Abstract group 4608.z: $C_{192}.S_4$

• Label: 4608.z
• Order: \( 2^{9} \cdot 3^{2} \)
• Exponent: \( 2^{6} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{6} \cdot 3 \)
• $\card{Z(G)}$: \( 2^{6} \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{10} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{7} \)
• Perm deg.: $259$
• Trans deg.: $768$
• Rank: $2$

Group information

Description:	$C_{192}.S_4$
Order:	\(4608\)\(\medspace = 2^{9} \cdot 3^{2} \)
Exponent:	\(192\)\(\medspace = 2^{6} \cdot 3 \)
Automorphism group:	$(C_8\times A_4).C_2^5$, of order \(3072\)\(\medspace = 2^{10} \cdot 3 \)
Composition factors:	$C_2$ x 9, $C_3$ x 2
Derived length:	$4$

This group is nonabelian and solvable.

Group statistics

Order	1	2	3	4	6	8	12	16	24	32	48	64	96	192
Elements	1	19	26	44	62	64	136	128	224	256	448	512	896	1792	4608
Conjugacy classes	1	3	5	8	9	12	22	24	36	48	72	96	144	288	768
Divisions	1	3	3	5	5	5	7	4	7	4	6	4	6	6	66
Autjugacy classes	1	3	3	4	5	5	6	4	7	4	6	4	6	6	64

Dimension	1	2	3	4	6	8	12	16	24	32	48	64	96	128	256
Irr. complex chars.	192	288	192	96	0	0	0	0	0	0	0	0	0	0	0	768
Irr. rational chars.	4	8	4	7	6	6	4	6	4	4	4	3	2	3	1	66

Minimal presentations

Permutation degree:	$259$
Transitive degree:	$768$
Rank:	$2$
Inequivalent generating pairs:	$1152$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	2	not computed	128
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d \mid d^{12}=1, a^{2}=cd^{3}, b^{96}=d^{6}, c^{2}=d^{6}, b^{a}= \!\cdots\! \rangle}$
Permutation group:	Degree $259$ $\langle(1,2,6,10,30,13,14,4)(3,7,18,24,54,33,34,12)(5,9,20,29,60,36,37,15)(8,25,48,21,16,19,44,26) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 65 & 157 \\ 120 & 180 \end{array}\right), \left(\begin{array}{rr} 5 & 0 \\ 0 & 5 \end{array}\right), \left(\begin{array}{rr} 79 & 127 \\ 87 & 113 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{193})$
Direct product:	$C_3$ $\, \times\, $ $(C_{64}.S_4)$
Semidirect product:	$(C_{96}.S_4)$ $\,\rtimes\,$ $C_2$	$(C_{32}.S_4)$ $\,\rtimes\,$ $C_6$	$(C_{64}.A_4)$ $\,\rtimes\,$ $C_6$	$(C_{192}.A_4)$ $\,\rtimes\,$ $C_2$	all 6
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{192}$ . $S_4$	$\GL(2,3)$ . $C_{96}$	$Q_8$ . $(S_3\times C_{96})$	$(C_{96}.S_4)$ . $C_2$	all 50

Elements of the group are displayed as matrices in $\GL_{2}(\F_{193})$.

Homology

Abelianization:	$C_{2} \times C_{96} \simeq C_{2} \times C_{32} \times C_{3}$
Schur multiplier:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 942 subgroups in 279 conjugacy classes, 60 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_{192}$	$G/Z \simeq$ $S_4$
Commutator:	$G' \simeq$ $\SL(2,3)$	$G/G' \simeq$ $C_2\times C_{96}$
Frattini:	$\Phi \simeq$ $C_{32}$	$G/\Phi \simeq$ $C_6\times S_4$
Fitting:	$\operatorname{Fit} \simeq$ $\OD_{128}:C_6$	$G/\operatorname{Fit} \simeq$ $S_3$
Radical:	$R \simeq$ $C_{192}.S_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_6$	$G/\operatorname{soc} \simeq$ Group 768.1086039
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_8.C_{32}$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$

