Abstract group 12700800.b: $A_7.S_7$

• Label: 12700800.b
• Order: \( 2^{7} \cdot 3^{4} \cdot 5^{2} \cdot 7^{2} \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2 \)
• $\card{Z(G)}$: 1
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3^{4} \cdot 5^{2} \cdot 7^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \)
• Perm deg.: $14$
• Trans deg.: $49$
• Rank: $2$

Group information

Description:	$A_7.S_7$
Order:	\(12700800\)\(\medspace = 2^{7} \cdot 3^{4} \cdot 5^{2} \cdot 7^{2} \)
Exponent:	\(420\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
Automorphism group:	$S_7\wr C_2$, of order \(50803200\)\(\medspace = 2^{9} \cdot 3^{4} \cdot 5^{2} \cdot 7^{2} \)
Composition factors:	$C_2$, $A_7$ x 2
Derived length:	$1$

This group is nonabelian, nonsolvable, and rational.

Group statistics

Order	1	2	3	4	5	6	7	10	12	14	15	20	21	28	30	35	42	60
Elements	1	27111	123200	627480	255024	2214240	519840	486864	2751840	151200	352800	846720	504000	907200	1481760	725760	302400	423360	12700800
Conjugacy classes	1	7	8	10	3	25	4	7	21	2	4	4	4	2	6	2	2	2	114
Divisions	1	7	8	10	3	25	4	7	21	2	4	4	4	2	6	2	2	2	114
Autjugacy classes	1	5	5	6	2	14	2	4	11	1	2	2	2	1	3	1	1	1	64

Dimension	1	6	14	15	20	21	35	36	84	90	120	126	196	200	210	225	280	294	300	315	420	441	490	525	700	735	1225
Irr. complex chars.	2	4	8	4	2	4	4	2	8	4	2	4	8	2	12	2	4	8	2	4	2	2	8	4	2	4	2	114
Irr. rational chars.	2	4	8	4	2	4	4	2	8	4	2	4	8	2	12	2	4	8	2	4	2	2	8	4	2	4	2	114

Minimal presentations

Permutation degree:	$14$
Transitive degree:	$49$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	36	36	36
Arbitrary	not computed	not computed	not computed

Constructions

Permutation group:	Degree $14$ $\langle(4,7,6)(11,14,13), (1,6,7,3,4)(10,12,13,11,14), (2,5,3), (1,2)(11,14,12,13) \!\cdots\! \rangle$
Direct product:	not computed
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$A_7$ . $S_7$ (2)	$A_7^2$ . $C_2$			more information

Elements of the group are displayed as permutations of degree 14.

Homology

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

Subgroups

There are 5 normal subgroups (3 characteristic).

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

Special subgroups

Center:	a subgroup isomorphic to $C_1$
Commutator:	a subgroup isomorphic to $A_7^2$
Frattini:	a subgroup isomorphic to $C_1$
Fitting:	not computed
Radical:	not computed
Socle:	not computed

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: inclusions not computed

Classes of subgroups up to automorphism

No diagram available: inclusions not computed

Normal subgroups

No diagram available: inclusions not computed

Normal subgroups up to automorphism

No diagram available: inclusions not computed

Classes of subgroups up to conjugation

All subgroups of index up to 12096 (order at least 1050) are shown, as well as all normal subgroups of any index.

