2.C_2^3$, $(C_2^3\times C_8):C_2$

Order 96: $C_4^2:C_6$ x 2, $C_4:S_4$ x 2, $C_8\times A_4$, $C_4^2:S_3$, $C_2^2\times S_4$, $C_2^3:C_{12}$, $C_2\times D_{24}$

Order 64: $C_2\times C_4\times C_8$ x 77, $C_8^2$ x 24, $D_8:C_2^2$ x 12, $D_8:C_4$ x 10, $C_2^3\times C_8$ x 9, $C_4^2.C_2^2$ x 8, $C_8.D_4$ x 8, $C_4:D_8$ x 8, $C_8:D_4$ x 8, $C_8\times D_4$ x 8, $C_4^2:C_2^2$ x 7, $C_8.C_2^3$ x 6, $C_8:D_4$ x 6, $C_4\times D_8$ x 6, $C_2^3.Q_8$ x 6, $C_2^2\times D_8$ x 5, $(C_2^2\times C_8):C_2$ x 4, $C_4^3$ x 4, $C_4:C_{16}$ x 4, $C_2\times \OD_{32}$ x 4, $C_8:Q_8$ x 4, $C_4:Q_{16}$ x 4, $C_8:D_4$ x 4, $C_4.D_8$ x 4, $C_4:\OD_{16}$ x 4, $C_2^2.D_8$ x 3, $D_4:C_8$ x 3, $C_2^2\times C_4^2$ x 3, $C_8:C_8$ x 3, $C_2^2.D_8$ x 3, $C_2^3:C_8$ x 2, $C_2^2\times Q_{16}$ x 2, $C_2^2\times \SD_{16}$ x 2, $C_2^2\times \OD_{16}$ x 2, $C_2^3:D_4$ x 2, $C_4^2:C_2^2$ x 2, $C_4^2.C_4$ x 2, $D_4:C_2^3$, $C_4^2.C_2^2$, $C_4^2:C_2^2$, $C_4^2:C_2^2$, $C_2^2\times C_{16}$

Order 48: $C_2\times S_4$ x 2, $C_4\times A_4$ x 2, $D_{24}$, $C_2^2\times A_4$, $C_2\times D_{12}$, $C_4^2:C_3$, $C_2\times C_{24}$

Order 32: $C_4\times C_8$ x 77, $C_2^2\times C_8$ x 56, $C_2\times C_4^2$ x 21, $C_2\times D_8$ x 15, $D_8:C_2$ x 12, $D_4:C_2^2$ x 8, $C_4:D_4$ x 8, $C_4\times D_4$ x 8, $C_2\times \OD_{16}$ x 7, $C_2\times Q_{16}$ x 6, $C_2\times \SD_{16}$ x 6, $\OD_{16}:C_2$ x 6, $C_4\wr C_2$ x 5, $C_2^2:C_8$ x 4, $C_2^2\times D_4$ x 4, $C_2^3\times C_4$ x 4, $C_4:Q_8$ x 4, $C_4:D_4$ x 4, $C_4^2:C_2$ x 4, $C_8.C_4$ x 4, $C_4:C_8$ x 4, $D_4:C_4$ x 3, $\OD_{32}$ x 3, $C_8:C_4$ x 3, $C_2^2.D_4$ x 2, $C_2^3:C_4$ x 2, $C_2\times C_{16}$ x 2, $C_2^2\times Q_8$

Order 24: $D_{12}$ x 2, $C_2\times A_4$ x 2, $C_2\times C_{12}$, $C_{24}$, $C_2\times D_6$, $S_4$

Order 16: $C_2\times C_8$ x 78, $C_2^2\times C_4$ x 28, $C_4^2$ x 25, $C_2\times D_4$ x 17, $D_4:C_2$ x 10, $D_8$ x 8, $\OD_{16}$ x 5, $Q_{16}$ x 4, $\SD_{16}$ x 4, $C_4:C_4$ x 4, $C_2^2:C_4$ x 4, $C_2\times Q_8$ x 4, $C_2^4$ x 2, $C_{16}$