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

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 4608: $C_{192}.S_4$

Order 2304: $C_{192}.A_4$, $C_{96}.S_4$, $C_{96}.S_4$

Order 1536: $D_8.C_{96}$, $C_{64}.S_4$

Order 1152: $S_3\times C_{192}$, $C_{48}.S_4$, $C_{48}.S_4$, $C_{96}.A_4$

Order 768: $\OD_{128}:C_6$ x 2, $\OD_{128}:C_6$ x 2, $C_{64}.A_4$ x 2, $D_8.C_{48}$, $C_8.C_{96}$, $C_4\times C_{192}$, $C_{32}.S_4$, $C_{32}.S_4$

Order 576: $\GL(2,3):C_{12}$, $C_{48}.A_4$, $C_{24}.S_4$, $C_3\times C_{192}$, $C_3:C_{192}$, $S_3\times C_{96}$

Order 512: $D_8.C_{32}$

Order 384: $C_3\times \OD_{128}$ x 3, $C_2\times C_{192}$ x 3, $C_{32}.A_4$ x 2, $\OD_{64}:C_6$ x 2, $\OD_{64}:C_6$ x 2, $S_3\times C_{64}$, $C_{16}.S_4$, $C_{16}.S_4$, $C_8.C_{48}$, $D_8.C_{24}$, $C_4\times C_{96}$

Order 288: $\GL(2,3):C_6$, $C_3\times C_{96}$, $C_3:C_{96}$, $C_{24}.A_4$, $\SL(2,3):C_{12}$, $S_3\times C_{48}$

Order 256: $\OD_{128}:C_2$ x 2, $\OD_{128}:C_2$ x 2, $D_8.C_{16}$, $C_8.C_{32}$, $C_4\times C_{64}$

Order 192: $C_{192}$ x 6, $C_3\times \OD_{64}$ x 3, $C_2\times C_{96}$ x 3, $\OD_{32}:C_6$ x 2, $C_{16}.A_4$ x 2, $\OD_{32}:C_6$ x 2, $\GL(2,3):C_4$, $D_8:C_{12}$, $S_3\times C_{32}$, $C_8.S_4$, $C_8.C_{24}$, $C_4\times C_{48}$, $C_3:C_{64}$

Order 144: $S_3\times C_{24}$, $C_3\times C_{48}$, $C_3:C_{48}$, $\SL(2,3):C_6$, $C_3\times \GL(2,3)$, $C_6.S_4$

Order 128: $\OD_{128}$ x 3, $C_2\times C_{64}$ x 3, $\OD_{64}:C_2$ x 2, $\OD_{64}:C_2$ x 2, $C_{16}.D_4$, $C_8.C_{16}$, $C_4\times C_{32}$

Order 96: $C_{96}$ x 6, $C_3\times \OD_{32}$ x 3, $C_2\times C_{48}$ x 3, $C_8.A_4$ x 2, $D_4:C_{12}$ x 2, $\OD_{16}:C_6$ x 2, $\Unitary(2,3)$, $C_8.C_{12}$, $C_4\times C_{24}$, $S_3\times C_{16}$, $\GL(2,3):C_2$, $D_8:C_6$, $C_3:C_{32}$

Order 72: $S_3\times C_{12}$, $C_3\times \SL(2,3)$, $C_3\times C_{24}$, $C_3:C_{24}$

Order 64: $C_{64}$ x 4, $\OD_{64}$ x 3, $C_2\times C_{32}$ x 3, $\OD_{32}:C_2$ x 2, $\OD_{32}:C_2$ x 2, $C_8.C_8$, $C_4\times C_{16}$, $D_8:C_4$

