Abstract group 4320.ch: $C_6:S_3\times S_5$

• Label: 4320.ch
• Order: \( 2^{5} \cdot 3^{3} \cdot 5 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3^{4} \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \cdot 3 \)
• Perm deg.: $13$
• Trans deg.: $90$
• Rank: $3$

Group information

Description:	$C_6:S_3\times S_5$
Order:	\(4320\)\(\medspace = 2^{5} \cdot 3^{3} \cdot 5 \)
Exponent:	\(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \)
Automorphism group:	$C_2^2\times \AGL(2,3).S_5$, of order \(207360\)\(\medspace = 2^{9} \cdot 3^{4} \cdot 5 \)
Composition factors:	$C_2$ x 3, $C_3$ x 2, $A_5$
Derived length:	$2$

This group is nonabelian, nonsolvable, and rational.

Group statistics

Order	1	2	3	4	5	6	10	12	15	30
Elements	1	519	188	600	24	1668	456	480	192	192	4320
Conjugacy classes	1	11	9	4	1	39	3	8	4	4	84
Divisions	1	11	9	4	1	39	3	8	4	4	84
Autjugacy classes	1	7	3	2	1	10	2	1	1	1	29

Dimension	1	2	4	5	6	8	10	12
Irr. complex chars.	8	16	8	8	4	16	16	8	84
Irr. rational chars.	8	16	8	8	4	16	16	8	84

Minimal presentations

Permutation degree:	$13$
Transitive degree:	$90$
Rank:	$3$
Inequivalent generating triples:	not computed

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

Permutation group:	Degree $13$ $\langle(6,7,8)(11,12,13), (1,2)(7,8)(12,13), (6,8)(11,12), (9,10), (11,12,13), (1,3)(2,4,5)(6,7)(9,10)(11,12)\rangle$
Direct product:	$C_2$ $\, \times\, $ $(C_3:S_3)$ $\, \times\, $ $S_5$
Semidirect product:	$(C_6\times S_5)$ $\,\rtimes\,$ $S_3$	$C_6$ $\,\rtimes\,$ $(S_3\times S_5)$	$(C_3\times S_5)$ $\,\rtimes\,$ $D_6$	$C_3$ $\,\rtimes\,$ $(D_6\times S_5)$	all 18
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

Elements of the group are displayed as permutations of degree 13.

Homology

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

Subgroups

There are 56936 subgroups in 1518 conjugacy classes, 56 normal (14 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $\GL(2,4):D_6$
Commutator:	$G' \simeq$ $C_3\times \GL(2,4)$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_6:S_3\times S_5$
Fitting:	$\operatorname{Fit} \simeq$ $C_3\times C_6$	$G/\operatorname{Fit} \simeq$ $C_2\times S_5$
Radical:	$R \simeq$ $C_6:S_3$	$G/R \simeq$ $S_5$
Socle:	$\operatorname{soc} \simeq$ $C_6\times \GL(2,4)$	$G/\operatorname{soc} \simeq$ $C_2^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2\times D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^3$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$

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

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Normal subgroups

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Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 4320: $C_6:S_3\times S_5$

Order 2160: $\GL(2,4):D_6$ x 4, $A_5:C_6^2$, $\GL(2,4):D_6$, $\GL(2,4):D_6$

Order 1440: $D_6\times S_5$ x 4

Order 1080: $\GL(2,4):S_3$ x 2, $C_3^2\times S_5$ x 2, $C_3^2:S_5$ x 2, $C_6\times \GL(2,4)$

Order 864: $C_6:S_3\times S_4$

Order 720: $S_3\times S_5$ x 16, $D_6\times A_5$ x 4, $C_6:S_5$ x 4, $C_6\times S_5$ x 4, $C_6:S_3\times F_5$

Order 540: $C_3\times \GL(2,4)$

Order 480: $C_2^2\times S_5$

Order 432: $C_6^2:D_6$ x 4, $C_3:D_6^2$, $C_6:S_3\times A_4$, $C_6^2:D_6$, $A_4:C_6^2$

