Abstract group 1152.156581: $C_4\times C_{12}\times S_4$

• Label: 1152.156581
• Order: \( 2^{7} \cdot 3^{2} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \cdot 3 \)
• $\card{Z(G)}$: \( 2^{4} \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{11} \cdot 3^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{8} \cdot 3 \)
• Perm deg.: $15$
• Trans deg.: $144$
• Rank: $3$

Group information

Description:	$C_4\times C_{12}\times S_4$
Order:	\(1152\)\(\medspace = 2^{7} \cdot 3^{2} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_2\times S_4\times C_2^4:C_3.D_4$, of order \(18432\)\(\medspace = 2^{11} \cdot 3^{2} \)
Composition factors:	$C_2$ x 7, $C_3$ x 2
Derived length:	$3$

This group is nonabelian and monomial (hence solvable).

Group statistics

Order	1	2	3	4	6	12
Elements	1	39	26	216	150	720	1152
Conjugacy classes	1	11	5	52	31	140	240
Divisions	1	11	3	28	17	40	100
Autjugacy classes	1	4	3	5	6	7	26

Dimension	1	2	3	4	6	8	12
Irr. complex chars.	96	48	96	0	0	0	0	240
Irr. rational chars.	8	24	8	22	20	6	12	100

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$144$
Rank:	$3$
Inequivalent generating triples:	$21840$

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:	${\langle a, b, c, d, e \mid a^{2}=b^{4}=c^{12}=d^{6}=e^{2}=[a,b]=[a,e]=[b,c]= \!\cdots\! \rangle}$
Permutation group:	Degree $15$ $\langle(8,9)(10,11)(13,15), (1,2,3,4)(8,10,9,11), (8,10,9,11), (5,6,7), (8,9)(10,11), (1,3)(2,4)(8,9)(10,11), (13,14,15), (12,13)(14,15), (12,14)(13,15)\rangle$
Direct product:	$C_4$ ${}^2$ $\, \times\, $ $C_3$ $\, \times\, $ $S_4$
Semidirect product:	$(A_4:C_4^2)$ $\,\rtimes\,$ $C_6$	$(A_4\times C_4^2)$ $\,\rtimes\,$ $C_6$	$(C_4\times A_4:C_{12})$ $\,\rtimes\,$ $C_2$	$A_4$ $\,\rtimes\,$ $(C_2\times C_4\times C_{12})$	all 16
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_2^4.D_6)$ . $C_6$	$C_2^4$ . $(C_6\times D_6)$	$C_6$ . $(C_2^4.D_6)$	$(C_2^3\times C_{12})$ . $D_6$	all 26
Aut. group:	$\Aut(Q_8\times C_{65})$	$\Aut(A_4\times C_{65})$	$\Aut(C_{65}\times \SL(2,3))$	$\Aut(S_4\times C_{65})$	all 5

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

Homology

Abelianization:	$C_{2} \times C_{4} \times C_{12} \simeq C_{2} \times C_{4}^{2} \times C_{3}$
Schur multiplier:	$C_{2}^{3} \times C_{4}$
Commutator length:	$1$

Subgroups

There are 3510 subgroups in 1024 conjugacy classes, 168 normal (22 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_4\times C_{12}$	$G/Z \simeq$ $S_4$
Commutator:	$G' \simeq$ $A_4$	$G/G' \simeq$ $C_2\times C_4\times C_{12}$
Frattini:	$\Phi \simeq$ $C_2^2$	$G/\Phi \simeq$ $C_2\times C_6\times S_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^2\times C_4\times C_{12}$	$G/\operatorname{Fit} \simeq$ $S_3$
Radical:	$R \simeq$ $C_4\times C_{12}\times S_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^3\times C_6$	$G/\operatorname{soc} \simeq$ $C_2\times D_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4\times C_4^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: subgroups only stored up to automorphism

Classes of subgroups up to automorphism

No diagram available

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

Order 1152: $C_4\times C_{12}\times S_4$

Order 576: $C_2\times C_{12}\times S_4$ x 3, $C_4\times A_4:C_{12}$ x 3, $C_2^2:C_{12}^2$

