Abstract group 12144.d: $S_4\times F_{23}$

• Label: 12144.d
• Order: \( 2^{4} \cdot 3 \cdot 11 \cdot 23 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 11 \cdot 23 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 11 \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{4} \cdot 3 \cdot 11 \cdot 23 \)
• $\card{\mathrm{Out}(G)}$: \( 1 \)
• Perm deg.: $27$
• Trans deg.: $92$
• Rank: $2$

Group information

Description:	$S_4\times F_{23}$
Order:	\(12144\)\(\medspace = 2^{4} \cdot 3 \cdot 11 \cdot 23 \)
Exponent:	\(3036\)\(\medspace = 2^{2} \cdot 3 \cdot 11 \cdot 23 \)
Automorphism group:	$S_4\times F_{23}$, of order \(12144\)\(\medspace = 2^{4} \cdot 3 \cdot 11 \cdot 23 \)
Composition factors:	$C_2$ x 4, $C_3$, $C_{11}$, $C_{23}$
Derived length:	$3$

This group is nonabelian and monomial (hence solvable).

Group statistics

Order	1	2	3	4	6	11	22	23	33	44	46	66	69	92
Elements	1	239	8	144	184	230	4370	22	1840	2760	198	1840	176	132	12144
Conjugacy classes	1	5	1	2	1	10	50	1	10	20	2	10	1	1	115
Divisions	1	5	1	2	1	1	5	1	1	2	2	1	1	1	25
Autjugacy classes	1	5	1	2	1	10	50	1	10	20	2	10	1	1	115

Dimension	1	2	3	10	20	22	30	44	66
Irr. complex chars.	44	22	44	0	0	2	0	1	2	115
Irr. rational chars.	4	2	4	4	2	2	4	1	2	25

Minimal presentations

Permutation degree:	$27$
Transitive degree:	$92$
Rank:	$2$
Inequivalent generating pairs:	$2160$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	66	66	66
Arbitrary	25	25	25

Constructions

Presentation:	$\langle a, b, c, d \mid a^{2}=b^{66}=c^{2}=d^{46}=[a,c]=[c,d]=1, b^{a}=b^{23}d^{23}, d^{a}=cd^{45}, c^{b}=d^{23}, d^{b}=cd^{21} \rangle$
Permutation group:	Degree $27$ $\langle(1,2,3,6,12,7,15,20,18,11,22,21,10,19,4,9,17,23,16,13,14,8,5), (2,4,10,20,15,9,6,13,19,8,16,5,11,21,17,23,18,14,12,22,3,7) \!\cdots\! \rangle$
Direct product:	$S_4$ $\, \times\, $ $F_{23}$
Semidirect product:	$A_4$ $\,\rtimes\,$ $(C_2\times F_{23})$	$(S_4\times D_{23})$ $\,\rtimes\,$ $C_{11}$	$D_{23}$ $\,\rtimes\,$ $(C_{11}\times S_4)$	$(C_{23}\times S_4)$ $\,\rtimes\,$ $C_{22}$	all 14
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Aut. group:	$\Aut(C_2\times D_{46})$	$\Aut(C_{23}:S_4)$	$\Aut(A_4\times D_{23})$	$\Aut(S_4\times D_{23})$	all 5

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

Homology

Abelianization:	$C_{2} \times C_{22} \simeq C_{2}^{2} \times C_{11}$
Schur multiplier:	$C_{2}^{2}$
Commutator length:	$1$

Subgroups

There are 4044 subgroups in 132 conjugacy classes, 22 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $S_4\times F_{23}$
Commutator:	$G' \simeq$ $A_4\times C_{23}$	$G/G' \simeq$ $C_2\times C_{22}$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $S_4\times F_{23}$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times C_{46}$	$G/\operatorname{Fit} \simeq$ $S_3\times C_{22}$
Radical:	$R \simeq$ $S_4\times F_{23}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_{46}$	$G/\operatorname{soc} \simeq$ $S_3\times C_{22}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
11-Sylow subgroup:	$P_{ 11 } \simeq$ $C_{11}$
23-Sylow subgroup:	$P_{ 23 } \simeq$ $C_{23}$

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 12144: $S_4\times F_{23}$

Order 6072: $S_4\times C_{23}:C_{11}$, $C_{23}:(C_{11}\times S_4)$, $F_{23}\times A_4$

Order 4048: $D_4\times F_{23}$

Order 3036: $A_4\times C_{23}:C_{11}$, $S_3\times F_{23}$

Order 2024: $C_{23}:(C_{11}\times D_4)$ x 2, $C_2^2\times F_{23}$ x 2, $C_{23}:(C_{11}\times D_4)$, $D_4\times C_{23}:C_{11}$, $C_4\times F_{23}$

