Abstract group 50400000000.b: $C_5^7.(C_2^7.S_7)$

• Label: 50400000000.b
• Order: \( 2^{11} \cdot 3^{2} \cdot 5^{8} \cdot 7 \)
• Exponent: \( 2^{3} \cdot 3 \cdot 5^{2} \cdot 7 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: 1
• $\card{\Aut(G)}$: \( 2^{12} \cdot 3^{2} \cdot 5^{8} \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2 \)
• Perm deg.: $35$
• Trans deg.: $35$
• Rank: $2$

Group information

Description:	$C_5^7.(C_2^7.S_7)$
Order:	\(50400000000\)\(\medspace = 2^{11} \cdot 3^{2} \cdot 5^{8} \cdot 7 \)
Exponent:	\(4200\)\(\medspace = 2^{3} \cdot 3 \cdot 5^{2} \cdot 7 \)
Automorphism group:	Group of order \(100800000000\)\(\medspace = 2^{12} \cdot 3^{2} \cdot 5^{8} \cdot 7 \)
Composition factors:	$C_2$ x 8, $C_5$ x 7, $A_7$
Derived length:	$1$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	5	6	7	8	10	12	14	15	20	24	25	30	35	40	50	60	100	120
Elements	1	4810895	2807000	495154800	126078124	1263745000	720000000	1360800000	4913860980	3666600000	3600000000	369068000	7028095200	1260000000	504000000	8689380000	2880000000	2839200000	2520000000	6308400000	1008000000	840000000	50400000000
Conjugacy classes	1	19	2	28	41	24	1	8	413	14	1	61	258	2	12	249	2	28	14	58	2	2	1240
Divisions	1	19	2	28	23	24	1	8	223	14	1	32	137	2	6	130	1	15	7	30	1	1	706
Autjugacy classes	1	19	2	28	23	24	1	8	223	14	1	32	137	2	6	130	1	15	7	30	1	1	706

Minimal presentations

Permutation degree:	$35$
Transitive degree:	$35$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of linear representations for this group have not been computed

Constructions

Permutation group:	Degree $35$ $\langle(1,30,19,6,21)(2,26,18,7,25,5,29,20,10,22)(3,27,17,8,24,4,28,16,9,23)(11,31) \!\cdots\! \rangle$
Transitive group:	35T316				more information
Direct product:	not computed
Semidirect product:	not computed
Trans. wreath product:	not computed
Possibly split product:	$C_5^7$ . $(C_2^7.S_7)$	$(C_5^7.C_2^6.C_2)$ . $S_7$	$(C_5^7.(C_2^6.S_7))$ . $C_2$	$(C_5^7.(C_2^6.S_7))$ . $C_2$	all 8

Elements of the group are displayed as permutations of degree 35.

Homology

Abelianization:	$C_{2}^{2} $
Schur multiplier:	$C_{2}^{3}$
Commutator length:	not computed

Subgroups

There are 10 normal subgroups, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	a subgroup isomorphic to $C_1$
Commutator:	a subgroup isomorphic to $C_5^7.C_2^6:A_7$
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 525 (order at least 96000000) are shown, as well as all normal subgroups of any index.

Order 50400000000: $C_5^7.(C_2^7.S_7)$

Order 25200000000: $C_5^7.(C_2^6.S_7)$, $C_5^7.(C_2^6.S_7)$, $C_5^7.(C_2^7.A_7)$

Order 12600000000: $C_5^7.C_2^6:A_7$

Order 7200000000: $C_5^7.C_2^6.C_2.A_6.C_2$

Order 3600000000: $C_5^7.C_2^6.A_6.C_2$ x 6, $C_5^7.C_2^6.C_2.A_6$

Order 2400000000: $C_5^7.C_2^6.C_2^2.S_5$

Order 1800000000: $C_5^7.C_2^5.A_6.C_2$ x 4, $C_5^7.C_2^6.A_6$ x 3

Order 1680000000: $C_5^7:(C_2^7.\GL(3,2))$

Order 1440000000: $C_5^7.C_2.C_2^6.D_6^2$, $C_5^6.C_2^6.C_2.A_6.C_2$

Order 1200000000: $C_5^7.C_2^6.C_2.S_5$ x 6, $C_5^7.C_2.C_2^6.S_5$ x 4, $C_5^7.D_4.C_2^4.S_5$ x 3, $C_5^7.C_4.C_2^5.S_5$ x 2, $C_5^7.C_2^5.C_2^3.A_5$