Order 384: $C_4^2\times S_4$, $C_4^3:C_6$

Order 288: $C_{12}\times S_4$ x 12, $C_2^3.C_6^2$ x 3, $C_2\times A_4:C_{12}$ x 3, $C_{12}^2:C_2$, $C_2\times C_6\times S_4$

Order 192: $C_4\times C_2^2:C_{12}$ x 6, $A_4:C_4^2$ x 3, $C_{12}:C_4^2$ x 3, $C_2^4.D_6$ x 3, $C_2\times D_4\times C_{12}$ x 3, $A_4\times C_4^2$ x 2, $C_2^2\times C_4\times C_{12}$ x 2, $C_4^2\times C_{12}$

Order 144: $C_6\times S_4$ x 6, $C_{12}\times A_4$ x 6, $A_4:C_{12}$ x 6, $C_{12}:C_{12}$ x 3, $C_{12}\times D_6$ x 3, $C_2^2:C_6^2$, $C_{12}^2$

Order 128: $D_4\times C_4^2$

Order 96: $D_4\times C_{12}$ x 24, $C_2\times C_4\times C_{12}$ x 17, $C_4\times S_4$ x 12, $C_6.C_4^2$ x 6, $C_2^3\times C_{12}$ x 6, $C_2^3:C_{12}$ x 6, $C_2^3:C_{12}$ x 6, $C_2^2.S_4$ x 3, $C_2\times C_4:C_{12}$ x 3, $S_3\times C_4^2$, $C_2^2\times S_4$, $C_{12}:C_2^3$

Order 72: $S_3\times C_{12}$ x 12, $C_3\times S_4$ x 4, $C_6\times A_4$ x 3, $C_6\times C_{12}$ x 3, $C_6:C_{12}$ x 3, $C_6\times D_6$

Order 64: $C_2^2:C_4^2$ x 6, $C_4:C_4^2$ x 3, $C_4^2:C_2^2$ x 3, $C_2^2\times C_4^2$ x 2, $C_4^3$

Order 48: $C_2^2\times C_{12}$ x 49, $C_4\times C_{12}$ x 31, $C_2^2:C_{12}$ x 24, $C_6\times D_4$ x 12, $C_4\times A_4$ x 12, $C_4:C_{12}$ x 12, $C_2\times S_4$ x 6, $A_4:C_4$ x 6, $C_4\times D_6$ x 3, $C_{12}:C_4$ x 3, $C_2^3\times C_6$ x 2, $C_2^2\times A_4$ x 2

Order 36: $C_3\times C_{12}$ x 6, $C_3:C_{12}$ x 6, $C_6\times S_3$ x 6, $C_6^2$, $C_3\times A_4$

Order 32: $C_4\times D_4$ x 24, $C_2\times C_4^2$ x 17, $C_2^3\times C_4$ x 6, $C_2^3:C_4$ x 6, $C_2.C_4^2$ x 6, $C_2^2.D_4$ x 3, $C_2^2\times D_4$

Order 24: $C_2\times C_{12}$ x 87, $C_2^2\times C_6$ x 17, $C_3\times D_4$ x 16, $C_4\times S_3$ x 12, $C_2\times A_4$ x 6, $S_4$ x 4, $C_6:C_4$ x 3, $C_2\times D_6$

Order 18: $C_3\times S_3$ x 4, $C_3\times C_6$ x 3

Order 16: $C_2^2\times C_4$ x 49, $C_4^2$ x 29, $C_2^2:C_4$ x 24, $C_4:C_4$ x 12, $C_2\times D_4$ x 12, $C_2^4$ x 2

Order 12: $C_{12}$ x 40, $C_2\times C_6$ x 30, $D_6$ x 6, $C_3:C_4$ x 6, $A_4$ x 2

