Abstract group 1152.154829: $C_{48}.S_4$

• Label: 1152.154829
• Order: \( 2^{7} \cdot 3^{2} \)
• Exponent: \( 2^{4} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3 \)
• $\card{Z(G)}$: \( 2 \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{10} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \)
• Perm deg.: $67$
• Trans deg.: $192$
• Rank: $2$

Group information

Description:	$C_{48}.S_4$
Order:	\(1152\)\(\medspace = 2^{7} \cdot 3^{2} \)
Exponent:	\(48\)\(\medspace = 2^{4} \cdot 3 \)
Automorphism group:	$(C_2\times C_4\times S_4).C_2^4$, of order \(3072\)\(\medspace = 2^{10} \cdot 3 \)
Composition factors:	$C_2$ x 7, $C_3$ x 2
Derived length:	$4$

This group is nonabelian and solvable.

Group statistics

Order	1	2	3	4	6	8	12	16	24	48
Elements	1	55	26	56	134	112	160	32	320	256	1152
Conjugacy classes	1	3	5	3	9	5	12	6	22	36	102
Divisions	1	3	3	3	5	4	5	2	6	4	36
Autjugacy classes	1	3	3	3	5	4	5	2	6	4	36

Dimension	1	2	3	4	6	8	12	16	24	32	64
Irr. complex chars.	12	33	12	36	9	0	0	0	0	0	0	102
Irr. rational chars.	4	7	4	5	5	3	2	2	1	2	1	36

Minimal presentations

Permutation degree:	$67$
Transitive degree:	$192$
Rank:	$2$
Inequivalent generating pairs:	$72$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	4	8	32
Arbitrary	4	8	18

Constructions

Presentation:	${\langle a, b, c, d \mid a^{2}=d^{12}=1, b^{24}=d^{6}, c^{2}=d^{6}, b^{a}= \!\cdots\! \rangle}$
Permutation group:	Degree $67$ $\langle(2,8)(3,14)(4,15)(5,21)(6,23)(9,18)(10,30)(11,35)(12,38)(13,29)(16,45)(17,32) \!\cdots\! \rangle$
Direct product:	$C_3$ $\, \times\, $ $(C_{16}.S_4)$
Semidirect product:	$(C_8.S_4)$ $\,\rtimes\,$ $C_6$	$(C_{24}.S_4)$ $\,\rtimes\,$ $C_2$	$(C_{48}.A_4)$ $\,\rtimes\,$ $C_2$	$(C_{16}.A_4)$ $\,\rtimes\,$ $C_6$	all 6
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{48}$ . $S_4$	$(C_8.S_4)$ . $C_6$	$C_8$ . $(C_6\times S_4)$	$C_6$ . $(C_8:S_4)$	all 22

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

Homology

Abelianization:	$C_{2} \times C_{6} \simeq C_{2}^{2} \times C_{3}$
Schur multiplier:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 1210 subgroups in 179 conjugacy classes, 32 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_6$	$G/Z \simeq$ $C_8:S_4$
Commutator:	$G' \simeq$ $C_8.A_4$	$G/G' \simeq$ $C_2\times C_6$
Frattini:	$\Phi \simeq$ $C_8$	$G/\Phi \simeq$ $C_6\times S_4$
Fitting:	$\operatorname{Fit} \simeq$ $\OD_{32}:C_6$	$G/\operatorname{Fit} \simeq$ $S_3$
Radical:	$R \simeq$ $C_{48}.S_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_6$	$G/\operatorname{soc} \simeq$ $C_8:S_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_{16}.D_4$
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 1152: $C_{48}.S_4$