Order 360: $S_3\times A_5$ x 8, $C_3:S_5$ x 8, $C_3\times S_5$ x 8, $C_3:S_3\times F_5$ x 4, $C_6\times A_5$ x 4, $C_{30}:D_6$, $C_2\times C_3^2:F_5$, $C_{30}:C_{12}$

Order 288: $D_6\times S_4$ x 4, $C_6^2:C_2^3$

Order 240: $C_2\times S_5$ x 6, $D_6\times F_5$ x 4, $C_2^2\times A_5$

Order 216: $C_6:S_3^2$ x 12, $C_6^2:C_6$ x 2, $C_3^2:S_4$ x 2, $C_3^2\times S_4$ x 2, $C_6^2:S_3$, $C_6^2:C_6$, $S_3\times C_6^2$, $C_6^2:C_6$

Order 180: $C_{15}:D_6$ x 4, $\GL(2,4)$ x 4, $C_3^2:F_5$ x 2, $C_3^2\times F_5$ x 2, $C_3:D_{30}$, $C_{30}:S_3$, $C_3^2\times D_{10}$

Order 144: $S_3\times S_4$ x 16, $C_6:S_4$ x 8, $C_{12}:D_6$ x 8, $D_6^2$ x 4, $A_4\times D_6$ x 4, $C_6\times S_4$ x 4, $C_6^2:C_2^2$ x 2, $C_6^2:C_2^2$ x 2, $C_4:C_6^2$, $C_6:D_{12}$, $C_{12}:D_6$

Order 120: $S_3\times F_5$ x 16, $S_3\times D_{10}$ x 4, $C_{30}:C_4$ x 4, $C_6\times F_5$ x 4, $S_5$ x 4, $C_2\times A_5$ x 3

Order 108: $C_3:S_3^2$ x 16, $C_3^2:D_6$ x 6, $C_3^2:D_6$ x 6, $C_3^2\times D_6$ x 6, $C_3\times C_6^2$, $C_3^2\times A_4$

Order 96: $D_4\times D_6$ x 4, $C_2^2\times S_4$

Order 90: $C_3:D_{15}$ x 2, $C_{15}:S_3$ x 2, $C_3^2\times D_5$ x 2, $C_3\times C_{30}$

Order 80: $C_2^2\times F_5$

Order 72: $S_3\times D_6$ x 48, $C_6:D_6$ x 23, $C_3:S_4$ x 16, $C_6\times D_6$ x 8, $C_6\times A_4$ x 8, $S_3\times A_4$ x 8, $C_3\times S_4$ x 8, $C_6^2:C_2$ x 8, $D_4\times C_3^2$ x 4, $C_3:D_{12}$ x 4, $C_{12}:S_3$ x 4, $C_2\times C_6^2$ x 2, $C_6\times C_{12}$, $C_6.D_6$

Order 60: $S_3\times D_5$ x 16, $C_{15}:C_4$ x 8, $C_3\times F_5$ x 8, $D_{30}$ x 4, $S_3\times C_{10}$ x 4, $C_3\times D_{10}$ x 4, $A_5$

Order 54: $C_3^2:S_3$ x 4, $C_3^2:C_6$ x 4, $S_3\times C_3^2$ x 4, $C_3^2\times C_6$ x 3

Order 48: $S_3\times D_4$ x 32, $C_2\times S_4$ x 10, $C_2^2\times D_6$ x 9, $C_6:D_4$ x 8, $C_6\times D_4$ x 4, $C_2\times D_{12}$ x 4, $C_4\times D_6$ x 4, $C_2^2\times A_4$

Order 45: $C_3\times C_{15}$

Order 40: $C_2\times F_5$ x 6, $C_2\times D_{10}$

Order 36: $C_6:S_3$ x 69, $S_3^2$ x 64, $C_6\times S_3$ x 48, $C_6^2$ x 15, $C_3\times A_4$ x 8, $C_3\times C_{12}$ x 2, $C_3^2:C_4$ x 2

