Abstract group 3072.cc: $(C_4\times C_8).\GL(2,\mathbb{Z}/4)$

• Label: 3072.cc
• Order: \( 2^{10} \cdot 3 \)
• Exponent: \( 2^{4} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \)
• $\card{Z(G)}$: \( 2^{4} \)
• $\card{\Aut(G)}$: \( 2^{15} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{9} \)
• Perm deg.: $40$
• Trans deg.: $192$
• Rank: $2$

Group information

Description:	$(C_4\times C_8).\GL(2,\mathbb{Z}/4)$
Order:	\(3072\)\(\medspace = 2^{10} \cdot 3 \)
Exponent:	\(48\)\(\medspace = 2^{4} \cdot 3 \)
Automorphism group:	$(C_4\times A_4).C_2^5.C_2^6$, of order \(98304\)\(\medspace = 2^{15} \cdot 3 \)
Composition factors:	$C_2$ x 10, $C_3$
Derived length:	$3$

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	48
Elements	1	31	8	96	56	128	192	1792	256	512	3072
Conjugacy classes	1	11	1	28	7	40	24	96	32	64	304
Divisions	1	11	1	16	5	16	8	16	8	4	86
Autjugacy classes	1	9	1	10	4	12	5	6	6	2	56

Dimension	1	2	3	4	6	8	12	16	24	32	48
Irr. complex chars.	32	184	32	0	56	0	0	0	0	0	0	304
Irr. rational chars.	4	14	4	20	10	8	10	6	2	4	4	86

Minimal presentations

Permutation degree:	$40$
Transitive degree:	$192$
Rank:	$2$
Inequivalent generating pairs:	$18$

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:	${\langle a, b, c, d, e \mid b^{48}=c^{2}=d^{2}=e^{2}=[a,c]=[a,d]=[b,d]=[c,d]= \!\cdots\! \rangle}$
Permutation group:	Degree $40$ $\langle(1,6)(2,11)(4,5)(7,19)(9,10)(12,27)(20,21)(28,29)(33,35)(34,36), (1,5,7,21,6,4,19,20) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 17 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 3 & 0 \\ 0 & 3 \end{array}\right), \left(\begin{array}{rr} 14 & 31 \\ 1 & 2 \end{array}\right), \left(\begin{array}{rr} 31 & 6 \\ 18 & 13 \end{array}\right), \left(\begin{array}{rr} 9 & 0 \\ 0 & 9 \end{array}\right), \left(\begin{array}{rr} 28 & 11 \\ 25 & 3 \end{array}\right), \left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 17 & 0 \\ 0 & 17 \end{array}\right), \left(\begin{array}{rr} 1 & 16 \\ 16 & 17 \end{array}\right), \left(\begin{array}{rr} 13 & 24 \\ 24 & 21 \end{array}\right), \left(\begin{array}{rr} 1 & 24 \\ 8 & 25 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/32\Z)$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$A_4$ $\,\rtimes\,$ $(C_8.(C_4\times C_8))$	$C_2^2$ $\,\rtimes\,$ $(C_{24}.(C_4\times C_8))$			more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_8$ . $(C_4^2.S_4)$	$C_8$ . $(C_4^2.S_4)$	$(C_2^4:C_{24})$ . $D_4$	$(C_2^3:C_{24})$ . $D_8$	all 116

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

Homology

Abelianization:	$C_{4} \times C_{8} $
Schur multiplier:	$C_{2}^{3}$
Commutator length:	$1$

Subgroups

There are 6050 subgroups in 1253 conjugacy classes, 175 normal (109 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_8$	$G/Z \simeq$ $C_8:S_4$
Commutator:	$G' \simeq$ $C_8\times A_4$	$G/G' \simeq$ $C_4\times C_8$
Frattini:	$\Phi \simeq$ $C_2\times C_4\times C_8$	$G/\Phi \simeq$ $C_2\times S_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^3\times C_4\times C_{16}$	$G/\operatorname{Fit} \simeq$ $S_3$
Radical:	$R \simeq$ $(C_4\times C_8).\GL(2,\mathbb{Z}/4)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^4$	$G/\operatorname{soc} \simeq$ $C_{24}:C_8$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $(C_2^3\times C_4\times C_{16}).C_2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available