Order 1518: $C_{69}:C_{22}$, $C_{69}:C_{22}$, $C_3\times F_{23}$

Order 1104: $S_4\times D_{23}$

Order 1012: $C_2\times F_{23}$ x 5, $C_{46}:C_{22}$ x 2, $C_{23}:C_{44}$, $C_{23}:C_{44}$

Order 759: $C_{23}:C_{33}$

Order 552: $A_4\times D_{23}$, $C_{23}:S_4$, $C_{23}\times S_4$

Order 528: $C_{22}\times S_4$

Order 506: $F_{23}$ x 3, $C_{23}:C_{22}$ x 2

Order 368: $D_4\times D_{23}$

Order 276: $A_4\times C_{23}$, $S_3\times D_{23}$

Order 264: $C_{11}\times S_4$ x 2, $A_4\times C_{22}$

Order 253: $C_{23}:C_{11}$

Order 184: $C_{23}:D_4$ x 2, $C_2\times D_{46}$ x 2, $D_4\times C_{23}$, $D_{92}$, $C_4\times D_{23}$

Order 176: $D_4\times C_{22}$

Order 138: $D_{69}$, $C_3\times D_{23}$, $S_3\times C_{23}$

Order 132: $S_3\times C_{22}$, $C_{11}\times A_4$

Order 92: $D_{46}$ x 5, $C_2\times C_{46}$ x 2, $C_{92}$, $C_{23}:C_4$

Order 88: $D_4\times C_{11}$ x 4, $C_2^2\times C_{22}$ x 2, $C_2\times C_{44}$

Order 69: $C_{69}$

Order 66: $S_3\times C_{11}$ x 2, $C_{66}$

Order 48: $C_2\times S_4$

Order 46: $D_{23}$ x 3, $C_{46}$ x 2

Order 44: $C_2\times C_{22}$ x 7, $C_{44}$ x 2

Order 33: $C_{33}$

Order 24: $S_4$ x 2, $C_2\times A_4$

Order 23: $C_{23}$

Order 22: $C_{22}$ x 5

Order 16: $C_2\times D_4$

Order 12: $D_6$, $A_4$

Order 11: $C_{11}$

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

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

Order 4: $C_2^2$ x 7, $C_4$ x 2

Order 3: $C_3$

Order 2: $C_2$ x 5

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 12144: $S_4\times F_{23}$

Order 6072: $S_4\times C_{23}:C_{11}$, $C_{23}:(C_{11}\times S_4)$, $F_{23}\times A_4$

Order 4048: $D_4\times F_{23}$

Order 3036: $A_4\times C_{23}:C_{11}$, $S_3\times F_{23}$

Order 2024: $C_{23}:(C_{11}\times D_4)$ x 2, $C_2^2\times F_{23}$ x 2, $C_{23}:(C_{11}\times D_4)$, $D_4\times C_{23}:C_{11}$, $C_4\times F_{23}$

Order 1518: $C_{69}:C_{22}$, $C_{69}:C_{22}$, $C_3\times F_{23}$

Order 1104: $S_4\times D_{23}$

Order 1012: $C_2\times F_{23}$ x 5, $C_{46}:C_{22}$ x 2, $C_{23}:C_{44}$, $C_{23}:C_{44}$

Order 759: $C_{23}:C_{33}$

Order 552: $A_4\times D_{23}$, $C_{23}:S_4$, $C_{23}\times S_4$

Order 528: $C_{22}\times S_4$

Order 506: $F_{23}$ x 3, $C_{23}:C_{22}$ x 2

Order 368: $D_4\times D_{23}$

Order 276: $A_4\times C_{23}$, $S_3\times D_{23}$

Order 264: $C_{11}\times S_4$ x 2, $A_4\times C_{22}$

Order 253: $C_{23}:C_{11}$

Order 184: $C_{23}:D_4$ x 2, $C_2\times D_{46}$ x 2, $D_4\times C_{23}$, $D_{92}$, $C_4\times D_{23}$