Order 900000000: $C_5^7.C_2^5.A_6$

Order 840000000: $C_5^7:(C_2^6:\GL(3,2))$

Order 787500000: $C_5^6.(D_5\times S_7)$

Order 720000000: $C_5^7.C_2^3.A_4^2.C_2^3$ x 10, $C_5^6.(C_2^6.S_6)$ x 4, $C_5^7.C_2^5.(C_6\times S_4).C_2$ x 3, $C_5^6.C_2^6.A_6.C_2$ x 2, $C_5^7.(C_2^3\times A_4).A_4.C_2^3$, $C_5^6.C_2^6.C_2.A_6$, $C_5^6.A_4^2.C_5.C_2^5.C_2$, $C_5^4.C_2.C_5^3.C_2^2.A_4^2.C_2^3$

Order 600000000: $C_5^7.C_2^6.S_5$ x 34, $C_5^7.C_2^6.C_2.A_5$ x 7, $C_5^7.C_2^5.C_2.S_5$ x 7, $C_5^7.C_4.C_2^4.S_5$, $C_5^7.C_2.C_2^6.A_5$

Order 480000000: $C_5^6.C_2^5.(D_5\times A_4).C_2^3$, $C_5^4.D_4^2.C_2\times C_5^3.A_4.C_2^2$, $C_5^4.C_2\wr A_4.C_2\times C_5^3:(C_2\times D_4)$

Order 420000000: $C_5^7:C_2\wr F_7$

Order 400000000: $C_5^7.C_2^4.D_{10}.C_2^4$

Order 393750000: $C_5^7.S_7$, $C_5^7.S_7$, $C_5^6.(D_5\times A_7)$

Order 360000000: $C_5^7.C_2.C_2^6.S_3^2$ x 9, $C_5^7.C_2^5.A_4.D_6$ x 6, $C_5^4.C_2.C_5^3.C_2^2.A_4^2.C_2^2$ x 6, $C_5^7.C_2.C_2^6.C_3.D_6$ x 4, $C_5^7.C_2.A_4.C_2^4.C_6.C_2$ x 4, $C_5^7.Q_8.A_4^2.C_2^2$ x 3, $C_5^6.A_4^2.C_5.C_2^4.C_2$ x 3, $C_5^6.A_4^2.C_2^3:D_{10}$ x 3, $C_5^6.A_4^2.C_2^2:D_{20}$ x 3, $C_5^7.(C_2^3\times A_4).A_4.C_2^2$ x 2, $C_5^6.C_2^4.C_{15}.C_6.C_2^4$ x 2, $C_5^6.A_4^2.D_{10}.C_2^3$ x 2, $C_5^6.C_2\wr A_6$ x 2, $C_5^6.(C_2^5.S_6)$ x 2, $C_5^6.C_2^5:S_6$ x 2, $C_5^7.C_2.C_2^6.C_6^2$, $C_5^6.C_2^6.A_6$, $C_5^6.C_2^4.C_5.C_3^2.C_2^4.C_2$, $C_5^6.A_4^2.C_2.C_{10}.C_2^3$, $C_5^4.C_2.C_5^2.C_2^4.C_5.D_6^2$

Order 300000000: $C_5^7.C_2^5.S_5$ x 36, $C_5^7.C_2^6.A_5$ x 12, $C_5^7.C_2^5.C_2.A_5$ x 2