Classes of subgroups up to automorphism

No diagram available

Normal subgroups

No diagram available

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

Order 3072: $(C_4\times C_8).\GL(2,\mathbb{Z}/4)$

Order 1536: $C_2\times A_4.\OD_{32}.C_2$ x 2, $C_2\times C_2^3.(C_4\times C_{24})$

Order 1024: $(C_2^3\times C_4\times C_{16}).C_2$

Order 768: $(C_4\times C_8).S_4$ x 6, $C_2\times A_4:\OD_{32}$ x 2, $C_2^4:C_{48}$ x 2, $C_4\times A_4\times C_{16}$ x 2, $C_{24}.(C_4\times C_8)$, $C_4\times C_2^3:C_{24}$

Order 512: $C_2\times (C_2\times C_4\times C_{16}).C_2$ x 4, $C_2\times (C_2^3\times C_8).C_2^2$ x 2, $C_2^3\times C_4\times C_{16}$

Order 384: $C_2^3:C_{48}$ x 8, $A_4:\OD_{32}$ x 6, $C_4\times C_8\times A_4$ x 4, $C_2\times A_4:C_{16}$ x 2, $C_2\times C_{24}.C_8$ x 2, $C_2^4:C_{24}$ x 2, $C_2\times C_4\times C_{48}$, $C_2\times A_4\times C_4^2$

Order 256: $C_4^2.\OD_{16}$ x 12, $C_2^2\times C_4\times C_{16}$ x 8, $C_8.(C_4\times C_8)$ x 8, $C_2^4.\OD_{16}$ x 4, $C_2^4\times C_{16}$ x 2, $C_2^4.\OD_{16}$ x 2, $C_2^3:\OD_{32}$ x 2, $C_2^3\times C_4\times C_8$

Order 192: $C_2^3:C_{24}$ x 12, $C_{24}.C_8$ x 6, $A_4\times C_{16}$ x 4, $A_4:C_{16}$ x 4, $A_4\times C_4^2$ x 3, $C_2^2\times C_{48}$ x 2, $C_6:\OD_{32}$ x 2, $C_4\times C_{48}$ x 2, $C_2^4:C_{12}$ x 2, $C_2\times C_4\times C_{24}$

Order 128: $C_2^3.\OD_{16}$ x 28, $C_2\times C_4\times C_{16}$ x 23, $C_2^3\times C_{16}$ x 20, $C_2^2:\OD_{32}$ x 12, $C_2^2\times C_4\times C_8$ x 12, $\OD_{32}:C_4$ x 8, $C_2^3:C_{16}$ x 4, $C_2^2\times \OD_{32}$ x 4, $C_2^4\times C_8$ x 2, $C_2^3\times C_4^2$

Order 96: $C_2^3:C_{12}$ x 12, $C_8\times A_4$ x 8, $C_2\times C_{48}$ x 8, $C_3:\OD_{32}$ x 6, $C_4\times C_{24}$ x 4, $C_6:C_{16}$ x 2, $C_2^2\times C_{24}$ x 2, $C_2^3\times A_4$, $C_2\times C_4\times C_{12}$

Order 64: $C_2^2\times C_{16}$ x 66, $C_2\times C_4\times C_8$ x 31, $C_2^3\times C_8$ x 28, $C_2\times \OD_{32}$ x 28, $C_8.C_8$ x 24, $C_2^2:C_{16}$ x 16, $C_4\times C_{16}$ x 12, $C_2^2\times C_4^2$ x 9, $C_2^4\times C_4$ x 2

Order 48: $C_2\times C_{24}$ x 12, $C_4\times A_4$ x 8, $C_2^2\times A_4$ x 5, $C_{48}$ x 4, $C_3:C_{16}$ x 4, $C_4\times C_{12}$ x 3, $C_2^2\times C_{12}$ x 2