Order 12: $D_6$ x 2, $C_{12}$ x 2, $C_2\times C_6$, $A_4$

Order 8: $C_2\times C_4$ x 42, $C_8$ x 25, $D_4$ x 12, $C_2^3$ x 10, $Q_8$ x 3

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

Order 4: $C_4$ x 15, $C_2^2$ x 14

Order 3: $C_3$

Order 2: $C_2$ x 6

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 6144: $C_4^3.(C_2^2\times S_4)$ ($C_1$)

Order 3072: $(C_4^2\times C_8).S_4$ ($C_2$) x 4, $C_2\times C_4^3.S_4$ ($C_2$) x 2, $C_4^3.(C_4\times A_4)$ ($C_2$)

Order 1536: $C_4^3.S_4$ ($C_2^2$) x 4, $(C_4^2\times C_8).A_4$ ($C_2^2$) x 2, $C_2\times C_4^3.A_4$ ($C_2^2$)

Order 1024: $C_2^4.C_4^3$ ($S_3$)

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

Order 512: $C_8^2:C_8$ ($D_6$) x 2, $C_4^3.C_2^3$ ($D_6$)

Order 384: $C_4^2.\SL(2,3)$ ($C_2\times D_4$)

Order 256: $C_2^2.C_8^2$ ($D_{12}$), $C_2\times C_4^2\times C_8$ ($S_4$), $C_4.C_8^2$ ($D_{12}$), $C_4.C_8^2$ ($C_2\times D_6$)

Order 128: $C_4^2\times C_8$ ($C_2\times S_4$) x 2, $C_2\times C_4^3$ ($C_2\times S_4$), $C_2.C_8^2$ ($C_2\times D_{12}$)

Order 64: $C_4^3$ ($C_2^2\times S_4$), $C_4^3$ ($C_4:S_4$), $C_2^3\times C_8$ ($C_4^2:S_3$), $C_2^2\times C_4^2$ ($C_4:S_4$)

Order 32: $C_2^2\times C_8$ ($C_4^2:D_6$) x 2, $C_2\times C_4^2$ ($C_8.S_4$) x 2, $C_2^3\times C_4$ ($C_4^2:D_6$), $C_2\times C_4^2$ ($C_2^3:D_{12}$)

Order 16: $C_2\times C_8$ ($C_8^2:S_3$), $C_4^2$ ($C_2\times C_8.S_4$), $C_2^4$ ($C_4^2:D_{12}$), $C_2^2\times C_4$ ($C_4^2:D_{12}$), $C_2^2\times C_4$ ($C_2^4.S_4$)

Order 8: $C_2^3$ ($C_2^3.C_4:S_4$) x 2, $C_8$ ($C_8^2:D_6$) x 2, $C_2^3$ ($C_4^3:D_6$), $C_2\times C_4$ ($C_8^2:D_6$)

Order 4: $C_2^2$ ($C_2\times C_2^3.C_4:S_4$), $C_2^2$ ($C_8^2:D_{12}$), $C_4$ ($C_2^2\times C_8^2.S_3$), $C_4$ ($C_8^2:D_{12}$)

Order 2: $C_2$ ($(C_4^2\times C_8).S_4$) x 2, $C_2$ ($C_8^2.D_{12}.C_2$)

Order 1: $C_1$ ($C_4^3.(C_2^2\times S_4)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 6144: $C_4^3.(C_2^2\times S_4)$ ($C_1$)

Order 3072: $(C_4^2\times C_8).S_4$ ($C_2$), $C_4^3.(C_4\times A_4)$ ($C_2$), $C_2\times C_4^3.S_4$ ($C_2$)

Order 1536: $C_4^3.S_4$ ($C_2^2$), $C_2\times C_4^3.A_4$ ($C_2^2$), $(C_4^2\times C_8).A_4$ ($C_2^2$)

Order 1024: $C_2^4.C_4^3$ ($S_3$)

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

Order 512: $C_4^3.C_2^3$ ($D_6$), $C_8^2:C_8$ ($D_6$)

Order 384: $C_4^2.\SL(2,3)$ ($C_2\times D_4$)

Order 256: $C_2^2.C_8^2$ ($D_{12}$), $C_2\times C_4^2\times C_8$ ($S_4$), $C_4.C_8^2$ ($D_{12}$), $C_4.C_8^2$ ($C_2\times D_6$)