Order 240000000: $C_5^7.(C_2\times D_4\times A_4).C_2^4$ x 18, $C_5^4.C_2.C_5^2.C_2^4.C_{30}.C_2^4$ x 15, $C_5^6.C_2^5.(C_2\times D_5\times S_4)$ x 11, $C_5^4.C_2.C_5^2.C_2^4.C_{60}.C_2^3$ x 6, $C_5^7.C_2.A_4.C_2^3.C_2^4$ x 4, $C_5^6.C_2^6.C_2.S_5$ x 3, $C_5^3:D_4\times C_5^4.C_2\wr A_4.C_2$ x 3, $C_5^7.A_4.C_2\wr C_4.C_2^2$ x 2, $C_5^3:C_2^3\times C_5^4.C_2\wr A_4.C_2$ x 2, $C_5^3:(C_2\times D_4)\times C_5^4.C_2^3.S_4$ x 2, $C_5^7.C_2^3.C_6.C_2^5.C_2$, $C_5^7.C_2.C_2^6.C_6.C_2^2$, $C_5^7.C_2.C_2^5.C_6.C_2^3$, $C_5^7.A_4.D_4^2.C_2^2$, $C_5^7.A_4.D_4:D_4.C_2^2$, $C_5^7.A_4.C_4:D_4.C_2^3$, $C_5^7.A_4.C_2^4.C_2^4$, $C_5^7.A_4.C_2^3.C_2^5$, $C_5^6.C_2^5.A_4.C_5.C_2^3$, $C_5^6.C_2^5.(C_5\times A_4).C_2^3$, $C_5^6.C_2^3.C_2^4.S_5$, $C_5^6.C_2^3.C_2^4.C_{15}.C_2^3$, $C_5^6.(C_2^4\times C_{10}).A_4.C_2^3$, $C_5^4.C_2\wr A_4.C_2\times C_5^3:D_4$, $C_5^3:(C_2\times D_4)\times C_5^4.C_2\wr A_4$, $C_5^3:(C_2\times C_4)\times C_5^4.C_2\wr A_4.C_2$

Order 225000000: $C_5^7.C_2^2.A_6.C_2$

Order 210000000: $C_5^7:C_2^4.\GL(3,2)$, $C_5^7:(C_2^4:\GL(3,2))$, $C_5^7:(C_2^6:F_7)$, $C_5^7:(C_2^6:F_7)$, $C_5^7:C_2\wr C_7:C_3$

Order 200000000: $C_5^7.C_2^4.(D_4\times F_5)$ x 8, $C_5^7.C_2^4.D_{10}.C_2^3$ x 3, $C_5^5.C_2^4.C_5.D_{10}^2.C_2$ x 2, $C_5^7.C_2^4.C_{10}.C_2^4$, $C_5^5.C_2^4.C_5^2.D_{10}.C_2^3$

Order 196875000: $C_5^7.A_7$

Order 180000000: $C_5^6.C_2^4.C_5.D_6^2$ x 18, $C_5^6.A_4^2.C_{10}.C_2^3$ x 7, $C_5^6.A_4^2.(D_4\times D_5)$ x 7, $C_5^6.C_2^4.C_5.D_6:D_6$ x 5, $C_5^4.C_2.C_{10}^2.(C_{15}\times A_4).C_2^3$ x 5, $C_5^6.A_4^2.D_{10}:C_4$ x 4, $C_5^4.C_2.C_{10}^2.(A_4\times D_{15}).C_2^2$ x 4, $C_5^4.C_2.C_5^3.C_2^2.A_4^2.C_2$ x 4, $C_5^7.C_2^5.A_4.C_6$ x 3, $C_5^6.C_2^4.C_{15}.C_6.C_2^3$ x 3, $C_5^6.C_2^4.C_5.(C_3:S_3.D_4)$ x 3, $C_5^6.A_4^2.C_{10}.C_2^2.C_2$ x 3, $C_5^7.C_2^5.A_4.S_3$ x 2, $C_5^7.C_2.A_4.C_2^4.C_6$ x 2, $C_5^7.(C_2^3\times A_4).A_4.C_2$ x 2, $C_5^6.C_2^4.C_5.C_3:S_3.C_2^3$ x 2, $C_5^6.A_4^2.D_{10}.C_2^2$ x 2, $C_5^6.A_4^2.(C_{10}.D_4)$ x 2, $C_5^4.C_2.C_{10}^2.(C_{15}\times S_4).C_2^2$ x 2, $C_5^4.C_2.C_5^2.C_2^4.C_{15}.D_6.C_2$ x 2, $C_5^7.Q_8.A_4^2.C_2$, $C_5^7.C_2^3.A_4^2.C_2$, $C_5^7.C_2.A_4^2.C_2^3$, $C_5^6.C_{30}.C_2^4.C_6.C_2^2$, $C_5^6.C_2^4.C_{30}.C_6.C_2^2$, $C_5^6.A_4^2.C_5.C_2^4$, $C_5^6.(C_5\times A_4).A_4.C_2^4$, $C_5^4.C_2.C_2^4.C_5^3.C_6^2.C_2$, $C_5^6.S_4^2:D_{10}$, $C_5^6.C_2^5:A_6$