Order 32: $C_2^2\times C_8$ x 86, $C_2\times C_{16}$ x 68, $C_2^3\times C_4$ x 28, $C_2\times C_4^2$ x 25, $\OD_{32}$ x 24, $C_4\times C_8$ x 20, $C_2^5$

Order 24: $C_2\times C_{12}$ x 12, $C_{24}$ x 8, $C_2\times A_4$ x 5, $C_2^2\times C_6$

Order 16: $C_2^2\times C_4$ x 93, $C_2\times C_8$ x 76, $C_4^2$ x 16, $C_{16}$ x 16, $C_2^4$ x 11

Order 12: $C_{12}$ x 8, $C_2\times C_6$ x 5, $A_4$

Order 8: $C_2\times C_4$ x 82, $C_2^3$ x 38, $C_8$ x 16

Order 6: $C_6$ x 5

Order 4: $C_2^2$ x 38, $C_4$ x 16

Order 3: $C_3$

Order 2: $C_2$ x 11

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 3072: $(C_4\times C_8).\GL(2,\mathbb{Z}/4)$

Order 1536: $C_2\times C_2^3.(C_4\times C_{24})$, $C_2\times A_4.\OD_{32}.C_2$

Order 1024: $(C_2^3\times C_4\times C_{16}).C_2$

Order 768: $(C_4\times C_8).S_4$ x 2, $C_2^4:C_{48}$ x 2, $C_{24}.(C_4\times C_8)$, $C_2\times A_4:\OD_{32}$, $C_4\times C_2^3:C_{24}$, $C_4\times A_4\times C_{16}$

Order 512: $C_2\times (C_2\times C_4\times C_{16}).C_2$ x 3, $C_2\times (C_2^3\times C_8).C_2^2$, $C_2^3\times C_4\times C_{16}$

Order 384: $C_2^3:C_{48}$ x 6, $C_4\times C_8\times A_4$ x 2, $A_4:\OD_{32}$ x 2, $C_2^4:C_{24}$ x 2, $C_2\times A_4:C_{16}$, $C_2\times C_4\times C_{48}$, $C_2\times C_{24}.C_8$, $C_2\times A_4\times C_4^2$

Order 256: $C_2^2\times C_4\times C_{16}$ x 6, $C_8.(C_4\times C_8)$ x 5, $C_4^2.\OD_{16}$ x 4, $C_2^4\times C_{16}$ x 2, $C_2^4.\OD_{16}$ x 2, $C_2^4.\OD_{16}$, $C_2^3\times C_4\times C_8$, $C_2^3:\OD_{32}$

Order 192: $C_2^3:C_{24}$ x 10, $A_4\times C_4^2$ x 2, $C_2^2\times C_{48}$ x 2, $C_{24}.C_8$ x 2, $A_4\times C_{16}$ x 2, $C_2^4:C_{12}$ x 2, $C_2\times C_4\times C_{24}$, $C_6:\OD_{32}$, $A_4:C_{16}$, $C_4\times C_{48}$

Order 128: $C_2^3\times C_{16}$ x 16, $C_2\times C_4\times C_{16}$ x 13, $C_2^3.\OD_{16}$ x 12, $C_2^2\times C_4\times C_8$ x 8, $C_2^2:\OD_{32}$ x 4, $C_2^3:C_{16}$ x 2, $C_2^4\times C_8$ x 2, $C_2^2\times \OD_{32}$ x 2, $\OD_{32}:C_4$ x 2, $C_2^3\times C_4^2$

Order 96: $C_2^3:C_{12}$ x 8, $C_8\times A_4$ x 6, $C_2\times C_{48}$ x 6, $C_4\times C_{24}$ x 2, $C_3:\OD_{32}$ x 2, $C_2^2\times C_{24}$ x 2, $C_2^3\times A_4$, $C_6:C_{16}$, $C_2\times C_4\times C_{12}$