Order 32: $C_2^2\times D_4$

Order 30: $D_{15}$ x 8, $C_3\times D_5$ x 8, $C_5\times S_3$ x 8, $C_{30}$ x 4

Order 27: $C_3^3$

Order 24: $C_2\times D_6$ x 78, $C_3:D_4$ x 32, $D_{12}$ x 16, $C_4\times S_3$ x 16, $C_3\times D_4$ x 16, $S_4$ x 12, $C_2^2\times C_6$ x 9, $C_2\times A_4$ x 7, $C_2\times C_{12}$ x 4, $C_6:C_4$ x 4

Order 20: $D_{10}$ x 6, $F_5$ x 4, $C_2\times C_{10}$

Order 18: $C_3:S_3$ x 38, $C_3\times S_3$ x 32, $C_3\times C_6$ x 29

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

Order 15: $C_{15}$ x 4

Order 12: $D_6$ x 136, $C_2\times C_6$ x 39, $C_{12}$ x 8, $C_3:C_4$ x 8, $A_4$ x 5

Order 10: $D_5$ x 4, $C_{10}$ x 3

Order 9: $C_3^2$ x 9

Order 8: $C_2^3$ x 17, $D_4$ x 16, $C_2\times C_4$ x 6

Order 6: $S_3$ x 48, $C_6$ x 39

Order 5: $C_5$

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

Order 3: $C_3$ x 9

Order 2: $C_2$ x 11

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 4320: $C_6:S_3\times S_5$

Order 2160: $A_5:C_6^2$, $\GL(2,4):D_6$, $\GL(2,4):D_6$, $\GL(2,4):D_6$

Order 1440: $D_6\times S_5$

Order 1080: $C_6\times \GL(2,4)$, $\GL(2,4):S_3$, $C_3^2\times S_5$, $C_3^2:S_5$

Order 864: $C_6:S_3\times S_4$

Order 720: $C_6:S_3\times F_5$, $D_6\times A_5$, $C_6:S_5$, $C_6\times S_5$, $S_3\times S_5$

Order 540: $C_3\times \GL(2,4)$

Order 480: $C_2^2\times S_5$

Order 432: $C_3:D_6^2$, $C_6:S_3\times A_4$, $C_6^2:D_6$, $A_4:C_6^2$, $C_6^2:D_6$

Order 360: $C_{30}:D_6$, $C_2\times C_3^2:F_5$, $C_{30}:C_{12}$, $C_3:S_3\times F_5$, $C_6\times A_5$, $S_3\times A_5$, $C_3:S_5$, $C_3\times S_5$

Order 288: $D_6\times S_4$, $C_6^2:C_2^3$

Order 240: $C_2\times S_5$ x 3, $D_6\times F_5$, $C_2^2\times A_5$

Order 216: $C_6:S_3^2$ x 6, $C_6^2:S_3$, $C_6^2:C_6$, $S_3\times C_6^2$, $C_6^2:C_6$, $C_6^2:C_6$, $C_3^2:S_4$, $C_3^2\times S_4$

Order 180: $C_{15}:D_6$ x 2, $C_3:D_{30}$, $C_{30}:S_3$, $C_3^2\times D_{10}$, $C_3^2:F_5$, $C_3^2\times F_5$, $\GL(2,4)$

Order 144: $C_6^2:C_2^2$ x 2, $C_6:S_4$ x 2, $C_6^2:C_2^2$ x 2, $C_{12}:D_6$ x 2, $D_6^2$, $A_4\times D_6$, $C_6\times S_4$, $S_3\times S_4$, $C_4:C_6^2$, $C_6:D_{12}$, $C_{12}:D_6$

Order 120: $C_2\times A_5$ x 2, $S_5$ x 2, $S_3\times D_{10}$, $C_{30}:C_4$, $C_6\times F_5$, $S_3\times F_5$