Order 576: $C_{24}.S_4$, $C_{24}.S_4$, $C_{48}.A_4$

Order 384: $C_{16}.S_4$, $C_{48}.D_4$

Order 288: $\GL(2,3):C_6$, $C_{12}.S_4$, $C_{24}.A_4$, $C_{48}:C_6$

Order 192: $C_{16}.A_4$ x 2, $C_8.S_4$, $C_8.S_4$, $C_6\times \SD_{32}$, $\OD_{32}:C_6$, $C_{24}.D_4$, $C_{24}.D_4$, $C_{16}.C_{12}$, $C_{12}.D_8$, $\OD_{32}:C_6$

Order 144: $C_3^2:Q_{16}$, $C_3\times D_{24}$, $C_3\times C_{48}$, $\SL(2,3):C_6$, $C_3\times \GL(2,3)$, $C_6.S_4$

Order 128: $C_{16}.D_4$

Order 96: $C_8.A_4$ x 2, $C_3\times \SD_{32}$ x 2, $C_8.C_{12}$ x 2, $C_{48}:C_2$, $C_3\times \OD_{32}$, $C_2\times C_{48}$, $C_{12}.D_4$, $\OD_{16}:C_6$, $\GL(2,3):C_2$, $C_4.S_4$, $Q_{16}:C_6$, $D_8:C_6$, $C_6\times Q_{16}$, $C_6\times D_8$, $\OD_{16}:C_6$

Order 72: $C_3\times D_{12}$, $C_3^2:Q_8$, $C_3\times \SL(2,3)$, $C_3\times C_{24}$

Order 64: $C_{16}.C_4$, $C_4.D_8$, $\OD_{32}:C_2$, $C_2\times \SD_{32}$, $\OD_{32}:C_2$, $C_8.D_4$, $C_8.D_4$

Order 48: $C_{48}$ x 4, $C_3\times Q_{16}$ x 3, $C_3\times D_8$ x 3, $C_3\times \OD_{16}$ x 3, $\SL(2,3):C_2$ x 2, $C_3\times \SD_{16}$ x 2, $C_3:Q_{16}$, $D_{24}$, $D_4:C_6$, $C_6\times Q_8$, $C_6\times D_4$, $\GL(2,3)$, $C_2.S_4$, $C_2\times C_{24}$

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

Order 32: $\SD_{32}$ x 2, $C_8.C_4$ x 2, $C_4.D_4$, $\OD_{16}:C_2$, $Q_{16}:C_2$, $D_8:C_2$, $C_2\times Q_{16}$, $C_2\times D_8$, $\OD_{16}:C_2$, $\OD_{32}$, $C_2\times C_{16}$

Order 24: $C_{24}$ x 6, $C_3\times Q_8$ x 3, $C_3\times D_4$ x 3, $C_2\times C_{12}$ x 2, $\SL(2,3)$ x 2, $D_{12}$, $C_3:Q_8$, $C_2^2\times C_6$

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

Order 16: $Q_{16}$ x 3, $D_8$ x 3, $\OD_{16}$ x 3, $\SD_{16}$ x 2, $C_{16}$ x 2, $C_2\times C_8$, $D_4:C_2$, $C_2\times Q_8$, $C_2\times D_4$

Order 12: $C_{12}$ x 5, $C_2\times C_6$ x 3, $D_6$, $C_3:C_4$

Order 9: $C_3^2$

Order 8: $C_8$ x 4, $Q_8$ x 3, $D_4$ x 3, $C_2\times C_4$ x 2, $C_2^3$

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

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

Order 3: $C_3$ x 3

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1152: $C_{48}.S_4$

Order 576: $C_{24}.S_4$, $C_{24}.S_4$, $C_{48}.A_4$

Order 384: $C_{16}.S_4$, $C_{48}.D_4$

Order 288: $\GL(2,3):C_6$, $C_{12}.S_4$, $C_{24}.A_4$, $C_{48}:C_6$

Order 192: $C_{16}.A_4$ x 2, $C_8.S_4$, $C_8.S_4$, $C_6\times \SD_{32}$, $\OD_{32}:C_6$, $C_{24}.D_4$, $C_{24}.D_4$, $C_{16}.C_{12}$, $C_{12}.D_8$, $\OD_{32}:C_6$