Order 64: $C_2^2\times C_{16}$ x 40, $C_2^3\times C_8$ x 24, $C_2\times C_4\times C_8$ x 19, $C_2\times \OD_{32}$ x 12, $C_8.C_8$ x 8, $C_2^2\times C_4^2$ x 7, $C_4\times C_{16}$ x 5, $C_2^2:C_{16}$ x 4, $C_2^4\times C_4$ x 2

Order 48: $C_2\times C_{24}$ x 10, $C_4\times A_4$ x 5, $C_2^2\times A_4$ x 4, $C_2^2\times C_{12}$ x 2, $C_4\times C_{12}$ x 2, $C_{48}$ x 2, $C_3:C_{16}$

Order 32: $C_2^2\times C_8$ x 70, $C_2\times C_{16}$ x 34, $C_2^3\times C_4$ x 20, $C_2\times C_4^2$ x 15, $C_4\times C_8$ x 10, $\OD_{32}$ x 8, $C_2^5$

Order 24: $C_2\times C_{12}$ x 8, $C_{24}$ x 6, $C_2\times A_4$ x 4, $C_2^2\times C_6$

Order 16: $C_2\times C_8$ x 60, $C_2^2\times C_4$ x 57, $C_2^4$ x 9, $C_4^2$ x 8, $C_{16}$ x 6

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

Order 8: $C_2\times C_4$ x 48, $C_2^3$ x 26, $C_8$ x 12

Order 6: $C_6$ x 4

Order 4: $C_2^2$ x 26, $C_4$ x 10

Order 3: $C_3$

Order 2: $C_2$ x 9

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 3072: $(C_4\times C_8).\GL(2,\mathbb{Z}/4)$ ($C_1$)

Order 1536: $C_2\times A_4.\OD_{32}.C_2$ ($C_2$) x 2, $C_2\times C_2^3.(C_4\times C_{24})$ ($C_2$)

Order 768: $(C_4\times C_8).S_4$ ($C_4$) x 4, $C_2^4:C_{48}$ ($C_4$) x 2, $C_4\times C_2^3:C_{24}$ ($C_2^2$)

Order 512: $C_2^3\times C_4\times C_{16}$ ($S_3$)

Order 384: $C_2^3:C_{48}$ ($C_8$) x 4, $C_4\times C_8\times A_4$ ($D_4$) x 2, $C_4\times C_8\times A_4$ ($C_2\times C_4$) x 2, $C_2^4:C_{24}$ ($D_4$), $C_2^4:C_{24}$ ($C_2\times C_4$), $C_2\times A_4\times C_4^2$ ($Q_8$)

Order 256: $C_2^4\times C_{16}$ ($C_3:C_4$) x 2, $C_2^3\times C_4\times C_8$ ($D_6$)

Order 192: $C_2^3:C_{24}$ ($C_2^2:C_4$) x 3, $A_4\times C_4^2$ ($C_4:C_4$) x 2, $C_2^3:C_{24}$ ($\SD_{16}$) x 2, $C_2^3:C_{24}$ ($\OD_{16}$) x 2, $C_2^3:C_{24}$ ($C_2\times C_8$) x 2, $C_2^4:C_{12}$ ($C_4:C_4$), $C_2^3:C_{24}$ ($Q_{16}$), $C_2^3:C_{24}$ ($D_8$), $C_2^3:C_{24}$ ($C_4^2$)

Order 128: $C_2^3\times C_{16}$ ($C_3:C_8$) x 4, $C_2^2\times C_4\times C_8$ ($C_3:D_4$) x 2, $C_2^2\times C_4\times C_8$ ($C_4\times S_3$) x 2, $C_2\times C_4\times C_{16}$ ($S_4$), $C_2^4\times C_8$ ($C_6:C_4$), $C_2^4\times C_8$ ($D_{12}$), $C_2^3\times C_4^2$ ($C_3:Q_8$)

