Abstract group 20160.l: $C_2^2\times S_7$

• Label: 20160.l
• Order: \( 2^{6} \cdot 3^{2} \cdot 5 \cdot 7 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3^{3} \cdot 5 \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 3 \)
• Perm deg.: $11$
• Trans deg.: $28$
• Rank: $3$

Group information

Description:	$C_2^2\times S_7$
Order:	\(20160\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 5 \cdot 7 \)
Exponent:	\(420\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
Automorphism group:	$S_4\times S_7$, of order \(120960\)\(\medspace = 2^{7} \cdot 3^{3} \cdot 5 \cdot 7 \)
Composition factors:	$C_2$ x 3, $A_7$
Derived length:	$1$

This group is nonabelian, nonsolvable, and rational.

Group statistics

Order	1	2	3	4	5	6	7	10	12	14
Elements	1	927	350	3360	504	6930	720	3528	1680	2160	20160
Conjugacy classes	1	15	2	8	1	18	1	7	4	3	60
Divisions	1	15	2	8	1	18	1	7	4	3	60
Autjugacy classes	1	5	2	3	1	6	1	2	1	1	23

Dimension	1	6	14	15	20	21	35
Irr. complex chars.	8	8	16	8	4	8	8	60
Irr. rational chars.	8	8	16	8	4	8	8	60

Minimal presentations

Permutation degree:	$11$
Transitive degree:	$28$
Rank:	$3$
Inequivalent generating triples:	$21720132$

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 $11$ $\langle(1,2,4,6)(8,9)(10,11), (1,3,2,5,4)(8,10)(9,11), (1,3,6,4,5,7,2)(8,11)(9,10)\rangle$
Transitive group:	28T431				more information
Direct product:	$C_2$ ${}^2$ $\, \times\, $ $S_7$
Semidirect product:	$A_7$ $\,\rtimes\,$ $C_2^3$	$(C_2^2\times A_7)$ $\,\rtimes\,$ $C_2$	$(C_2\times A_7)$ $\,\rtimes\,$ $C_2^2$		more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Aut. group:	$\Aut(S_3\times C_4^2:D_{12})$	$\Aut(C_2^5:(A_4\times S_3^2))$	$\Aut(C_{1405}:C_{10})$	$\Aut(C_2^8.F_8)$	all 17

Elements of the group are displayed as permutations of degree 11.

Homology

Abelianization:	$C_{2}^{3} $
Schur multiplier:	$C_{2}^{4}$
Commutator length:	$1$

Subgroups

There are 264796 subgroups in 1692 conjugacy classes, 21 normal (5 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $S_7$
Commutator:	$G' \simeq$ $A_7$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_2^2\times S_7$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^2$	$G/\operatorname{Fit} \simeq$ $S_7$
Radical:	$R \simeq$ $C_2^2$	$G/R \simeq$ $S_7$
Socle:	$\operatorname{soc} \simeq$ $C_2^2\times A_7$	$G/\operatorname{soc} \simeq$ $C_2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4\times C_2^3$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7$

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

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Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 20160: $C_2^2\times S_7$

Order 10080: $C_2\times S_7$ x 6, $C_2^2\times A_7$

Order 5040: $S_7$ x 4, $C_2\times A_7$ x 3

Order 2880: $C_2^2\times S_6$

Order 2520: $A_7$

Order 1440: $C_2\times S_6$ x 6, $C_2^2\times A_6$

Order 960: $C_2^3\times S_5$

Order 720: $S_6$ x 4, $C_2\times A_6$ x 3

Order 672: $C_2^2\times \GL(3,2)$

Order 576: $C_2^2:D_6^2$

Order 480: $C_2^2\times S_5$ x 15, $C_2^3\times A_5$

Order 360: $A_6$

Order 336: $C_2\times \GL(3,2)$ x 3

Order 288: $D_6\times S_4$ x 12, $C_2\times A_4\times D_6$, $C_2\times C_6:S_4$, $C_2\times C_6\times S_4$, $C_6^2:D_4$

Order 240: $C_2\times S_5$ x 34, $C_2^2\times A_5$ x 8

Order 192: $C_2^3\times S_4$ x 2, $C_{12}:C_2^4$

Order 168: $C_2^2\times F_7$, $\PSL(2,7)$

Order 160: $C_2^3\times F_5$

Order 144: $S_3\times S_4$ x 16, $S_3^2:C_2^2$ x 12, $A_4\times D_6$ x 6, $C_6:S_4$ x 6, $C_6\times S_4$ x 6, $D_6^2$ x 2, $C_2^2:C_6^2$, $C_6^2:C_4$

Order 120: $S_5$ x 12, $C_2\times A_5$ x 10

Order 96: $C_2^2\times S_4$ x 29, $D_4\times D_6$ x 24, $C_2^3\times D_6$ x 2, $C_2^3\times A_4$ x 2, $C_2^3:D_6$ x 2, $C_{12}:C_2^3$, $C_2^2\times D_{12}$, $C_{12}:C_2^3$

