Abstract group 720.769: $C_6\times S_5$

• Label: 720.769
• Order: \( 2^{4} \cdot 3^{2} \cdot 5 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3 \)
• $\card{Z(G)}$: \( 2 \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{5} \cdot 3 \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \)
• Perm deg.: $10$
• Trans deg.: $30$
• Rank: $2$

Group information

Description:	$C_6\times S_5$
Order:	\(720\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \)
Exponent:	\(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \)
Automorphism group:	$C_2^2\times S_5$, of order \(480\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \)
Composition factors:	$C_2$ x 2, $C_3$, $A_5$
Derived length:	$1$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	5	6	10	12	15	30
Elements	1	51	62	60	24	282	24	120	48	48	720
Conjugacy classes	1	5	5	2	1	19	1	4	2	2	42
Divisions	1	5	3	2	1	11	1	2	1	1	28
Autjugacy classes	1	4	3	1	1	8	1	1	1	1	22

Dimension	1	2	4	5	6	8	10	12
Irr. complex chars.	12	0	12	12	6	0	0	0	42
Irr. rational chars.	4	4	4	4	2	4	4	2	28

Minimal presentations

Permutation degree:	$10$
Transitive degree:	$30$
Rank:	$2$
Inequivalent generating pairs:	$228$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	4	8	8
Arbitrary	4	6	6

Constructions

Permutation group:	Degree $10$ $\langle(1,2)(3,4,5)(7,8,10), (3,5,4)(6,9,8,10)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrrrr} -1 & -1 & -1 & -1 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrrrr} -1 & -1 & -1 & -1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 0 & 1 & 0 \end{array}\right) \right\rangle \subseteq \GL_{6}(\Z)$
Transitive group:	30T180	36T1243			more information
Direct product:	$C_2$ $\, \times\, $ $C_3$ $\, \times\, $ $S_5$
Semidirect product:	$(C_6\times A_5)$ $\,\rtimes\,$ $C_2$	$(C_2\times A_5)$ $\,\rtimes\,$ $C_6$	$A_5$ $\,\rtimes\,$ $(C_2\times C_6)$	$\GL(2,4)$ $\,\rtimes\,$ $C_2^2$	more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Aut. group:	$\Aut(C_7\times A_5)$	$\Aut(C_9\times A_5)$	$\Aut(C_7\times \SL(2,5))$	$\Aut(C_7\times S_5)$	all 8

Elements of the group are displayed as permutations of degree 10.

Homology

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

Subgroups

There are 1190 subgroups in 121 conjugacy classes, 14 normal (10 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_6$	$G/Z \simeq$ $S_5$
Commutator:	$G' \simeq$ $A_5$	$G/G' \simeq$ $C_2\times C_6$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_6\times S_5$
Fitting:	$\operatorname{Fit} \simeq$ $C_6$	$G/\operatorname{Fit} \simeq$ $S_5$
Radical:	$R \simeq$ $C_6$	$G/R \simeq$ $S_5$
Socle:	$\operatorname{soc} \simeq$ $C_6\times A_5$	$G/\operatorname{soc} \simeq$ $C_2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$

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 720: $C_6\times S_5$

Order 360: $C_3\times S_5$ x 2, $C_6\times A_5$

Order 240: $C_2\times S_5$

Order 180: $\GL(2,4)$

Order 144: $C_6\times S_4$

Order 120: $S_5$ x 2, $C_6\times F_5$, $C_2\times A_5$

Order 72: $C_3\times S_4$ x 2, $C_6\times D_6$, $C_6\times A_4$

Order 60: $C_3\times F_5$ x 2, $A_5$, $C_3\times D_{10}$

Order 48: $C_2\times S_4$, $C_6\times D_4$

Order 40: $C_2\times F_5$

Order 36: $C_6\times S_3$ x 6, $C_6^2$, $C_3\times A_4$

Order 30: $C_3\times D_5$ x 2, $C_{30}$

Order 24: $C_3\times D_4$ x 4, $C_2^2\times C_6$ x 2, $C_2\times A_4$ x 2, $S_4$ x 2, $C_2\times C_{12}$, $C_2\times D_6$

Order 20: $F_5$ x 2, $D_{10}$

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

Order 16: $C_2\times D_4$

Order 15: $C_{15}$

Order 12: $C_2\times C_6$ x 9, $D_6$ x 6, $A_4$ x 2, $C_{12}$ x 2

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

Order 9: $C_3^2$

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

Order 6: $C_6$ x 11, $S_3$ x 4

Order 5: $C_5$

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

Order 3: $C_3$ x 3

Order 2: $C_2$ x 5

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 720: $C_6\times S_5$

Order 360: $C_6\times A_5$, $C_3\times S_5$

Order 240: $C_2\times S_5$

Order 180: $\GL(2,4)$

Order 144: $C_6\times S_4$

Order 120: $C_6\times F_5$, $C_2\times A_5$, $S_5$

Order 72: $C_6\times D_6$, $C_6\times A_4$, $C_3\times S_4$

Order 60: $C_3\times F_5$, $A_5$, $C_3\times D_{10}$

Order 48: $C_2\times S_4$, $C_6\times D_4$

Order 40: $C_2\times F_5$

Order 36: $C_6\times S_3$ x 4, $C_6^2$, $C_3\times A_4$

Order 30: $C_3\times D_5$ x 2, $C_{30}$

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

Order 20: $D_{10}$, $F_5$

Order 18: $C_3\times S_3$ x 3, $C_3\times C_6$ x 2

Order 16: $C_2\times D_4$

Order 15: $C_{15}$

Order 12: $C_2\times C_6$ x 8, $D_6$ x 4, $A_4$ x 2, $C_{12}$

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

Order 9: $C_3^2$

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

Order 6: $C_6$ x 8, $S_3$ x 3

Order 5: $C_5$

Order 4: $C_2^2$ x 6, $C_4$

Order 3: $C_3$ x 3

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 720: $C_6\times S_5$ ($C_1$)

Order 360: $C_3\times S_5$ ($C_2$) x 2, $C_6\times A_5$ ($C_2$)

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

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

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

Order 60: $A_5$ ($C_2\times C_6$)

Order 6: $C_6$ ($S_5$)

Order 3: $C_3$ ($C_2\times S_5$)

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

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

Normal subgroups up to automorphism (quotient in parentheses)

Order 720: $C_6\times S_5$ ($C_1$)

Order 360: $C_6\times A_5$ ($C_2$), $C_3\times S_5$ ($C_2$)

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

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

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

Order 60: $A_5$ ($C_2\times C_6$)

Order 6: $C_6$ ($S_5$)

Order 3: $C_3$ ($C_2\times S_5$)

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

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

Series

Derived series	$C_6\times S_5$	$\rhd$	$A_5$
Chief series	$C_6\times S_5$	$\rhd$	$C_6\times A_5$	$\rhd$	$C_6$	$\rhd$	$C_3$	$\rhd$	$C_1$
Lower central series	$C_6\times S_5$	$\rhd$	$A_5$
Upper central series	$C_1$	$\lhd$	$C_6$

Supergroups

This group is a maximal subgroup of 36 larger groups in the database.

This group is a maximal quotient of 34 larger groups in the database.

Character theory

Complex character table

See the $42 \times 42$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $28 \times 28$ rational character table.