Order 96: $C_8\times A_4$ ($D_4:C_4$) x 2, $C_8\times A_4$ ($C_2^2:C_8$) x 2, $C_8\times A_4$ ($Q_8:C_4$) x 2, $C_2^3:C_{12}$ ($C_4:C_8$) x 2, $C_8\times A_4$ ($C_8:C_4$), $C_8\times A_4$ ($C_4\times C_8$), $C_2^3\times A_4$ ($C_8.C_4$), $C_2^3:C_{12}$ ($C_2.C_4^2$), $C_2^3:C_{12}$ ($C_8.C_4$), $C_2^3:C_{12}$ ($C_8:C_4$), $C_2^3:C_{12}$ ($C_8:C_4$)

Order 64: $C_2^3\times C_8$ ($C_6:C_8$) x 2, $C_2^3\times C_8$ ($C_{24}:C_2$) x 2, $C_2^3\times C_8$ ($D_6:C_4$) x 2, $C_2^3\times C_8$ ($C_3:\OD_{16}$) x 2, $C_2^2\times C_4^2$ ($C_6.D_4$) x 2, $C_2^2\times C_{16}$ ($A_4:C_4$) x 2, $C_2\times C_4\times C_8$ ($C_2\times S_4$), $C_2^4\times C_4$ ($C_{12}:C_4$), $C_2^3\times C_8$ ($C_3:Q_{16}$), $C_2^3\times C_8$ ($D_{24}$), $C_2^3\times C_8$ ($C_6.D_4$), $C_2^3\times C_8$ ($C_{12}:C_4$)

Order 48: $C_2^2\times A_4$ ($C_4.C_4^2$), $C_2^2\times A_4$ ($C_8:C_8$), $C_2^2\times A_4$ ($C_8:C_8$), $C_4\times A_4$ ($C_4.C_4^2$), $C_4\times A_4$ ($C_2^2.C_4^2$), $C_4\times A_4$ ($C_8:C_8$), $C_4\times A_4$ ($C_8:C_8$)

Order 32: $C_2\times C_{16}$ ($A_4:C_8$) x 4, $C_2^3\times C_4$ ($C_{12}:C_8$) x 2, $C_2^2\times C_8$ ($C_{12}.D_4$) x 2, $C_2^2\times C_8$ ($D_{12}:C_4$) x 2, $C_2^2\times C_8$ ($C_6.Q_{16}$) x 2, $C_4\times C_8$ ($\GL(2,\mathbb{Z}/4)$) x 2, $C_4\times C_8$ ($C_4\times S_4$) x 2, $C_2^5$ ($C_{24}.C_4$), $C_2^3\times C_4$ ($C_6.C_4^2$), $C_2^3\times C_4$ ($C_{24}.C_4$), $C_2^3\times C_4$ ($C_{24}:C_4$), $C_2^3\times C_4$ ($C_{24}:C_4$), $C_2^2\times C_8$ ($C_{12}:C_8$), $C_2^2\times C_8$ ($C_2^2.S_4$), $C_2^2\times C_8$ ($C_4:S_4$), $C_2^2\times C_8$ ($C_6.C_4^2$), $C_2\times C_4^2$ ($A_4:Q_8$)

Order 24: $C_2\times A_4$ ($C_{16}.C_8$) x 2, $C_2\times A_4$ ($C_4^2.D_4$)

Order 16: $C_2\times C_8$ ($C_2.\GL(2,\mathbb{Z}/4)$) x 2, $C_2\times C_8$ ($A_4:\OD_{16}$) x 2, $C_2\times C_8$ ($C_2\times A_4:C_8$) x 2, $C_2\times C_8$ ($C_8:S_4$) x 2, $C_4^2$ ($C_2.\GL(2,\mathbb{Z}/4)$) x 2, $C_2\times C_8$ ($C_2.\GL(2,\mathbb{Z}/4)$), $C_2\times C_8$ ($A_4:C_4^2$), $C_2\times C_8$ ($C_8:S_4$), $C_2\times C_8$ ($A_4:Q_{16}$), $C_2^4$ ($C_{24}:C_8$), $C_2^4$ ($C_{24}:C_8$), $C_2^4$ ($C_{12}.C_4^2$), $C_2^2\times C_4$ ($C_2^3.D_{12}$), $C_2^2\times C_4$ ($(C_2\times C_{12}):C_8$), $C_2^2\times C_4$ ($C_{24}:C_8$), $C_2^2\times C_4$ ($C_{24}:C_8$), $C_2^2\times C_4$ ($C_{12}.C_4^2$)