Order 12700800: $A_7.S_7$

Order 6350400: $A_7^2$

Order 1814400: $A_6:S_7$ x 2

Order 907200: $A_6\times A_7$ x 2

Order 604800: $S_5\times S_7$ x 2

Order 423360: $A_7\times \GL(3,2)$ x 2

Order 362880: $C_3:(S_4\times S_7)$ x 2

Order 302400: $A_5:S_7$ x 4, $A_5\times S_7$ x 2, $S_5\times A_7$ x 2

Order 259200: $A_6:S_6$

Order 181440: $C_3:S_4\times A_7$ x 2, $C_3\times A_4:S_7$ x 2, $A_4\times C_3:S_7$ x 2, $A_7:\SOPlus(4,2)$ x 2

Order 151200: $A_5\times A_7$ x 4

Order 129600: $A_6^2$

Order 120960: $S_4\times S_7$ x 4, $C_3:(D_4\times S_7)$ x 2

Order 105840: $A_7:F_7$ x 2

Order 100800: $F_5\times S_7$ x 2

Order 90720: $A_7:S_3^2$ x 4, $C_3\times A_4\times A_7$ x 2, $C_3^2:C_4\times A_7$ x 2

Order 86400: $S_5\times S_6$ x 2

Order 60480: $S_4\times A_7$ x 6, $A_4:S_7$ x 4, $A_4\times S_7$ x 4, $C_3:D_4\times A_7$ x 2, $C_3:(C_4\times S_7)$ x 2, $(C_6\times S_7):C_2$ x 2, $D_6:S_7$ x 2, $C_3\times C_2^2:S_7$ x 2, $C_2\times C_6:S_7$ x 2, $A_6\times \GL(3,2)$ x 2, $D_6\times S_7$ x 2

Order 52920: $C_7:C_3\times A_7$ x 2

Order 51840: $C_3:(S_4\times S_6)$ x 2

Order 50400: $A_7:F_5$ x 2, $D_5\times S_7$ x 2, $F_5\times A_7$ x 2

Order 45360: $C_3^2:S_7$ x 4, $C_3:S_3\times A_7$ x 2

Order 43200: $A_5:S_6$ x 4, $A_5\times S_6$ x 2, $S_5\times A_6$ x 2

Order 40320: $D_4\times S_7$ x 2

Order 35280: $C_7:S_7$ x 2

Order 30240: $A_4\times A_7$ x 6, $S_3\times S_7$ x 6, $C_6:S_7$ x 4, $A_7:C_{12}$ x 2, $C_3:C_4\times A_7$ x 2, $C_2\times C_6\times A_7$ x 2, $D_6\times A_7$ x 2, $C_6\times S_7$ x 2

Order 28800: $C_2\times S_5^2$

Order 28224: $\GL(3,2)^2$ x 2

Order 25920: $A_6:\SOPlus(4,2)$ x 2, $C_3:S_4\times A_6$ x 2, $A_4\times C_3:S_6$ x 2, $C_3\times A_4:S_6$ x 2

Order 25200: $C_5:S_7$ x 2, $C_5\times S_7$ x 2, $D_5\times A_7$ x 2

Order 22680: $C_3^2\times A_7$ x 2

Order 21600: $A_5\times A_6$ x 4

Order 20160: $C_2^2\times S_7$ x 4, $C_2^2:S_7$ x 4, $S_5\times \GL(3,2)$ x 2, $C_4:S_7$ x 2, $C_4\times S_7$ x 2, $D_4\times A_7$ x 2

Order 17640: $C_7\times A_7$ x 2

Order 17280: $S_4\times S_6$ x 4, $C_3:(D_4\times S_6)$ x 2, $S_3\times S_4\times S_5$ x 2

Order 15120: $C_3:S_7$ x 6, $S_3\times A_7$ x 4, $C_3\times S_7$ x 4, $A_6:F_7$ x 2, $C_6\times A_7$ x 2

Order 14400: $S_5^2$ x 6, $A_5^2:C_2^2$ x 2, $F_5\times S_6$ x 2, $A_5^2:C_2^2$

Order 12960: $A_6:S_3^2$ x 4, $C_3\times A_4\times A_6$ x 2, $C_3^2:C_4\times A_6$ x 2

Order 12600: $C_5\times A_7$ x 2

Order 12096: $C_3:S_4\times \GL(3,2)$ x 2

Order 10368: $C_3:(S_3\times S_4^2)$

Order 10080: $C_2\times S_7$ x 6, $A_5\times \GL(3,2)$ x 4, $C_2^2\times A_7$ x 4, $C_4\times A_7$ x 2, $A_7:C_4$ x 2