Order 84: $C_2\times F_7$ x 6, $C_{14}:C_6$

Order 80: $C_2^2\times F_5$ x 14, $C_2^2\times D_{10}$

Order 72: $S_3\times D_6$ x 18, $\SOPlus(4,2)$ x 16, $C_2\times C_3^2:C_4$ x 6, $S_3\times A_4$ x 4, $C_3:S_4$ x 4, $C_3\times S_4$ x 4, $C_6\times A_4$ x 3, $C_6\times D_6$ x 2, $C_6:D_6$

Order 64: $D_4\times C_2^3$

Order 60: $A_5$ x 2

Order 56: $C_2\times D_{14}$

Order 48: $S_3\times D_4$ x 64, $C_2\times S_4$ x 62, $C_2^2\times D_6$ x 36, $C_6:D_4$ x 24, $C_2^2\times A_4$ x 15, $C_6\times D_4$ x 12, $C_2\times D_{12}$ x 12, $C_4\times D_6$ x 12, $C_2^3\times C_6$ x 2, $C_2^2\times C_{12}$, $C_6.C_2^3$

Order 42: $F_7$ x 4, $C_7:C_6$ x 3

Order 40: $C_2\times F_5$ x 28, $C_2\times D_{10}$ x 14, $C_2^2\times C_{10}$

Order 36: $S_3^2$ x 20, $C_6\times S_3$ x 12, $C_6:S_3$ x 6, $C_3^2:C_4$ x 4, $C_6^2$, $C_3\times A_4$

Order 32: $C_2^2\times D_4$ x 28, $C_2^5$ x 2, $C_2^3\times C_4$

Order 28: $D_{14}$ x 6, $C_2\times C_{14}$

Order 24: $C_2\times D_6$ x 131, $C_3:D_4$ x 32, $S_4$ x 20, $C_2^2\times C_6$ x 18, $C_2\times A_4$ x 17, $D_{12}$ x 16, $C_4\times S_3$ x 16, $C_3\times D_4$ x 16, $C_2\times C_{12}$ x 6, $C_6:C_4$ x 6

Order 21: $C_7:C_3$

Order 20: $D_{10}$ x 28, $F_5$ x 8, $C_2\times C_{10}$ x 7

Order 18: $C_3\times S_3$ x 8, $C_3:S_3$ x 4, $C_3\times C_6$ x 3

Order 16: $C_2\times D_4$ x 112, $C_2^4$ x 29, $C_2^2\times C_4$ x 14

Order 14: $D_7$ x 4, $C_{14}$ x 3

Order 12: $D_6$ x 114, $C_2\times C_6$ x 35, $C_{12}$ x 4, $C_3:C_4$ x 4, $A_4$ x 3

Order 10: $D_5$ x 8, $C_{10}$ x 7

Order 9: $C_3^2$

Order 8: $C_2^3$ x 85, $D_4$ x 64, $C_2\times C_4$ x 28

Order 7: $C_7$

Order 6: $S_3$ x 20, $C_6$ x 18

Order 5: $C_5$

Order 4: $C_2^2$ x 65, $C_4$ x 8

Order 3: $C_3$ x 2

Order 2: $C_2$ x 15

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 20160: $C_2^2\times S_7$

Order 10080: $C_2^2\times A_7$, $C_2\times S_7$

Order 5040: $S_7$, $C_2\times A_7$

Order 2880: $C_2^2\times S_6$

Order 2520: $A_7$

Order 1440: $C_2^2\times A_6$, $C_2\times S_6$

Order 960: $C_2^3\times S_5$

Order 720: $C_2\times A_6$, $S_6$

Order 672: $C_2^2\times \GL(3,2)$

Order 576: $C_2^2:D_6^2$

Order 480: $C_2^2\times S_5$ x 5, $C_2^3\times A_5$

Order 360: $A_6$

Order 336: $C_2\times \GL(3,2)$

Order 288: $D_6\times S_4$ x 2, $C_2\times A_4\times D_6$, $C_2\times C_6:S_4$, $C_2\times C_6\times S_4$, $C_6^2:D_4$

Order 240: $C_2\times S_5$ x 6, $C_2^2\times A_5$ x 3

Order 192: $C_2^3\times S_4$ x 2, $C_{12}:C_2^4$

Order 168: $C_2^2\times F_7$, $\PSL(2,7)$

Order 160: $C_2^3\times F_5$

Order 144: $D_6^2$ x 2, $C_6:S_4$ x 2, $S_3^2:C_2^2$ x 2, $S_3\times S_4$ x 2, $C_2^2:C_6^2$, $C_6^2:C_4$, $A_4\times D_6$, $C_6\times S_4$

Order 120: $S_5$ x 4, $C_2\times A_5$ x 3

Order 96: $C_2^2\times S_4$ x 9, $D_4\times D_6$ x 4, $C_2^3\times D_6$ x 2, $C_2^3\times A_4$ x 2, $C_2^3:D_6$ x 2, $C_{12}:C_2^3$, $C_2^2\times D_{12}$, $C_{12}:C_2^3$