Order 160000000: $C_5^6.C_{10}.C_2^5.C_2^5$

Order 157500000: $C_5^6.(C_2\times S_7)$

Order 150000000: $C_5^7.C_2^5.A_5$ x 8, $C_5^7.C_2^4.S_5$ x 8, $C_5^7.D_4.C_2.S_5$

Order 144000000: $C_2\times C_5^6.C_2^4.D_6\wr C_2$

Order 140000000: $C_5^7:C_2\wr D_7$

Order 120000000: $C_5^4.C_2.C_5^2.C_2^4.C_{30}.C_2^3$ x 46, $C_5^7.A_4.C_2^3.C_2^4$ x 40, $C_5^7.C_2.A_4.C_2\wr C_2^2$ x 32, $C_5^6.C_2^6.S_5$ x 25, $C_5^4.C_2^3.C_{10}^2.C_{30}.C_2^3$ x 24, $C_5^7.A_4.C_2\wr C_4.C_2$ x 16, $C_5^4.C_2.C_5^2.C_2^4.D_{15}:D_4$ x 16, $C_5^7.(C_2^3\times A_4).D_4.C_2$ x 15, $C_5^4.C_2^3.C_{10}^2.C_{15}.C_2^4$ x 14, $C_5^7.A_4.D_4:D_4.C_2$ x 12, $C_5^7.A_4.C_2^4.C_2^3$ x 12, $C_5^6.C_{10}.C_2^5.C_6.C_2^2$ x 11, $C_5^4.C_2.C_5^2.C_2^4.C_{60}.C_2^2$ x 11, $C_5^6.C_2^5.(D_5\times S_4)$ x 10, $C_5^7.A_4.C_4^2:C_4.C_2$ x 8, $C_5^7.A_4.C_2^2.C_2^5$ x 8, $C_5^4.C_2.C_{10}^2.A_4.C_{10}.C_2^3$ x 8, $C_5^4.C_2.C_{10}^2.(C_{20}\times S_4).C_2$ x 8, $C_5^4.C_2.C_5^2.C_2^4.C_{20}:D_6$ x 8, $C_5^7.(C_2\times D_4\times A_4).C_2^3$ x 7, $C_2\times C_5^6.C_2^5.S_5$ x 7, $C_5^7.C_2^5.C_6.C_2^3$ x 6, $C_5^7.C_2.C_2^6.D_6$ x 6, $C_5^6.C_2^5.C_2^2.D_{30}$ x 6, $C_5\times C_5^6.C_2^5.A_4.C_2^2$ x 6, $C_5^4.C_2^3.C_{10}^2.C_{60}.C_2^2$ x 5, $C_5^4.C_2.C_2^4.C_5^3.C_6.C_2^3$ x 5, $C_5^7.C_2.C_2^5.C_6.C_2^2$ x 4, $C_5^7.A_4.C_2^3.C_4.C_2^2$ x 4, $C_5^7.A_4.C_2.C_2^3.C_2^3$ x 4, $C_5^6.C_2^5.S_4\times D_5$ x 4, $C_5^6.C_2\wr S_5$ x 4, $C_5^7.A_4.D_4^2.C_2$ x 3, $C_5^6.C_4.C_2^4.S_5$ x 3, $C_5^6.C_2^6.C_2.A_5$ x 3, $C_5^6.C_2^5.A_4.C_2\times D_5$ x 3, $C_5^6.C_2^3.C_5.C_2^4.C_6.C_2$ x 3, $C_5^4.C_2.C_{10}^2.A_4.C_{20}.C_2^2$ x 3, $C_5^7.C_2^3.C_6.C_2^4.C_2$ x 2, $C_5^7.C_2^3.(D_4\times A_4).C_2$ x 2, $C_5^7.A_4.C_4^2.C_2^3$ x 2, $C_5^7.A_4.C_4:D_4.C_2^2$ x 2, $C_5^7.A_4.C_2^6.C_2$ x 2, $C_5^7.A_4.C_2^5.C_2^2$ x 2, $C_5^7.A_4.C_2^3.C_2^3.C_2$ x 2, $C_5^6.C_2^4.C_{30}.C_2^4$ x 2, $C_5^6.C_2^3.C_2^4.C_5.D_6$ x 2, $C_5^5.C_{10}^2.C_6.C_2^6$ x 2, $C_5^4.C_{10}.C_{10}^2.A_4.C_2^4$ x 2, $C_5^7.C_4.C_2.A_4.C_2^4$, $C_5^7.C_2^5.A_4.C_2^2$, $C_5^7.C_2^4.C_6.C_2^4$, $C_5^7.C_2.C_2^4.C_6.C_2^3$, $C_5^7.A_4.C_4.C_2^5$, $C_5^7.A_4.C_2^2.D_4.C_2^2$, $C_5^7.A_4.C_2.C_2^6$, $C_5^7.A_4.C_2.C_2^5.C_2$, $C_5^6.C_{30}.C_2^3.C_2^5$, $C_5^6.C_4^2.C_{30}.C_2^4$, $C_5^6.C_4.C_2^5.A_5$, $C_5^6.C_2^5.C_2.S_5$, $C_5^6.C_2.C_2^5.C_{30}.C_2^2$, $C_5^5.C_{10}^2.C_6.C_2^5.C_2$, $C_5^4.C_{10}.C_{10}^2.A_4.C_4.C_2^2$, $C_5\times C_5^6.C_2^3.C_2^4.D_6$