Order 12: $A_4$ ($C_8.(C_4\times C_8)$)

Order 8: $C_2^3$ ($C_{48}.C_8$) x 2, $C_2\times C_4$ ($C_4^2.S_4$) x 2, $C_8$ ($C_4.\GL(2,\mathbb{Z}/4)$) x 2, $C_8$ ($C_4.\GL(2,\mathbb{Z}/4)$) x 2, $C_8$ ($C_4.\GL(2,\mathbb{Z}/4)$) x 2, $C_2^3$ ($(C_2\times C_8).S_4$), $C_2^3$ ($(C_2\times C_{24}):C_8$), $C_2\times C_4$ ($C_2^2.\GL(2,\mathbb{Z}/4)$), $C_2\times C_4$ ($C_2^3.D_{24}$), $C_2\times C_4$ ($C_2^4.D_{12}$), $C_2\times C_4$ ($(C_2\times C_8).S_4$), $C_8$ ($C_4^2.S_4$), $C_8$ ($C_4^2.S_4$)

Order 4: $C_2^2$ ($C_{24}.(C_4\times C_8)$), $C_2^2$ ($(C_2\times C_4).\GL(2,\mathbb{Z}/4)$), $C_2^2$ ($A_4.C_4:C_8.C_2$), $C_2^2$ ($A_4.C_4:C_8.C_2$), $C_4$ ($A_4.(C_2.D_4:C_4)$), $C_4$ ($(C_2^3\times A_4).C_4.C_2$), $C_4$ ($A_4.C_4:C_8.C_2$), $C_4$ ($A_4.C_4:C_8.C_2$)

Order 2: $C_2$ ($A_4.(C_8.C_8).C_2$), $C_2$ ($(C_2\times C_8\times A_4).D_4$), $C_2$ ($C_2^3.(C_4.D_{24})$)

Order 1: $C_1$ ($(C_4\times C_8).\GL(2,\mathbb{Z}/4)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 3072: $(C_4\times C_8).\GL(2,\mathbb{Z}/4)$ ($C_1$)

Order 1536: $C_2\times C_2^3.(C_4\times C_{24})$ ($C_2$), $C_2\times A_4.\OD_{32}.C_2$ ($C_2$)

Order 768: $C_2^4:C_{48}$ ($C_4$) x 2, $(C_4\times C_8).S_4$ ($C_4$), $C_4\times C_2^3:C_{24}$ ($C_2^2$)

Order 512: $C_2^3\times C_4\times C_{16}$ ($S_3$)

Order 384: $C_2^3:C_{48}$ ($C_8$) x 2, $C_4\times C_8\times A_4$ ($D_4$), $C_4\times C_8\times A_4$ ($C_2\times C_4$), $C_2^4:C_{24}$ ($D_4$), $C_2^4:C_{24}$ ($C_2\times C_4$), $C_2\times A_4\times C_4^2$ ($Q_8$)

Order 256: $C_2^4\times C_{16}$ ($C_3:C_4$) x 2, $C_2^3\times C_4\times C_8$ ($D_6$)

Order 192: $C_2^3:C_{24}$ ($\OD_{16}$) x 2, $C_2^3:C_{24}$ ($C_2\times C_8$) x 2, $C_2^3:C_{24}$ ($C_2^2:C_4$) x 2, $A_4\times C_4^2$ ($C_4:C_4$), $C_2^4:C_{12}$ ($C_4:C_4$), $C_2^3:C_{24}$ ($Q_{16}$), $C_2^3:C_{24}$ ($\SD_{16}$), $C_2^3:C_{24}$ ($D_8$), $C_2^3:C_{24}$ ($C_4^2$)