Order 8640: $S_4\times A_6$ x 6, $A_4:S_6$ x 4, $A_4\times S_6$ x 4, $C_3:(S_4\times S_5)$ x 4, $D_6\times S_6$ x 2, $S_5\times \SOPlus(4,2)$ x 2, $C_3:(C_4\times S_6)$ x 2, $(C_6\times S_6):C_2$ x 2, $D_6:S_6$ x 2, $S_3\times A_4\times S_5$ x 2, $C_3\times C_2^2:S_6$ x 2, $C_3:D_4\times A_6$ x 2, $C_2\times C_6:S_6$ x 2, $S_3\times S_4\times A_5$ x 2, $C_3:S_4\times S_5$ x 2, $S_4\times C_3:S_5$ x 2, $C_3\times S_4\times S_5$ x 2, $S_3\times A_4:S_5$ x 2

Order 7560: $C_3\times A_7$ x 4, $C_7:C_3\times A_6$ x 2

Order 7200: $A_5\times S_5$ x 8, $\PSOPlus(4,5)$ x 5, $A_6:F_5$ x 2, $D_5\times S_6$ x 2, $F_5\times A_6$ x 2, $C_2\times A_5^2$

Order 6480: $C_3^2:S_6$ x 4, $C_3:S_3\times A_6$ x 2

Order 6048: $C_3^2:C_4\times \GL(3,2)$ x 2, $C_3\times A_4\times \GL(3,2)$ x 2

Order 5760: $C_2\times S_4\times S_5$ x 4, $S_3\times D_4\times S_5$ x 2, $D_4\times S_6$ x 2

Order 5184: $C_3^2:S_4^2$ x 2, $A_4^2:S_3^2$ x 2, $C_3^2:S_4^2$ x 2, $C_3^3:(D_4\times S_4)$ x 2, $(A_4\times C_6^2):D_6$

Order 5040: $S_7$ x 6, $C_7:S_6$ x 2, $F_7\times S_5$ x 2, $C_2\times A_7$ x 2

Order 4800: $C_2\times F_5\times S_5$ x 2

Order 4320: $A_4\times A_6$ x 6, $S_3\times S_6$ x 6, $\GL(2,4):S_4$ x 4, $\GL(2,4):S_4$ x 4, $A_4\times C_3:S_5$ x 4, $C_6:S_6$ x 4, $A_5:\SOPlus(4,2)$ x 4, $S_3^2\times S_5$ x 4, $S_3^2:S_5$ x 4, $A_6:C_{12}$ x 2, $C_3:C_4\times A_6$ x 2, $C_2\times C_6\times A_6$ x 2, $\GL(2,4):S_4$ x 2, $S_4\times \GL(2,4)$ x 2, $S_3\times A_4\times A_5$ x 2, $C_3\times A_4\times S_5$ x 2, $D_6\times A_6$ x 2, $C_6\times S_6$ x 2, $C_3^2:C_4\times S_5$ x 2, $A_5\times \SOPlus(4,2)$ x 2