Order 108: $C_3:S_3^2$ x 6, $C_3^2\times D_6$ x 4, $C_3^2:D_6$ x 3, $C_3^2:D_6$ x 3, $C_3\times C_6^2$, $C_3^2\times A_4$

Order 96: $C_2^2\times S_4$, $D_4\times D_6$

Order 90: $C_3^2\times D_5$ x 2, $C_3:D_{15}$, $C_{15}:S_3$, $C_3\times C_{30}$

Order 80: $C_2^2\times F_5$

Order 72: $C_6:D_6$ x 11, $S_3\times D_6$ x 6, $C_6^2:C_2$ x 3, $C_2\times C_6^2$ x 2, $C_6\times D_6$ x 2, $C_6\times A_4$ x 2, $C_3:S_4$ x 2, $D_4\times C_3^2$ x 2, $S_3\times A_4$, $C_3\times S_4$, $C_6\times C_{12}$, $C_6.D_6$, $C_3:D_{12}$, $C_{12}:S_3$

Order 60: $S_3\times D_5$ x 2, $C_{15}:C_4$, $C_3\times F_5$, $A_5$, $D_{30}$, $S_3\times C_{10}$, $C_3\times D_{10}$

Order 54: $S_3\times C_3^2$ x 3, $C_3^2\times C_6$ x 2, $C_3^2:S_3$ x 2, $C_3^2:C_6$ x 2

Order 48: $C_2\times S_4$ x 4, $C_2^2\times D_6$ x 3, $C_6:D_4$ x 2, $S_3\times D_4$ x 2, $C_2^2\times A_4$, $C_6\times D_4$, $C_2\times D_{12}$, $C_4\times D_6$

Order 45: $C_3\times C_{15}$

Order 40: $C_2\times F_5$ x 3, $C_2\times D_{10}$

Order 36: $C_6:S_3$ x 17, $C_6^2$ x 8, $C_6\times S_3$ x 7, $S_3^2$ x 6, $C_3\times A_4$ x 2, $C_3\times C_{12}$, $C_3^2:C_4$

Order 32: $C_2^2\times D_4$

Order 30: $C_3\times D_5$ x 2, $C_{30}$, $D_{15}$, $C_5\times S_3$

Order 27: $C_3^3$

Order 24: $C_2\times D_6$ x 18, $C_3:D_4$ x 3, $C_2^2\times C_6$ x 3, $C_2\times A_4$ x 3, $S_4$ x 3, $C_3\times D_4$ x 2, $C_2\times C_{12}$, $C_6:C_4$, $D_{12}$, $C_4\times S_3$

Order 20: $D_{10}$ x 4, $F_5$ x 2, $C_2\times C_{10}$

Order 18: $C_3\times C_6$ x 8, $C_3:S_3$ x 7, $C_3\times S_3$ x 5

Order 16: $C_2\times D_4$ x 6, $C_2^4$ x 2, $C_2^2\times C_4$

Order 15: $C_{15}$

Order 12: $D_6$ x 27, $C_2\times C_6$ x 11, $A_4$ x 2, $C_{12}$, $C_3:C_4$

Order 10: $D_5$ x 3, $C_{10}$ x 2

Order 9: $C_3^2$ x 3

Order 8: $C_2^3$ x 11, $D_4$ x 6, $C_2\times C_4$ x 3

Order 6: $C_6$ x 10, $S_3$ x 10

Order 5: $C_5$

Order 4: $C_2^2$ x 17, $C_4$ x 2

Order 3: $C_3$ x 3

Order 2: $C_2$ x 7

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 4320: $C_6:S_3\times S_5$ ($C_1$)

Order 2160: $\GL(2,4):D_6$ ($C_2$) x 4, $A_5:C_6^2$ ($C_2$), $\GL(2,4):D_6$ ($C_2$), $\GL(2,4):D_6$ ($C_2$)