Order 128: $C_2^3\times C_{16}$ ($C_3:C_8$) x 2, $C_2\times C_4\times C_{16}$ ($S_4$), $C_2^4\times C_8$ ($C_6:C_4$), $C_2^4\times C_8$ ($D_{12}$), $C_2^3\times C_4^2$ ($C_3:Q_8$), $C_2^2\times C_4\times C_8$ ($C_3:D_4$), $C_2^2\times C_4\times C_8$ ($C_4\times S_3$)

Order 96: $C_8\times A_4$ ($C_2^2:C_8$) x 2, $C_2^3:C_{12}$ ($C_4:C_8$) x 2, $C_8\times A_4$ ($D_4:C_4$), $C_8\times A_4$ ($C_8:C_4$), $C_8\times A_4$ ($C_4\times C_8$), $C_8\times A_4$ ($Q_8:C_4$), $C_2^3\times A_4$ ($C_8.C_4$), $C_2^3:C_{12}$ ($C_2.C_4^2$), $C_2^3:C_{12}$ ($C_8.C_4$), $C_2^3:C_{12}$ ($C_8:C_4$), $C_2^3:C_{12}$ ($C_8:C_4$)

Order 64: $C_2^3\times C_8$ ($C_6:C_8$) x 2, $C_2^3\times C_8$ ($C_3:\OD_{16}$) x 2, $C_2^2\times C_{16}$ ($A_4:C_4$) x 2, $C_2\times C_4\times C_8$ ($C_2\times S_4$), $C_2^4\times C_4$ ($C_{12}:C_4$), $C_2^3\times C_8$ ($C_3:Q_{16}$), $C_2^3\times C_8$ ($D_{24}$), $C_2^3\times C_8$ ($C_{24}:C_2$), $C_2^3\times C_8$ ($C_6.D_4$), $C_2^3\times C_8$ ($D_6:C_4$), $C_2^3\times C_8$ ($C_{12}:C_4$), $C_2^2\times C_4^2$ ($C_6.D_4$)

Order 48: $C_2^2\times A_4$ ($C_4.C_4^2$), $C_2^2\times A_4$ ($C_8:C_8$), $C_2^2\times A_4$ ($C_8:C_8$), $C_4\times A_4$ ($C_4.C_4^2$), $C_4\times A_4$ ($C_2^2.C_4^2$), $C_4\times A_4$ ($C_8:C_8$), $C_4\times A_4$ ($C_8:C_8$)

Order 32: $C_2^3\times C_4$ ($C_{12}:C_8$) x 2, $C_2^2\times C_8$ ($C_{12}.D_4$) x 2, $C_2\times C_{16}$ ($A_4:C_8$) x 2, $C_2^5$ ($C_{24}.C_4$), $C_2^3\times C_4$ ($C_6.C_4^2$), $C_2^3\times C_4$ ($C_{24}.C_4$), $C_2^3\times C_4$ ($C_{24}:C_4$), $C_2^3\times C_4$ ($C_{24}:C_4$), $C_2^2\times C_8$ ($C_{12}:C_8$), $C_2^2\times C_8$ ($D_{12}:C_4$), $C_2^2\times C_8$ ($C_6.Q_{16}$), $C_2^2\times C_8$ ($C_2^2.S_4$), $C_2^2\times C_8$ ($C_4:S_4$), $C_2^2\times C_8$ ($C_6.C_4^2$), $C_4\times C_8$ ($\GL(2,\mathbb{Z}/4)$), $C_4\times C_8$ ($C_4\times S_4$), $C_2\times C_4^2$ ($A_4:Q_8$)

Order 24: $C_2\times A_4$ ($C_4^2.D_4$), $C_2\times A_4$ ($C_{16}.C_8$)