Order 48: $C_{48}$ x 6, $C_3\times \OD_{16}$ x 3, $C_2\times C_{24}$ x 3, $D_4:C_6$ x 2, $\SL(2,3):C_2$ x 2, $C_3\times \SD_{16}$ x 2, $S_3\times C_8$, $\GL(2,3)$, $C_2.S_4$, $C_3\times Q_{16}$, $C_3\times D_8$, $C_4\times C_{12}$, $C_3:C_{16}$

Order 36: $C_3\times C_{12}$, $C_3:C_{12}$, $C_6\times S_3$

Order 32: $C_{32}$ x 4, $\OD_{32}$ x 3, $C_2\times C_{16}$ x 3, $\OD_{16}:C_2$ x 2, $C_4\wr C_2$ x 2, $D_8:C_2$, $C_4\times C_8$, $C_8.C_4$

Order 24: $C_{24}$ x 7, $C_2\times C_{12}$ x 3, $C_3\times D_4$ x 3, $\SL(2,3)$ x 2, $C_3\times Q_8$ x 2, $C_4\times S_3$, $C_3:C_8$

Order 18: $C_3\times S_3$ x 2, $C_3\times C_6$

Order 16: $C_{16}$ x 4, $\OD_{16}$ x 3, $C_2\times C_8$ x 3, $\SD_{16}$ x 2, $D_4:C_2$ x 2, $Q_{16}$, $D_8$, $C_4^2$

Order 12: $C_{12}$ x 7, $C_2\times C_6$ x 2, $D_6$, $C_3:C_4$

Order 9: $C_3^2$

Order 8: $C_8$ x 5, $D_4$ x 3, $C_2\times C_4$ x 3, $Q_8$ x 2

Order 6: $C_6$ x 5, $S_3$ x 2

Order 4: $C_4$ x 5, $C_2^2$ x 2

Order 3: $C_3$ x 3

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 4608: $C_{192}.S_4$

Order 2304: $C_{192}.A_4$, $C_{96}.S_4$, $C_{96}.S_4$

Order 1536: $D_8.C_{96}$, $C_{64}.S_4$

Order 1152: $S_3\times C_{192}$, $C_{48}.S_4$, $C_{48}.S_4$, $C_{96}.A_4$

Order 768: $\OD_{128}:C_6$ x 2, $\OD_{128}:C_6$ x 2, $C_{64}.A_4$ x 2, $D_8.C_{48}$, $C_8.C_{96}$, $C_4\times C_{192}$, $C_{32}.S_4$, $C_{32}.S_4$

Order 576: $\GL(2,3):C_{12}$, $C_{48}.A_4$, $C_{24}.S_4$, $C_3\times C_{192}$, $C_3:C_{192}$, $S_3\times C_{96}$

Order 512: $D_8.C_{32}$

Order 384: $C_3\times \OD_{128}$ x 3, $C_2\times C_{192}$ x 3, $C_{32}.A_4$ x 2, $\OD_{64}:C_6$ x 2, $\OD_{64}:C_6$ x 2, $S_3\times C_{64}$, $C_{16}.S_4$, $C_{16}.S_4$, $C_8.C_{48}$, $D_8.C_{24}$, $C_4\times C_{96}$

Order 288: $\GL(2,3):C_6$, $C_3\times C_{96}$, $C_3:C_{96}$, $C_{24}.A_4$, $\SL(2,3):C_{12}$, $S_3\times C_{48}$

Order 256: $\OD_{128}:C_2$ x 2, $\OD_{128}:C_2$ x 2, $D_8.C_{16}$, $C_8.C_{32}$, $C_4\times C_{64}$

Order 192: $C_{192}$ x 6, $C_3\times \OD_{64}$ x 3, $C_2\times C_{96}$ x 3, $\OD_{32}:C_6$ x 2, $C_{16}.A_4$ x 2, $\OD_{32}:C_6$ x 2, $\GL(2,3):C_4$, $D_8:C_{12}$, $S_3\times C_{32}$, $C_8.S_4$, $C_8.C_{24}$, $C_4\times C_{48}$, $C_3:C_{64}$