Order 9: $C_3^2$

Order 8: $C_2\times C_4$ x 81, $C_2^3$ x 17, $D_4$ x 16

Order 6: $C_6$ x 17, $S_3$ x 4

Order 4: $C_2^2$ x 28, $C_4$ x 28

Order 3: $C_3$ x 3

Order 2: $C_2$ x 11

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1152: $C_4\times C_{12}\times S_4$

Order 576: $C_2\times C_{12}\times S_4$, $C_2^2:C_{12}^2$, $C_4\times A_4:C_{12}$

Order 384: $C_4^2\times S_4$, $C_4^3:C_6$

Order 288: $C_2^3.C_6^2$, $C_2\times A_4:C_{12}$, $C_{12}\times S_4$, $C_{12}^2:C_2$, $C_2\times C_6\times S_4$

Order 192: $A_4\times C_4^2$ x 2, $C_4\times C_2^2:C_{12}$ x 2, $C_2^2\times C_4\times C_{12}$ x 2, $A_4:C_4^2$, $C_{12}:C_4^2$, $C_4^2\times C_{12}$, $C_2^4.D_6$, $C_2\times D_4\times C_{12}$

Order 144: $C_{12}:C_{12}$, $C_2^2:C_6^2$, $C_6\times S_4$, $C_{12}\times D_6$, $C_{12}\times A_4$, $A_4:C_{12}$, $C_{12}^2$

Order 128: $D_4\times C_4^2$

Order 96: $C_2\times C_4\times C_{12}$ x 7, $C_2^3\times C_{12}$ x 2, $C_2^3:C_{12}$ x 2, $D_4\times C_{12}$ x 2, $C_2^3:C_{12}$ x 2, $S_3\times C_4^2$, $C_6.C_4^2$, $C_2^2\times S_4$, $C_{12}:C_2^3$, $C_2^2.S_4$, $C_4\times S_4$, $C_2\times C_4:C_{12}$

Order 72: $C_6\times D_6$, $C_6\times A_4$, $C_3\times S_4$, $C_6\times C_{12}$, $C_6:C_{12}$, $S_3\times C_{12}$

Order 64: $C_2^2:C_4^2$ x 2, $C_2^2\times C_4^2$ x 2, $C_4:C_4^2$, $C_4^3$, $C_4^2:C_2^2$

Order 48: $C_2^2\times C_{12}$ x 11, $C_4\times C_{12}$ x 9, $C_2^2:C_{12}$ x 3, $C_2^3\times C_6$ x 2, $C_2^2\times A_4$ x 2, $C_6\times D_4$ x 2, $C_4\times A_4$ x 2, $C_2\times S_4$, $C_4\times D_6$, $A_4:C_4$, $C_4:C_{12}$, $C_{12}:C_4$

Order 36: $C_3\times C_{12}$, $C_3:C_{12}$, $C_6^2$, $C_6\times S_3$, $C_3\times A_4$

Order 32: $C_2\times C_4^2$ x 7, $C_2^3\times C_4$ x 2, $C_4\times D_4$ x 2, $C_2^3:C_4$ x 2, $C_2^2\times D_4$, $C_2^2.D_4$, $C_2.C_4^2$

Order 24: $C_2\times C_{12}$ x 16, $C_2^2\times C_6$ x 6, $C_2\times A_4$ x 2, $C_3\times D_4$ x 2, $C_6:C_4$, $C_4\times S_3$, $C_2\times D_6$, $S_4$

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

Order 16: $C_2^2\times C_4$ x 11, $C_4^2$ x 7, $C_2^2:C_4$ x 3, $C_2^4$ x 2, $C_2\times D_4$ x 2, $C_4:C_4$

Order 12: $C_2\times C_6$ x 11, $C_{12}$ x 7, $A_4$ x 2, $D_6$, $C_3:C_4$

Order 9: $C_3^2$

Order 8: $C_2\times C_4$ x 14, $C_2^3$ x 6, $D_4$ x 2

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

Order 4: $C_2^2$ x 9, $C_4$ x 5

Order 3: $C_3$ x 3

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1152: $C_4\times C_{12}\times S_4$ ($C_1$)