Order 4032: $S_4\times \GL(3,2)$ x 8, $C_3:D_4\times \PSL(2,7)$ x 2

Order 3600: $A_5^2$ x 4, $C_5\times S_6$ x 2, $C_5:S_6$ x 2, $D_5\times A_6$ x 2

Order 3528: $C_7:C_3\times \GL(3,2)$ x 2

Order 3456: $S_3\times S_4^2$ x 4, $\GL(2,\mathbb{Z}/4):S_3^2$ x 2

Order 3360: $F_5\times \GL(3,2)$ x 2

Order 3240: $C_3^2\times A_6$ x 2

Order 3024: $C_{21}:(C_6\times S_4)$ x 2, $C_3:S_3\times \GL(3,2)$ x 2

Order 2880: $S_4\times S_5$ x 22, $C_3:(D_4\times S_5)$ x 6, $S_3\times C_2^2:S_5$ x 4, $C_3:D_4\times S_5$ x 4, $C_2\times S_4\times A_5$ x 4, $C_2\times A_4\times S_5$ x 4, $C_2\times A_4:S_5$ x 4, $C_2^2:S_6$ x 4, $C_2^2\times S_6$ x 4, $\GL(2,4):C_2^4$ x 4, $S_3\times F_5\times S_4$ x 2, $D_{12}:S_5$ x 2, $S_3\times C_4:S_5$ x 2, $D_{12}\times S_5$ x 2, $S_3\times D_4\times A_5$ x 2, $C_4\times S_3\times S_5$ x 2, $D_4\times C_3:S_5$ x 2, $C_3\times D_4\times S_5$ x 2, $C_4:S_6$ x 2, $C_4\times S_6$ x 2, $D_4\times A_6$ x 2

Order 2592: $A_4\times C_3^2:S_4$ x 4, $A_4:S_3^3$ x 4, $C_3^3:\GL(2,\mathbb{Z}/4)$ x 4, $A_4\times C_3^2:D_{12}$ x 2, $C_3\times C_6^2:D_{12}$ x 2, $C_3^3:(C_4\times S_4)$ x 2, $(A_4\times C_6^2):S_3$, $C_3:S_3\times A_4^2$, $C_3^2\times \PSOPlus(4,3)$, $C_3^3:C_{12}:D_4$

Order 2520: $A_5:F_7$ x 4, $A_7$ x 4, $C_7\times A_6$ x 2, $F_7\times A_5$ x 2, $C_7:C_3\times S_5$ x 2

Order 2400: $F_5\times S_5$ x 10, $D_{10}\times S_5$ x 2, $C_2\times F_5\times A_5$ x 2, $D_{10}.S_5$ x 2

Order 2160: $A_5:S_3^2$ x 8, $C_3:S_6$ x 6, $A_4\times \GL(2,4)$ x 4, $S_3^2\times A_5$ x 4, $C_3^2:C_4\times A_5$ x 4, $A_5:S_3^2$ x 4, $\GL(2,4):D_6$ x 4, $S_3\times A_6$ x 4, $C_3\times S_6$ x 4, $\GL(2,4):D_6$ x 2, $C_6\times A_6$ x 2, $(C_3\times \GL(2,4)):C_4$ x 2

Order 2016: $A_4\times \GL(3,2)$ x 8, $C_3:C_4\times \GL(3,2)$ x 2, $C_2\times C_6\times \GL(3,2)$ x 2, $D_6\times \GL(3,2)$ x 2

Order 1920: $C_2\times D_4\times S_5$ x 2

Order 1800: $C_5\times A_6$ x 2

Order 1728: $C_3:S_4^2$ x 12, $A_4^2:D_6$ x 8, $A_4^2:D_6$ x 5, $C_3:S_4^2$ x 4, $C_3\times S_4^2$ x 4, $D_6^2:D_6$ x 4, $C_6^2:(C_2\times S_4)$ x 4, $C_3\times \GL(2,\mathbb{Z}/4):S_3$ x 4, $C_2^3.S_3^3$ x 4, $C_2^3.S_3^3$ x 4, $A_4:D_6^2$ x 2, $A_4:D_6^2$ x 2, $A_4\times D_6:D_6$ x 2, $C_3:S_3\times \GL(2,\mathbb{Z}/4)$ x 2, $C_3^3:D_4^2$ x 2, $C_2^3.S_3^3$ x 2, $C_2^3.S_3^3$ x 2, $C_2^3.S_3^3$ x 2, $C_6^2:(C_2\times S_4)$