Order 144: $C_3^2:Q_{16}$, $C_3\times D_{24}$, $C_3\times C_{48}$, $\SL(2,3):C_6$, $C_3\times \GL(2,3)$, $C_6.S_4$

Order 128: $C_{16}.D_4$

Order 96: $C_8.A_4$ x 2, $C_3\times \SD_{32}$ x 2, $C_8.C_{12}$ x 2, $C_{48}:C_2$, $C_3\times \OD_{32}$, $C_2\times C_{48}$, $C_{12}.D_4$, $\OD_{16}:C_6$, $\GL(2,3):C_2$, $C_4.S_4$, $Q_{16}:C_6$, $D_8:C_6$, $C_6\times Q_{16}$, $C_6\times D_8$, $\OD_{16}:C_6$

Order 72: $C_3\times D_{12}$, $C_3^2:Q_8$, $C_3\times \SL(2,3)$, $C_3\times C_{24}$

Order 64: $C_{16}.C_4$, $C_4.D_8$, $\OD_{32}:C_2$, $C_2\times \SD_{32}$, $\OD_{32}:C_2$, $C_8.D_4$, $C_8.D_4$

Order 48: $C_{48}$ x 4, $C_3\times Q_{16}$ x 3, $C_3\times D_8$ x 3, $C_3\times \OD_{16}$ x 3, $\SL(2,3):C_2$ x 2, $C_3\times \SD_{16}$ x 2, $C_3:Q_{16}$, $D_{24}$, $D_4:C_6$, $C_6\times Q_8$, $C_6\times D_4$, $\GL(2,3)$, $C_2.S_4$, $C_2\times C_{24}$

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

Order 32: $\SD_{32}$ x 2, $C_8.C_4$ x 2, $C_4.D_4$, $\OD_{16}:C_2$, $Q_{16}:C_2$, $D_8:C_2$, $C_2\times Q_{16}$, $C_2\times D_8$, $\OD_{16}:C_2$, $\OD_{32}$, $C_2\times C_{16}$

Order 24: $C_{24}$ x 6, $C_3\times Q_8$ x 3, $C_3\times D_4$ x 3, $C_2\times C_{12}$ x 2, $\SL(2,3)$ x 2, $D_{12}$, $C_3:Q_8$, $C_2^2\times C_6$

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

Order 16: $Q_{16}$ x 3, $D_8$ x 3, $\OD_{16}$ x 3, $\SD_{16}$ x 2, $C_{16}$ x 2, $C_2\times C_8$, $D_4:C_2$, $C_2\times Q_8$, $C_2\times D_4$

Order 12: $C_{12}$ x 5, $C_2\times C_6$ x 3, $D_6$, $C_3:C_4$

Order 9: $C_3^2$

Order 8: $C_8$ x 4, $Q_8$ x 3, $D_4$ x 3, $C_2\times C_4$ x 2, $C_2^3$

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

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

Order 3: $C_3$ x 3

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1152: $C_{48}.S_4$ ($C_1$)

Order 576: $C_{24}.S_4$ ($C_2$), $C_{24}.S_4$ ($C_2$), $C_{48}.A_4$ ($C_2$)

Order 384: $C_{16}.S_4$ ($C_3$)

Order 288: $C_{24}.A_4$ ($C_2^2$)

Order 192: $C_8.S_4$ ($C_6$), $C_8.S_4$ ($C_6$), $\OD_{32}:C_6$ ($S_3$), $C_{16}.A_4$ ($C_6$)

Order 144: $\SL(2,3):C_6$ ($D_4$)

Order 96: $C_8.A_4$ ($C_2\times C_6$), $\OD_{16}:C_6$ ($D_6$)

Order 72: $C_3\times \SL(2,3)$ ($D_8$)

Order 64: $\OD_{32}:C_2$ ($C_3\times S_3$)