Order 144: $S_3\times C_{24}$, $C_3\times C_{48}$, $C_3:C_{48}$, $\SL(2,3):C_6$, $C_3\times \GL(2,3)$, $C_6.S_4$

Order 128: $\OD_{128}$ x 3, $C_2\times C_{64}$ x 3, $\OD_{64}:C_2$ x 2, $\OD_{64}:C_2$ x 2, $C_{16}.D_4$, $C_8.C_{16}$, $C_4\times C_{32}$

Order 96: $C_{96}$ x 6, $C_3\times \OD_{32}$ x 3, $C_2\times C_{48}$ x 3, $C_8.A_4$ x 2, $D_4:C_{12}$ x 2, $\OD_{16}:C_6$ x 2, $\Unitary(2,3)$, $C_8.C_{12}$, $C_4\times C_{24}$, $S_3\times C_{16}$, $\GL(2,3):C_2$, $D_8:C_6$, $C_3:C_{32}$

Order 72: $S_3\times C_{12}$, $C_3\times \SL(2,3)$, $C_3\times C_{24}$, $C_3:C_{24}$

Order 64: $C_{64}$ x 4, $\OD_{64}$ x 3, $C_2\times C_{32}$ x 3, $\OD_{32}:C_2$ x 2, $\OD_{32}:C_2$ x 2, $C_8.C_8$, $C_4\times C_{16}$, $D_8:C_4$

Order 48: $C_{48}$ x 6, $C_3\times \OD_{16}$ x 3, $C_2\times C_{24}$ x 3, $D_4:C_6$ x 2, $\SL(2,3):C_2$ x 2, $C_3\times \SD_{16}$ x 2, $S_3\times C_8$, $\GL(2,3)$, $C_2.S_4$, $C_3\times Q_{16}$, $C_3\times D_8$, $C_4\times C_{12}$, $C_3:C_{16}$

Order 36: $C_3\times C_{12}$, $C_3:C_{12}$, $C_6\times S_3$

Order 32: $C_{32}$ x 4, $\OD_{32}$ x 3, $C_2\times C_{16}$ x 3, $\OD_{16}:C_2$ x 2, $C_4\wr C_2$ x 2, $D_8:C_2$, $C_4\times C_8$, $C_8.C_4$

Order 24: $C_{24}$ x 7, $C_2\times C_{12}$ x 3, $C_3\times D_4$ x 3, $\SL(2,3)$ x 2, $C_3\times Q_8$ x 2, $C_4\times S_3$, $C_3:C_8$

Order 18: $C_3\times C_6$, $C_3\times S_3$

Order 16: $C_{16}$ x 4, $\OD_{16}$ x 3, $C_2\times C_8$ x 3, $\SD_{16}$ x 2, $D_4:C_2$ x 2, $Q_{16}$, $D_8$, $C_4^2$

Order 12: $C_{12}$ x 6, $C_2\times C_6$ x 2, $D_6$, $C_3:C_4$

Order 9: $C_3^2$

Order 8: $C_8$ x 5, $D_4$ x 3, $C_2\times C_4$ x 3, $Q_8$ x 2

Order 6: $C_6$ x 5, $S_3$

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

Order 3: $C_3$ x 3

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 4608: $C_{192}.S_4$ ($C_1$)

Order 2304: $C_{192}.A_4$ ($C_2$), $C_{96}.S_4$ ($C_2$), $C_{96}.S_4$ ($C_2$)

Order 1536: $C_{64}.S_4$ ($C_3$)

Order 1152: $C_{48}.S_4$ ($C_4$), $C_{48}.S_4$ ($C_4$), $C_{96}.A_4$ ($C_2^2$)

Order 768: $\OD_{128}:C_6$ ($S_3$), $C_{32}.S_4$ ($C_6$), $C_{32}.S_4$ ($C_6$), $C_{64}.A_4$ ($C_6$)