Order 176: $D_4\times C_{22}$

Order 138: $D_{69}$, $C_3\times D_{23}$, $S_3\times C_{23}$

Order 132: $S_3\times C_{22}$, $C_{11}\times A_4$

Order 92: $D_{46}$ x 5, $C_2\times C_{46}$ x 2, $C_{92}$, $C_{23}:C_4$

Order 88: $D_4\times C_{11}$ x 4, $C_2^2\times C_{22}$ x 2, $C_2\times C_{44}$

Order 69: $C_{69}$

Order 66: $S_3\times C_{11}$ x 2, $C_{66}$

Order 48: $C_2\times S_4$

Order 46: $D_{23}$ x 3, $C_{46}$ x 2

Order 44: $C_2\times C_{22}$ x 7, $C_{44}$ x 2

Order 33: $C_{33}$

Order 24: $S_4$ x 2, $C_2\times A_4$

Order 23: $C_{23}$

Order 22: $C_{22}$ x 5

Order 16: $C_2\times D_4$

Order 12: $D_6$, $A_4$

Order 11: $C_{11}$

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

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

Order 4: $C_2^2$ x 7, $C_4$ x 2

Order 3: $C_3$

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 12144: $S_4\times F_{23}$ ($C_1$)

Order 6072: $S_4\times C_{23}:C_{11}$ ($C_2$), $C_{23}:(C_{11}\times S_4)$ ($C_2$), $F_{23}\times A_4$ ($C_2$)

Order 3036: $A_4\times C_{23}:C_{11}$ ($C_2^2$)

Order 2024: $C_2^2\times F_{23}$ ($S_3$)

Order 1104: $S_4\times D_{23}$ ($C_{11}$)

Order 1012: $C_{46}:C_{22}$ ($D_6$)

Order 552: $A_4\times D_{23}$ ($C_{22}$), $C_{23}:S_4$ ($C_{22}$), $C_{23}\times S_4$ ($C_{22}$)

Order 506: $F_{23}$ ($S_4$)

Order 276: $A_4\times C_{23}$ ($C_2\times C_{22}$)

Order 253: $C_{23}:C_{11}$ ($C_2\times S_4$)

Order 184: $C_2\times D_{46}$ ($S_3\times C_{11}$)

Order 92: $C_2\times C_{46}$ ($S_3\times C_{22}$)

Order 46: $D_{23}$ ($C_{11}\times S_4$)

Order 24: $S_4$ ($F_{23}$)

Order 23: $C_{23}$ ($C_{22}\times S_4$)

Order 12: $A_4$ ($C_2\times F_{23}$)

Order 4: $C_2^2$ ($S_3\times F_{23}$)

Order 1: $C_1$ ($S_4\times F_{23}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 12144: $S_4\times F_{23}$ ($C_1$)

Order 6072: $S_4\times C_{23}:C_{11}$ ($C_2$), $C_{23}:(C_{11}\times S_4)$ ($C_2$), $F_{23}\times A_4$ ($C_2$)

Order 3036: $A_4\times C_{23}:C_{11}$ ($C_2^2$)

Order 2024: $C_2^2\times F_{23}$ ($S_3$)

Order 1104: $S_4\times D_{23}$ ($C_{11}$)

Order 1012: $C_{46}:C_{22}$ ($D_6$)

Order 552: $A_4\times D_{23}$ ($C_{22}$), $C_{23}:S_4$ ($C_{22}$), $C_{23}\times S_4$ ($C_{22}$)

Order 506: $F_{23}$ ($S_4$)

Order 276: $A_4\times C_{23}$ ($C_2\times C_{22}$)

Order 253: $C_{23}:C_{11}$ ($C_2\times S_4$)

Order 184: $C_2\times D_{46}$ ($S_3\times C_{11}$)

Order 92: $C_2\times C_{46}$ ($S_3\times C_{22}$)

Order 46: $D_{23}$ ($C_{11}\times S_4$)

Order 24: $S_4$ ($F_{23}$)

Order 23: $C_{23}$ ($C_{22}\times S_4$)

Order 12: $A_4$ ($C_2\times F_{23}$)

Order 4: $C_2^2$ ($S_3\times F_{23}$)

Order 1: $C_1$ ($S_4\times F_{23}$)

Series

Derived series

$S_4\times F_{23}$

$\rhd$

$A_4\times C_{23}$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Chief series

$S_4\times F_{23}$

$\rhd$

$C_{23}:(C_{11}\times S_4)$

$\rhd$

$A_4\times C_{23}:C_{11}$

$\rhd$

$C_{46}:C_{22}$

$\rhd$

$C_2\times C_{46}$

$\rhd$

$C_{23}$

$\rhd$

$C_1$

Lower central series

$S_4\times F_{23}$

$\rhd$

$A_4\times C_{23}$

Upper central series

$C_1$

Character theory

Complex character table

See the $115 \times 115$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $25 \times 25$ rational character table.