Abstract group 787500000.b: $C_5^6.(D_5\times S_7)$

• Label: 787500000.b
• Order: \( 2^{5} \cdot 3^{2} \cdot 5^{8} \cdot 7 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: 1
• $\card{\Aut(G)}$: \( 2^{8} \cdot 3^{2} \cdot 5^{8} \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \)
• Perm deg.: $35$
• Trans deg.: $35$
• Rank: $2$

Group information

Description:	$C_5^6.(D_5\times S_7)$
Order:	\(787500000\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 5^{8} \cdot 7 \)
Exponent:	\(2100\)\(\medspace = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Automorphism group:	Group of order \(6300000000\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 5^{8} \cdot 7 \)
Composition factors:	$C_2$ x 2, $C_5$ x 7, $A_7$
Derived length:	$2$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	5	6	7	10	12	14	15	20	25	30	35	50	60
Elements	1	815855	176750	5670000	7953124	50496250	11250000	121981020	7875000	56250000	27167000	125580000	31500000	206535000	45000000	31500000	57750000	787500000
Conjugacy classes	1	7	2	4	178	8	1	694	2	1	215	160	30	247	2	10	14	1576
Divisions	1	7	2	4	90	8	1	350	2	1	108	80	15	124	1	5	7	806

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 $35$ $\langle(1,23,14)(2,22,15,5,24,13)(3,21,11,4,25,12)(6,17,27,35,7,16,28,34,8,20,29,33,9,19,30,32,10,18,26,31) \!\cdots\! \rangle$
Transitive group:	35T260				more information
Direct product:	not computed
Semidirect product:	not computed
Trans. wreath product:	not computed
Possibly split product:	$(C_5^6.S_7)$ . $D_5$	$C_5^6$ . $(D_5\times S_7)$	$(C_5^7.S_7)$ . $C_2$	$(C_5^7.S_7)$ . $C_2$	all 10

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

Homology

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

Subgroups

There are 12 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:	not computed
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 1008 (order at least 781250) are shown, as well as all normal subgroups of any index.

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

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

Order 196875000: $C_5^7.A_7$

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

Order 112500000: $C_5^7.C_2.A_6.C_2$

Order 78750000: $C_5^6.S_7$, $C_5^6.S_7$, $C_5^6.(C_2\times A_7)$

Order 56250000: $C_5^7.A_6.C_2$ x 2, $C_5^7.C_2.A_6$

Order 39375000: $C_5^6.A_7$

Order 37500000: $C_5^6.D_{10}.S_5$

Order 28125000: $C_5^7.A_6$

Order 26250000: $C_5^7:(C_2\times \GL(3,2))$

Order 22500000: $C_5^5.(D_5\times S_6)$ x 6, $C_5^5.C_{10}^2.S_3^2.C_2$

Order 18750000: $C_5^7.C_2.S_5$ x 7, $C_5^6.D_{10}.A_5$

Order 13125000: $C_5^7:\PSL(2,7)$

Order 11250000: $C_5^6.S_6$ x 6, $C_5\wr A_6.C_2$ x 6, $C_5^6.S_6$ x 6, $C_5^5.C_{10}^2.S_3^2$ x 3, $C_5^7.(S_3\times A_4).C_2$, $C_5^5.C_{10}^2.C_3.D_6$, $C_5^5.(C_3\times A_4).C_{10}^2$, $C_5^4.C_{15}^2.(D_4\times D_5)$, $C_5^3.C_{10}^2.C_5^2.(C_6\times S_3)$

Order 9375000: $C_5^7.S_5$ x 6, $C_5^7.C_2.A_5$ x 4

Order 7500000: $C_5^5.D_{10}.S_5$ x 7, $C_5^7.D_6.C_2^3$, $C_5^5.C_{10}^2.D_6.C_2$, $C_5^3.C_{10}^2.C_5^2.D_6.C_2$

Order 6562500: $C_5^7:(C_2\times F_7)$

Order 6250000: $C_5^4.C_5^3.D_{10}.C_2^2$

Order 5625000: $C_5\wr A_6$ x 6, $C_5^7.S_3^2.C_2$, $C_5^7.C_3:S_3.C_2^2$, $C_5^7.C_3.D_6.C_2$, $C_5^7.(S_3\times A_4)$, $C_5^7.(C_6\times A_4)$, $C_5^5.C_{10}^2.C_3.S_3$, $C_5^5.C_3.C_{10}^2.C_6$, $C_5^5.(C_{15}\times A_4).C_{10}$, $C_5^4.C_{15}^2.D_{20}$, $C_5^4.C_{15}^2.C_5.D_4$, $C_5^4.C_{15}^2.(C_5\times D_4)$, $C_5^4.C_3^2.C_{10}.C_{10}^2$, $C_5^3:C_2^2.C_5^3.C_{15}.C_6$, $C_5^3.C_{10}^2.C_{15}.C_{30}$