Order 112500000: $C_5^7.C_2.A_6.C_2$ x 6, $C_5^6.D_{10}.A_6$

Order 105000000: $C_5^7:C_2^3.\GL(3,2)$, $C_5^7:(C_2^3:\GL(3,2))$, $C_5^7:(C_2^3:F_8):C_3$

Order 100000000: $C_5^5.C_2^4.C_5^3.C_2^3.C_2$ x 20, $C_5^7.C_2^4.C_2^2:F_5$ x 8, $C_5^7.C_2^4.(D_4\times D_5)$ x 8, $C_5^7.C_2^4.C_4:F_5$ x 4, $C_5^7.C_2^4.C_{10}.C_2^3$ x 3, $C_5^5.C_2^4.C_5^3.C_4^2$ x 2, $C_5^5.C_2^4.C_5.F_5^2$ x 2, $C_5^5.C_2^4.C_5.D_{10}^2$ x 2, $C_5^7.C_2^5.C_5.C_2^3$, $C_5^5.C_2^4.C_5^3.C_2.C_2^3$

Order 96000000: $C_5^5.D_4.C_2^5.S_5$, $C_5^4.C_2.C_{10}^2.A_4.D_4.C_2^3$, $C_2\times C_5^6.C_2^5.A_4.C_2^3$

Order 10000000: $C_5^7.C_2^6.C_2$

Order 5000000: $C_5^7.C_2^6$

Order 156250: $C_5^7.C_2$

Order 78125: $C_5^7$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 50400000000: $C_5^7.(C_2^7.S_7)$ ($C_1$)

Order 25200000000: $C_5^7.(C_2^6.S_7)$ ($C_2$), $C_5^7.(C_2^6.S_7)$ ($C_2$), $C_5^7.(C_2^7.A_7)$ ($C_2$)

Order 12600000000: $C_5^7.C_2^6:A_7$ ($C_2^2$)

Order 10000000: $C_5^7.C_2^6.C_2$ ($S_7$)

Order 5000000: $C_5^7.C_2^6$ ($C_2\times S_7$)

Order 156250: $C_5^7.C_2$ ($C_2^6.S_7$)

Order 78125: $C_5^7$ ($C_2^7.S_7$)

Order 1: $C_1$ ($C_5^7.(C_2^7.S_7)$)

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 3 larger groups in the database.

This group is a maximal quotient of 0 larger groups in the database.

Character theory

Complex character table

The $1240 \times 1240$ character table is not available for this group.

Rational character table

The $706 \times 706$ rational character table is not available for this group.