Order 1680: $D_7\times S_5$ x 2, $D_5\times \GL(3,2)$ x 2

Order 1512: $A_4\times C_{21}:C_6$ x 2, $C_{21}:(C_3\times S_4)$ x 2, $C_3\times A_4:F_7$ x 2, $C_3^2:C_{28}:C_6$ x 2, $C_3^2\times \GL(3,2)$ x 2

Order 1440: $D_6\times S_5$ x 26, $S_4\times A_5$ x 16, $A_4:S_5$ x 14, $A_4\times S_5$ x 14, $\GL(2,4):C_2^3$ x 6, $C_2\times S_6$ x 6, $\GL(2,4):D_4$ x 6, $\GL(2,4):D_4$ x 6, $\GL(2,4):D_4$ x 6, $D_6:S_5$ x 6, $\GL(2,4):C_2^3$ x 4, $\GL(2,4):C_2^3$ x 4, $C_2^3:\GL(2,4)$ x 4, $C_2^2\times A_6$ x 4, $\GL(2,4):D_4$ x 4, $D_6:S_5$ x 4, $C_3:(C_4\times S_5)$ x 4, $S_3\times D_5\times S_4$ x 2, $S_3\times A_4\times F_5$ x 2, $S_3\times A_4:F_5$ x 2, $C_{15}:(C_4\times S_4)$ x 2, $C_{15}:C_4\times S_4$ x 2, $F_5\times C_3:S_4$ x 2, $C_3\times F_5\times S_4$ x 2, $F_5\times \SOPlus(4,2)$ x 2, $D_4\times \GL(2,4)$ x 2, $D_{12}\times A_5$ x 2, $C_4\times S_3\times A_5$ x 2, $C_{12}:S_5$ x 2, $C_{12}:S_5$ x 2, $C_{12}:S_5$ x 2, $C_{12}\times S_5$ x 2, $C_3:C_4\times S_5$ x 2, $D_6.S_5$ x 2, $C_4\times A_6$ x 2, $A_6:C_4$ x 2

Order 1344: $D_4\times \GL(3,2)$ x 2

Order 1296: $C_6^2:S_3^2$ x 8, $C_6^2:S_3^2$ x 6, $A_4\times C_3:S_3^2$ x 4, $C_3^3:(S_3\times D_4)$ x 4, $C_3\times C_6^2:C_{12}$ x 2, $(A_4\times C_3^3):C_4$ x 2, $C_3^4:(C_2\times D_4)$ x 2, $C_3^2\times A_4^2$, $C_3^4:C_4^2$

Order 1260: $C_7:\GL(2,4)$ x 4

Order 1200: $D_5\times S_5$ x 10, $F_5\times A_5$ x 6, $A_5:F_5$ x 6, $D_{10}\times A_5$ x 2, $C_{10}\times S_5$ x 2, $C_{10}:S_5$ x 2

Order 1176: $C_7\times \PSL(2,7)$ x 2

Order 1152: $\GL(2,\mathbb{Z}/4):D_6$ x 6, $C_2\times S_4^2$ x 4, $D_6^2:C_2^3$

Order 1080: $C_3^2:S_5$ x 8, $S_3\times \GL(2,4)$ x 4, $\GL(2,4):S_3$ x 4, $C_3\times A_6$ x 4, $C_3^2\times S_5$ x 2, $C_3^2:S_5$ x 2

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 12700800: $A_7.S_7$ ($C_1$)

Order 6350400: $A_7^2$ ($C_2$)

Order 2520: $A_7$ ($S_7$) x 2

Order 1: $C_1$ ($A_7.S_7$)

Normal subgroups up to automorphism (quotient in parentheses)

not computed

Series

Derived series	not computed
Chief series	not computed
Lower central series	not computed
Upper central series	not computed

Supergroups

This group is a maximal subgroup of 4 larger groups in the database.

This group is a maximal quotient of 0 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 $114 \times 114$ rational character table. Alternatively, you may search for characters of this group with desired properties.