Order 576: $C_2\times C_{12}\times S_4$ ($C_2$) x 3, $C_4\times A_4:C_{12}$ ($C_2$) x 3, $C_2^2:C_{12}^2$ ($C_2$)

Order 384: $C_4^2\times S_4$ ($C_3$)

Order 288: $C_{12}\times S_4$ ($C_4$) x 12, $C_2^3.C_6^2$ ($C_2^2$) x 3, $C_2\times A_4:C_{12}$ ($C_2^2$) x 3, $C_2\times C_6\times S_4$ ($C_2^2$)

Order 192: $A_4:C_4^2$ ($C_6$) x 3, $C_2^4.D_6$ ($C_6$) x 3, $A_4\times C_4^2$ ($C_6$), $C_2^2\times C_4\times C_{12}$ ($S_3$)

Order 144: $C_6\times S_4$ ($C_2\times C_4$) x 6, $C_{12}\times A_4$ ($C_2\times C_4$) x 6, $A_4:C_{12}$ ($C_2\times C_4$) x 6, $C_2^2:C_6^2$ ($C_2^3$)

Order 96: $C_4\times S_4$ ($C_{12}$) x 12, $C_2^3\times C_{12}$ ($D_6$) x 3, $C_2^3:C_{12}$ ($C_2\times C_6$) x 3, $C_2^2.S_4$ ($C_2\times C_6$) x 3, $C_2^2\times S_4$ ($C_2\times C_6$)

Order 72: $C_3\times S_4$ ($C_4^2$) x 4, $C_6\times A_4$ ($C_2^2\times C_4$) x 3

Order 64: $C_2^2\times C_4^2$ ($C_3\times S_3$)

Order 48: $C_2\times S_4$ ($C_2\times C_{12}$) x 6, $C_2^2\times C_{12}$ ($C_4\times S_3$) x 6, $C_4\times A_4$ ($C_2\times C_{12}$) x 6, $A_4:C_4$ ($C_2\times C_{12}$) x 6, $C_2^3\times C_6$ ($C_2\times D_6$), $C_2^2\times A_4$ ($C_2^2\times C_6$), $C_4\times C_{12}$ ($S_4$)

Order 36: $C_3\times A_4$ ($C_2\times C_4^2$)

Order 32: $C_2^3\times C_4$ ($C_6\times S_3$) x 3

Order 24: $S_4$ ($C_4\times C_{12}$) x 4, $C_2\times C_{12}$ ($C_2\times S_4$) x 3, $C_2^2\times C_6$ ($C_4\times D_6$) x 3, $C_2\times A_4$ ($C_2^2\times C_{12}$) x 3

Order 16: $C_2^2\times C_4$ ($S_3\times C_{12}$) x 6, $C_4^2$ ($C_3\times S_4$), $C_2^4$ ($C_6\times D_6$)

Order 12: $C_{12}$ ($C_4\times S_4$) x 6, $C_2\times C_6$ ($S_3\times C_4^2$), $C_2\times C_6$ ($C_2^2\times S_4$), $A_4$ ($C_2\times C_4\times C_{12}$)

Order 8: $C_2^3$ ($C_{12}\times D_6$) x 3, $C_2\times C_4$ ($C_6\times S_4$) x 3

Order 6: $C_6$ ($C_2^4.D_6$) x 3

Order 4: $C_4$ ($C_{12}\times S_4$) x 6, $C_2^2$ ($C_{12}^2:C_2$), $C_2^2$ ($C_2\times C_6\times S_4$)

Order 3: $C_3$ ($C_4^2\times S_4$)

Order 2: $C_2$ ($C_2\times C_{12}\times S_4$) x 3

Order 1: $C_1$ ($C_4\times C_{12}\times S_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1152: $C_4\times C_{12}\times S_4$ ($C_1$)

Order 576: $C_2\times C_{12}\times S_4$ ($C_2$), $C_2^2:C_{12}^2$ ($C_2$), $C_4\times A_4:C_{12}$ ($C_2$)