Order 84: $C_{14}:C_6$, $C_2\times F_7$

Order 80: $C_2^2\times F_5$ x 4, $C_2^2\times D_{10}$

Order 72: $S_3\times D_6$ x 4, $C_6\times D_6$ x 2, $C_2\times C_3^2:C_4$ x 2, $C_3:S_4$ x 2, $\SOPlus(4,2)$ x 2, $C_6:D_6$, $C_6\times A_4$, $S_3\times A_4$, $C_3\times S_4$

Order 64: $D_4\times C_2^3$

Order 60: $A_5$ x 2

Order 56: $C_2\times D_{14}$

Order 48: $C_2\times S_4$ x 12, $C_2^2\times D_6$ x 11, $C_6:D_4$ x 6, $C_2^2\times A_4$ x 5, $S_3\times D_4$ x 5, $C_2^3\times C_6$ x 2, $C_6\times D_4$ x 2, $C_2\times D_{12}$ x 2, $C_4\times D_6$ x 2, $C_2^2\times C_{12}$, $C_6.C_2^3$

Order 42: $C_7:C_6$, $F_7$

Order 40: $C_2\times F_5$ x 5, $C_2\times D_{10}$ x 4, $C_2^2\times C_{10}$

Order 36: $S_3^2$ x 4, $C_3^2:C_4$ x 2, $C_6:S_3$ x 2, $C_6\times S_3$ x 2, $C_6^2$, $C_3\times A_4$

Order 32: $C_2^2\times D_4$ x 8, $C_2^5$ x 2, $C_2^3\times C_4$

Order 28: $C_2\times C_{14}$, $D_{14}$

Order 24: $C_2\times D_6$ x 25, $S_4$ x 8, $C_3:D_4$ x 7, $C_2^2\times C_6$ x 7, $C_2\times A_4$ x 5, $C_6:C_4$ x 2, $D_{12}$ x 2, $C_4\times S_3$ x 2, $C_3\times D_4$ x 2, $C_2\times C_{12}$

Order 21: $C_7:C_3$

Order 20: $D_{10}$ x 5, $F_5$ x 3, $C_2\times C_{10}$ x 2

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

Order 16: $C_2\times D_4$ x 15, $C_2^4$ x 9, $C_2^2\times C_4$ x 4

Order 14: $C_{14}$, $D_7$

Order 12: $D_6$ x 21, $C_2\times C_6$ x 11, $A_4$ x 3, $C_3:C_4$ x 2, $C_{12}$

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

Order 9: $C_3^2$

Order 8: $C_2^3$ x 19, $D_4$ x 11, $C_2\times C_4$ x 5

Order 7: $C_7$

Order 6: $S_3$ x 7, $C_6$ x 6

Order 5: $C_5$

Order 4: $C_2^2$ x 17, $C_4$ x 3

Order 3: $C_3$ x 2

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 20160: $C_2^2\times S_7$ ($C_1$)

Order 10080: $C_2\times S_7$ ($C_2$) x 6, $C_2^2\times A_7$ ($C_2$)

Order 5040: $S_7$ ($C_2^2$) x 4, $C_2\times A_7$ ($C_2^2$) x 3

Order 2520: $A_7$ ($C_2^3$)

Order 4: $C_2^2$ ($S_7$)

Order 2: $C_2$ ($C_2\times S_7$) x 3

Order 1: $C_1$ ($C_2^2\times S_7$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 20160: $C_2^2\times S_7$ ($C_1$)

Order 10080: $C_2^2\times A_7$ ($C_2$), $C_2\times S_7$ ($C_2$)

Order 5040: $S_7$ ($C_2^2$), $C_2\times A_7$ ($C_2^2$)

Order 2520: $A_7$ ($C_2^3$)

Order 4: $C_2^2$ ($S_7$)

Order 2: $C_2$ ($C_2\times S_7$)

Order 1: $C_1$ ($C_2^2\times S_7$)

Series

Derived series

$C_2^2\times S_7$

$\rhd$

$C_2^2\times S_7$

$\rhd$

$A_7$

$\rhd$

$A_7$

Chief series

$C_2^2\times S_7$

$\rhd$

$C_2^2\times S_7$

$\rhd$

$C_2^2\times A_7$

$\rhd$

$C_2^2\times A_7$

$\rhd$

$C_2^2$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_2$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$C_2^2\times S_7$

$\rhd$

$C_2^2\times S_7$

$\rhd$

$A_7$

$\rhd$

$A_7$

Upper central series

$C_1$

$\lhd$

$C_1$

$\lhd$

$C_2^2$

$\lhd$

$C_2^2$

Supergroups

This group is a maximal subgroup of 17 larger groups in the database.

This group is a maximal quotient of 18 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 $60 \times 60$ rational character table. Alternatively, you may search for characters of this group with desired properties.