Order 48: $D_4:C_6$ ($D_{12}$), $\SL(2,3):C_2$ ($C_3\times D_4$), $C_{48}$ ($S_4$)

Order 32: $\OD_{16}:C_2$ ($C_6\times S_3$)

Order 24: $\SL(2,3)$ ($C_3\times D_8$), $C_{24}$ ($C_2\times S_4$), $C_3\times Q_8$ ($D_{24}$)

Order 16: $D_4:C_2$ ($C_3\times D_{12}$), $C_{16}$ ($C_3\times S_4$)

Order 12: $C_{12}$ ($C_4:S_4$)

Order 8: $Q_8$ ($C_3\times D_{24}$), $C_8$ ($C_6\times S_4$)

Order 6: $C_6$ ($C_8:S_4$)

Order 4: $C_4$ ($C_{12}:S_4$)

Order 3: $C_3$ ($C_{16}.S_4$)

Order 2: $C_2$ ($C_{24}:S_4$)

Order 1: $C_1$ ($C_{48}.S_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1152: $C_{48}.S_4$ ($C_1$)

Order 576: $C_{24}.S_4$ ($C_2$), $C_{24}.S_4$ ($C_2$), $C_{48}.A_4$ ($C_2$)

Order 384: $C_{16}.S_4$ ($C_3$)

Order 288: $C_{24}.A_4$ ($C_2^2$)

Order 192: $C_8.S_4$ ($C_6$), $C_8.S_4$ ($C_6$), $\OD_{32}:C_6$ ($S_3$), $C_{16}.A_4$ ($C_6$)

Order 144: $\SL(2,3):C_6$ ($D_4$)

Order 96: $C_8.A_4$ ($C_2\times C_6$), $\OD_{16}:C_6$ ($D_6$)

Order 72: $C_3\times \SL(2,3)$ ($D_8$)

Order 64: $\OD_{32}:C_2$ ($C_3\times S_3$)

Order 48: $D_4:C_6$ ($D_{12}$), $\SL(2,3):C_2$ ($C_3\times D_4$), $C_{48}$ ($S_4$)

Order 32: $\OD_{16}:C_2$ ($C_6\times S_3$)

Order 24: $\SL(2,3)$ ($C_3\times D_8$), $C_{24}$ ($C_2\times S_4$), $C_3\times Q_8$ ($D_{24}$)

Order 16: $D_4:C_2$ ($C_3\times D_{12}$), $C_{16}$ ($C_3\times S_4$)

Order 12: $C_{12}$ ($C_4:S_4$)

Order 8: $Q_8$ ($C_3\times D_{24}$), $C_8$ ($C_6\times S_4$)

Order 6: $C_6$ ($C_8:S_4$)

Order 4: $C_4$ ($C_{12}:S_4$)

Order 3: $C_3$ ($C_{16}.S_4$)

Order 2: $C_2$ ($C_{24}:S_4$)

Order 1: $C_1$ ($C_{48}.S_4$)

Series

Derived series

$C_{48}.S_4$

$\rhd$

$C_8.A_4$

$\rhd$

$Q_8$

$\rhd$

$C_2$

$\rhd$

$C_1$

Chief series

$C_{48}.S_4$

$\rhd$

$C_{24}.S_4$

$\rhd$

$C_{24}.A_4$

$\rhd$

$\SL(2,3):C_6$

$\rhd$

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

$\rhd$

$C_3\times Q_8$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_{48}.S_4$

$\rhd$

$C_8.A_4$

$\rhd$

$\SL(2,3):C_2$

$\rhd$

$\SL(2,3)$

Upper central series

$C_1$

$\lhd$

$C_6$

$\lhd$

$C_{12}$

$\lhd$

$C_{24}$

$\lhd$

$C_{48}$

Supergroups

This group is a maximal subgroup of 1 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 $102 \times 102$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $36 \times 36$ rational character table.