Order 128: $C_2\times C_4^3$ ($C_2\times S_4$), $C_2.C_8^2$ ($C_2\times D_{12}$), $C_4^2\times C_8$ ($C_2\times S_4$)

Order 64: $C_4^3$ ($C_2^2\times S_4$), $C_4^3$ ($C_4:S_4$), $C_2^3\times C_8$ ($C_4^2:S_3$), $C_2^2\times C_4^2$ ($C_4:S_4$)

Order 32: $C_2^3\times C_4$ ($C_4^2:D_6$), $C_2^2\times C_8$ ($C_4^2:D_6$), $C_2\times C_4^2$ ($C_8.S_4$), $C_2\times C_4^2$ ($C_2^3:D_{12}$)

Order 16: $C_2\times C_8$ ($C_8^2:S_3$), $C_4^2$ ($C_2\times C_8.S_4$), $C_2^4$ ($C_4^2:D_{12}$), $C_2^2\times C_4$ ($C_4^2:D_{12}$), $C_2^2\times C_4$ ($C_2^4.S_4$)

Order 8: $C_2^3$ ($C_4^3:D_6$), $C_2^3$ ($C_2^3.C_4:S_4$), $C_2\times C_4$ ($C_8^2:D_6$), $C_8$ ($C_8^2:D_6$)

Order 4: $C_2^2$ ($C_2\times C_2^3.C_4:S_4$), $C_2^2$ ($C_8^2:D_{12}$), $C_4$ ($C_2^2\times C_8^2.S_3$), $C_4$ ($C_8^2:D_{12}$)

Order 2: $C_2$ ($C_8^2.D_{12}.C_2$), $C_2$ ($(C_4^2\times C_8).S_4$)

Order 1: $C_1$ ($C_4^3.(C_2^2\times S_4)$)

Series

Derived series

$C_4^3.(C_2^2\times S_4)$

$\rhd$

$C_4^3.(C_2^2\times S_4)$

$\rhd$

$C_4^3.A_4$

$\rhd$

$C_4^3.A_4$

$\rhd$

$C_2.C_8^2$

$\rhd$

$C_2.C_8^2$

$\rhd$

$C_2$

$\rhd$

$C_2$

$\rhd$

$C_1$

$\rhd$

$C_1$

Chief series

$C_4^3.(C_2^2\times S_4)$

$\rhd$

$C_4^3.(C_2^2\times S_4)$

$\rhd$

$(C_4^2\times C_8).S_4$

$\rhd$

$(C_4^2\times C_8).S_4$

$\rhd$

$C_4^3.S_4$

$\rhd$

$C_4^3.S_4$

$\rhd$

$C_4^3.A_4$

$\rhd$

$C_4^3.A_4$

$\rhd$

$C_4^2.\SL(2,3)$

$\rhd$

$C_4^2.\SL(2,3)$

$\rhd$

$C_2.C_8^2$

$\rhd$

$C_2.C_8^2$

$\rhd$

$C_2\times C_4^2$

$\rhd$

$C_2\times C_4^2$

$\rhd$

$C_4^2$

$\rhd$

$C_4^2$

$\rhd$

$C_2^3$

$\rhd$

$C_2^2$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$C_4^3.(C_2^2\times S_4)$

$\rhd$

$C_4^3.(C_2^2\times S_4)$

$\rhd$

$C_4^3.A_4$

$\rhd$

$C_4^3.A_4$

$\rhd$

$C_4^2.\SL(2,3)$

$\rhd$

$C_4^2.\SL(2,3)$

Upper central series

$C_1$

$\lhd$

$C_1$

$\lhd$

$C_2^2$

$\lhd$

$C_2^2$

$\lhd$

$C_2\times C_4$

$\lhd$

$C_2\times C_4$

$\lhd$

$C_2\times C_8$

$\lhd$

$C_2\times C_8$

Supergroups

This group is a maximal subgroup of 3 larger groups in the database.

This group is a maximal quotient of 0 larger groups in the database.

Character theory

Complex character table

See the $176 \times 176$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $88 \times 88$ rational character table.