Order 16: $C_2\times C_8$ ($A_4:\OD_{16}$) x 2, $C_2\times C_8$ ($C_2\times A_4:C_8$) x 2, $C_2\times C_8$ ($C_2.\GL(2,\mathbb{Z}/4)$), $C_2\times C_8$ ($C_2.\GL(2,\mathbb{Z}/4)$), $C_2\times C_8$ ($A_4:C_4^2$), $C_2\times C_8$ ($C_8:S_4$), $C_2\times C_8$ ($C_8:S_4$), $C_2\times C_8$ ($A_4:Q_{16}$), $C_4^2$ ($C_2.\GL(2,\mathbb{Z}/4)$), $C_2^4$ ($C_{24}:C_8$), $C_2^4$ ($C_{24}:C_8$), $C_2^4$ ($C_{12}.C_4^2$), $C_2^2\times C_4$ ($C_2^3.D_{12}$), $C_2^2\times C_4$ ($(C_2\times C_{12}):C_8$), $C_2^2\times C_4$ ($C_{24}:C_8$), $C_2^2\times C_4$ ($C_{24}:C_8$), $C_2^2\times C_4$ ($C_{12}.C_4^2$)

Order 12: $A_4$ ($C_8.(C_4\times C_8)$)

Order 8: $C_2\times C_4$ ($C_4^2.S_4$) x 2, $C_8$ ($C_4.\GL(2,\mathbb{Z}/4)$) x 2, $C_2^3$ ($C_{48}.C_8$), $C_2^3$ ($(C_2\times C_8).S_4$), $C_2^3$ ($(C_2\times C_{24}):C_8$), $C_2\times C_4$ ($C_2^2.\GL(2,\mathbb{Z}/4)$), $C_2\times C_4$ ($C_2^3.D_{24}$), $C_2\times C_4$ ($C_2^4.D_{12}$), $C_2\times C_4$ ($(C_2\times C_8).S_4$), $C_8$ ($C_4.\GL(2,\mathbb{Z}/4)$), $C_8$ ($C_4.\GL(2,\mathbb{Z}/4)$), $C_8$ ($C_4^2.S_4$), $C_8$ ($C_4^2.S_4$)

Order 4: $C_2^2$ ($C_{24}.(C_4\times C_8)$), $C_2^2$ ($(C_2\times C_4).\GL(2,\mathbb{Z}/4)$), $C_2^2$ ($A_4.C_4:C_8.C_2$), $C_2^2$ ($A_4.C_4:C_8.C_2$), $C_4$ ($A_4.(C_2.D_4:C_4)$), $C_4$ ($(C_2^3\times A_4).C_4.C_2$), $C_4$ ($A_4.C_4:C_8.C_2$), $C_4$ ($A_4.C_4:C_8.C_2$)

Order 2: $C_2$ ($A_4.(C_8.C_8).C_2$), $C_2$ ($C_2^3.(C_4.D_{24})$)

Order 1: $C_1$ ($(C_4\times C_8).\GL(2,\mathbb{Z}/4)$)

Series

Derived series

$(C_4\times C_8).\GL(2,\mathbb{Z}/4)$

$\rhd$

$C_8\times A_4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Chief series

$(C_4\times C_8).\GL(2,\mathbb{Z}/4)$

$\rhd$

$C_2\times A_4.\OD_{32}.C_2$

$\rhd$

$C_4\times C_2^3:C_{24}$

$\rhd$

$C_2^4:C_{24}$

$\rhd$

$C_2^4:C_{12}$

$\rhd$

$C_2^3:C_{12}$

$\rhd$

$C_2^2\times A_4$

$\rhd$

$C_2\times A_4$

$\rhd$

$C_2^3$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$(C_4\times C_8).\GL(2,\mathbb{Z}/4)$

$\rhd$

$C_8\times A_4$

$\rhd$

$C_4\times A_4$

$\rhd$

$C_2\times A_4$

$\rhd$

$A_4$

Upper central series

$C_1$

$\lhd$

$C_2\times C_8$

$\lhd$

$C_2^2\times C_8$

$\lhd$

$C_2\times C_4\times C_8$

$\lhd$

$C_2\times C_4\times C_{16}$

Supergroups

This group is a maximal subgroup of 1 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 $304 \times 304$ 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 $86 \times 86$ rational character table.