Order 576: $\GL(2,3):C_{12}$ ($C_8$), $C_{48}.A_4$ ($C_2\times C_4$), $C_{24}.S_4$ ($C_8$)

Order 384: $C_{32}.A_4$ ($C_2\times C_6$), $C_{16}.S_4$ ($C_{12}$), $C_{16}.S_4$ ($C_{12}$), $\OD_{64}:C_6$ ($D_6$)

Order 288: $\GL(2,3):C_6$ ($C_{16}$), $C_{24}.A_4$ ($C_2\times C_8$), $\SL(2,3):C_{12}$ ($C_{16}$)

Order 256: $\OD_{128}:C_2$ ($C_3\times S_3$)

Order 192: $\GL(2,3):C_4$ ($C_{24}$), $\OD_{32}:C_6$ ($C_4\times S_3$), $C_{16}.A_4$ ($C_2\times C_{12}$), $C_{192}$ ($S_4$), $C_8.S_4$ ($C_{24}$)

Order 144: $\SL(2,3):C_6$ ($C_2\times C_{16}$), $C_3\times \GL(2,3)$ ($C_{32}$), $C_6.S_4$ ($C_{32}$)

Order 128: $\OD_{64}:C_2$ ($C_6\times S_3$)

Order 96: $C_8.A_4$ ($C_2\times C_{24}$), $\Unitary(2,3)$ ($C_{48}$), $C_{96}$ ($C_2\times S_4$), $\GL(2,3):C_2$ ($C_{48}$), $\OD_{16}:C_6$ ($S_3\times C_8$)

Order 72: $C_3\times \SL(2,3)$ ($C_2\times C_{32}$)

Order 64: $\OD_{32}:C_2$ ($S_3\times C_{12}$), $C_{64}$ ($C_3\times S_4$)

Order 48: $D_4:C_6$ ($S_3\times C_{16}$), $\SL(2,3):C_2$ ($C_2\times C_{48}$), $\GL(2,3)$ ($C_{96}$), $C_2.S_4$ ($C_{96}$), $C_{48}$ ($C_4\times S_4$)

Order 32: $\OD_{16}:C_2$ ($S_3\times C_{24}$), $C_{32}$ ($C_6\times S_4$)

Order 24: $\SL(2,3)$ ($C_2\times C_{96}$), $C_{24}$ ($C_8\times S_4$), $C_3\times Q_8$ ($S_3\times C_{32}$)

Order 16: $D_4:C_2$ ($S_3\times C_{48}$), $C_{16}$ ($C_{12}\times S_4$)

Order 12: $C_{12}$ ($C_{16}\times S_4$)

Order 8: $Q_8$ ($S_3\times C_{96}$), $C_8$ ($C_{24}\times S_4$)

Order 6: $C_6$ ($C_{32}\times S_4$)

Order 4: $C_4$ ($C_{48}\times S_4$)

Order 3: $C_3$ ($C_{64}.S_4$)

Order 2: $C_2$ ($C_{96}\times S_4$)

Order 1: $C_1$ ($C_{192}.S_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 4608: $C_{192}.S_4$ ($C_1$)

Order 2304: $C_{192}.A_4$ ($C_2$), $C_{96}.S_4$ ($C_2$), $C_{96}.S_4$ ($C_2$)

Order 1536: $C_{64}.S_4$ ($C_3$)

Order 1152: $C_{48}.S_4$ ($C_4$), $C_{48}.S_4$ ($C_4$), $C_{96}.A_4$ ($C_2^2$)

Order 768: $\OD_{128}:C_6$ ($S_3$), $C_{32}.S_4$ ($C_6$), $C_{32}.S_4$ ($C_6$), $C_{64}.A_4$ ($C_6$)