Order 1080: $\GL(2,4):S_3$ ($C_2^2$) x 2, $C_3^2\times S_5$ ($C_2^2$) x 2, $C_3^2:S_5$ ($C_2^2$) x 2, $C_6\times \GL(2,4)$ ($C_2^2$)

Order 720: $C_6\times S_5$ ($S_3$) x 4

Order 540: $C_3\times \GL(2,4)$ ($C_2^3$)

Order 360: $C_3\times S_5$ ($D_6$) x 8, $C_6\times A_5$ ($D_6$) x 4

Order 240: $C_2\times S_5$ ($C_3:S_3$)

Order 180: $\GL(2,4)$ ($C_2\times D_6$) x 4

Order 120: $S_5$ ($C_6:S_3$) x 2, $C_2\times A_5$ ($C_6:S_3$)

Order 60: $A_5$ ($C_6:D_6$)

Order 36: $C_6:S_3$ ($S_5$)

Order 18: $C_3:S_3$ ($C_2\times S_5$) x 2, $C_3\times C_6$ ($C_2\times S_5$)

Order 9: $C_3^2$ ($C_2^2\times S_5$)

Order 6: $C_6$ ($S_3\times S_5$) x 4

Order 3: $C_3$ ($D_6\times S_5$) x 4

Order 2: $C_2$ ($\GL(2,4):D_6$)

Order 1: $C_1$ ($C_6:S_3\times S_5$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 4320: $C_6:S_3\times S_5$ ($C_1$)

Order 2160: $A_5:C_6^2$ ($C_2$), $\GL(2,4):D_6$ ($C_2$), $\GL(2,4):D_6$ ($C_2$), $\GL(2,4):D_6$ ($C_2$)

Order 1080: $C_6\times \GL(2,4)$ ($C_2^2$), $\GL(2,4):S_3$ ($C_2^2$), $C_3^2\times S_5$ ($C_2^2$), $C_3^2:S_5$ ($C_2^2$)

Order 720: $C_6\times S_5$ ($S_3$)

Order 540: $C_3\times \GL(2,4)$ ($C_2^3$)

Order 360: $C_6\times A_5$ ($D_6$), $C_3\times S_5$ ($D_6$)

Order 240: $C_2\times S_5$ ($C_3:S_3$)

Order 180: $\GL(2,4)$ ($C_2\times D_6$)

Order 120: $C_2\times A_5$ ($C_6:S_3$), $S_5$ ($C_6:S_3$)

Order 60: $A_5$ ($C_6:D_6$)

Order 36: $C_6:S_3$ ($S_5$)

Order 18: $C_3\times C_6$ ($C_2\times S_5$), $C_3:S_3$ ($C_2\times S_5$)

Order 9: $C_3^2$ ($C_2^2\times S_5$)

Order 6: $C_6$ ($S_3\times S_5$)

Order 3: $C_3$ ($D_6\times S_5$)

Order 2: $C_2$ ($\GL(2,4):D_6$)

Order 1: $C_1$ ($C_6:S_3\times S_5$)

Series

Derived series

$C_6:S_3\times S_5$

$\rhd$

$C_3\times \GL(2,4)$

$\rhd$

$A_5$

Chief series

$C_6:S_3\times S_5$

$\rhd$

$\GL(2,4):D_6$

$\rhd$

$C_6:S_3$

$\rhd$

$C_3:S_3$

$\rhd$

$C_3^2$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_6:S_3\times S_5$

$\rhd$

$C_3\times \GL(2,4)$

Upper central series

$C_1$

$\lhd$

$C_2$

Supergroups

This group is a maximal subgroup of 22 larger groups in the database.

This group is a maximal quotient of 10 larger groups in the database.

Character theory

Complex character table

Every character has rational values, so the complex character table is the same as the rational character table below.

Rational character table

See the $84 \times 84$ rational character table. Alternatively, you may search for characters of this group with desired properties.