Order 384: $C_4^2\times S_4$ ($C_3$)

Order 288: $C_2^3.C_6^2$ ($C_2^2$), $C_2\times A_4:C_{12}$ ($C_2^2$), $C_{12}\times S_4$ ($C_4$), $C_2\times C_6\times S_4$ ($C_2^2$)

Order 192: $A_4\times C_4^2$ ($C_6$), $A_4:C_4^2$ ($C_6$), $C_2^4.D_6$ ($C_6$), $C_2^2\times C_4\times C_{12}$ ($S_3$)

Order 144: $C_2^2:C_6^2$ ($C_2^3$), $C_6\times S_4$ ($C_2\times C_4$), $C_{12}\times A_4$ ($C_2\times C_4$), $A_4:C_{12}$ ($C_2\times C_4$)

Order 96: $C_2^2\times S_4$ ($C_2\times C_6$), $C_2^3\times C_{12}$ ($D_6$), $C_2^3:C_{12}$ ($C_2\times C_6$), $C_2^2.S_4$ ($C_2\times C_6$), $C_4\times S_4$ ($C_{12}$)

Order 72: $C_6\times A_4$ ($C_2^2\times C_4$), $C_3\times S_4$ ($C_4^2$)

Order 64: $C_2^2\times C_4^2$ ($C_3\times S_3$)

Order 48: $C_2^3\times C_6$ ($C_2\times D_6$), $C_2^2\times A_4$ ($C_2^2\times C_6$), $C_2\times S_4$ ($C_2\times C_{12}$), $C_2^2\times C_{12}$ ($C_4\times S_3$), $C_4\times A_4$ ($C_2\times C_{12}$), $A_4:C_4$ ($C_2\times C_{12}$), $C_4\times C_{12}$ ($S_4$)

Order 36: $C_3\times A_4$ ($C_2\times C_4^2$)

Order 32: $C_2^3\times C_4$ ($C_6\times S_3$)

Order 24: $C_2\times C_{12}$ ($C_2\times S_4$), $C_2^2\times C_6$ ($C_4\times D_6$), $C_2\times A_4$ ($C_2^2\times C_{12}$), $S_4$ ($C_4\times C_{12}$)

Order 16: $C_4^2$ ($C_3\times S_4$), $C_2^4$ ($C_6\times D_6$), $C_2^2\times C_4$ ($S_3\times C_{12}$)

Order 12: $C_2\times C_6$ ($S_3\times C_4^2$), $C_2\times C_6$ ($C_2^2\times S_4$), $A_4$ ($C_2\times C_4\times C_{12}$), $C_{12}$ ($C_4\times S_4$)

Order 8: $C_2^3$ ($C_{12}\times D_6$), $C_2\times C_4$ ($C_6\times S_4$)

Order 6: $C_6$ ($C_2^4.D_6$)

Order 4: $C_2^2$ ($C_{12}^2:C_2$), $C_2^2$ ($C_2\times C_6\times S_4$), $C_4$ ($C_{12}\times S_4$)

Order 3: $C_3$ ($C_4^2\times S_4$)

Order 2: $C_2$ ($C_2\times C_{12}\times S_4$)

Order 1: $C_1$ ($C_4\times C_{12}\times S_4$)

Series

Derived series

$C_4\times C_{12}\times S_4$

$\rhd$

$A_4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Chief series

$C_4\times C_{12}\times S_4$

$\rhd$

$C_2\times C_{12}\times S_4$

$\rhd$

$C_2\times C_6\times S_4$

$\rhd$

$C_2^2:C_6^2$

$\rhd$

$C_2^2\times A_4$

$\rhd$

$C_2\times A_4$

$\rhd$

$A_4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Lower central series

$C_4\times C_{12}\times S_4$

$\rhd$

$A_4$

Upper central series

$C_1$

$\lhd$

$C_4\times C_{12}$

Supergroups

This group is a maximal subgroup of 2 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 $240 \times 240$ 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 $100 \times 100$ rational character table.