Order 576: $\GL(2,3):C_{12}$ ($C_8$), $C_{48}.A_4$ ($C_2\times C_4$), $C_{24}.S_4$ ($C_8$)

Order 384: $C_{32}.A_4$ ($C_2\times C_6$), $C_{16}.S_4$ ($C_{12}$), $C_{16}.S_4$ ($C_{12}$), $\OD_{64}:C_6$ ($D_6$)

Order 288: $\GL(2,3):C_6$ ($C_{16}$), $C_{24}.A_4$ ($C_2\times C_8$), $\SL(2,3):C_{12}$ ($C_{16}$)

Order 256: $\OD_{128}:C_2$ ($C_3\times S_3$)

Order 192: $\GL(2,3):C_4$ ($C_{24}$), $\OD_{32}:C_6$ ($C_4\times S_3$), $C_{16}.A_4$ ($C_2\times C_{12}$), $C_{192}$ ($S_4$), $C_8.S_4$ ($C_{24}$)

Order 144: $\SL(2,3):C_6$ ($C_2\times C_{16}$), $C_3\times \GL(2,3)$ ($C_{32}$), $C_6.S_4$ ($C_{32}$)

Order 128: $\OD_{64}:C_2$ ($C_6\times S_3$)

Order 96: $C_8.A_4$ ($C_2\times C_{24}$), $\Unitary(2,3)$ ($C_{48}$), $C_{96}$ ($C_2\times S_4$), $\GL(2,3):C_2$ ($C_{48}$), $\OD_{16}:C_6$ ($S_3\times C_8$)

Order 72: $C_3\times \SL(2,3)$ ($C_2\times C_{32}$)

Order 64: $\OD_{32}:C_2$ ($S_3\times C_{12}$), $C_{64}$ ($C_3\times S_4$)

Order 48: $D_4:C_6$ ($S_3\times C_{16}$), $\SL(2,3):C_2$ ($C_2\times C_{48}$), $\GL(2,3)$ ($C_{96}$), $C_2.S_4$ ($C_{96}$), $C_{48}$ ($C_4\times S_4$)

Order 32: $\OD_{16}:C_2$ ($S_3\times C_{24}$), $C_{32}$ ($C_6\times S_4$)

Order 24: $\SL(2,3)$ ($C_2\times C_{96}$), $C_{24}$ ($C_8\times S_4$), $C_3\times Q_8$ ($S_3\times C_{32}$)

Order 16: $D_4:C_2$ ($S_3\times C_{48}$), $C_{16}$ ($C_{12}\times S_4$)

Order 12: $C_{12}$ ($C_{16}\times S_4$)

Order 8: $Q_8$ ($S_3\times C_{96}$), $C_8$ ($C_{24}\times S_4$)

Order 6: $C_6$ ($C_{32}\times S_4$)

Order 4: $C_4$ ($C_{48}\times S_4$)

Order 3: $C_3$ ($C_{64}.S_4$)

Order 2: $C_2$ ($C_{96}\times S_4$)

Order 1: $C_1$ ($C_{192}.S_4$)

Series

Derived series

$C_{192}.S_4$

$\rhd$

$\SL(2,3)$

$\rhd$

$Q_8$

$\rhd$

$C_2$

$\rhd$

$C_1$

Chief series

$C_{192}.S_4$

$\rhd$

$C_{192}.A_4$

$\rhd$

$C_{96}.A_4$

$\rhd$

$C_{48}.A_4$

$\rhd$

$C_{24}.A_4$

$\rhd$

$\SL(2,3):C_6$

$\rhd$

$C_3\times \SL(2,3)$

$\rhd$

$\SL(2,3)$

$\rhd$

$Q_8$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_{192}.S_4$

$\rhd$

$\SL(2,3)$

Upper central series

$C_1$

$\lhd$

$C_{192}$

Character theory

Complex character table

The $768 \times 768$ character table is not available for this group.

Rational character table

See the $66 \times 66$ rational character table.