Order 5250000: $C_5^6:(C_2\times \GL(3,2))$

Order 4687500: $C_5^7.A_5$ x 2

Order 4500000: $C_5^5.(C_{15}\times A_4).C_2^3$ x 4, $C_5^5.(C_5\times A_4).D_6.C_2$ x 2, $C_5^5.(C_2\times S_6)$

Order 3750000: $C_5^6.C_2.S_5$ x 52, $C_5^7.D_6.C_2^2$ x 7, $C_5^5.D_{10}.A_5$ x 6, $C_5^5.C_{10}^2.D_6$ x 6, $C_5^6.(C_2\times S_5)$ x 6, $C_5^7.C_6.C_2^3$ x 3, $C_5^3.C_{10}^2.C_5^2.D_6$ x 3, $C_5^7.A_4.C_2^2$ x 2, $C_5^6.D_{15}:D_4$ x 2, $C_5^6.(C_2\times S_5)$ x 2, $C_5^6.C_{30}.C_2^3$, $C_5^6.C_{20}:D_6$, $C_5^6.C_2^2.A_5$, $C_5^5.A_4.C_{10}^2$, $C_5^3:S_3\times C_5^4:D_4$, $C_5^3.C_{10}^2.C_{15}.C_{10}.C_2$, $C_5^3.C_{10}^2.C_5.D_{30}$, $C_5^3.(C_5^2\times A_4).C_{10}^2$

Order 3281250: $C_5^7:F_7$, $C_5\wr F_7$, $C_5^7:C_7:C_6$

Order 3125000: $C_5^7.D_{10}.C_2$ x 2, $C_5^5.C_5^2.D_{10}.C_2$ x 2, $C_5^6.C_5.D_{10}.C_2$, $C_5^6.(C_{10}\times F_5)$, $C_5^5.C_5.C_{10}^2.C_2$

Order 2812500: $C_5^7.S_3^2$ x 3, $C_5^7.(C_6\times S_3)$ x 2, $C_5^5.C_{15}^2.C_4$ x 2, $C_5^7.C_3.D_6$, $C_5^6.(C_5\times S_3^2)$, $C_5^5.C_{15}^2.C_2^2$, $C_5^5.C_{10}^2.C_3^2$, $C_5^4.C_{15}^2.C_{10}.C_2$

Order 2625000: $C_5^6:\GL(3,2)$

Order 2500000: $C_5^7.C_2^4.C_2$

Order 2250000: $C_5^5.C_2^2.C_{15}.D_6$ x 13, $C_5^5.(C_5\times A_4).D_6$ x 8, $C_5^6.(S_3\times A_4).C_2$ x 6, $C_5\wr S_3^2.C_2^2$ x 6, $C_5^3:C_2^2.C_5^3.(C_6\times S_3)$ x 5, $C_5^5.(C_5\times S_3\times A_4).C_2$ x 4, $C_5^5.(C_{15}\times S_4).C_2$ x 3, $C_5^4.C_{15}^2.C_2^3.C_2$, $C_5^3:S_4\times C_5^3:S_3$, $C_5^3.C_{10}^2.C_{15}.C_6.C_2$, $C_5^3.A_4.C_{10}\times C_5^2:S_3$, $C_5^5:S_6$, $C_5^5:S_6$, $C_5^5:(C_2\times A_6)$

Order 2187500: $C_5^7:D_{14}$

Order 1875000: $C_5^6.S_5$ x 52, $C_5^6.C_2.A_5$ x 32, $C_5^7.D_6.C_2$ x 15, $C_5^7.C_6.C_2^2$ x 7, $C_5^6.C_{30}.C_2^2$ x 6, $C_5^6.S_5$ x 6, $C_5^6.S_5$ x 6, $C_5^6.(C_2\times A_5)$ x 6, $C_5^7.A_4.C_2$ x 4, $C_5^5.C_{10}^2.S_3$ x 3, $C_5^5.C_6.C_{10}^2$ x 3, $C_5^7.D_{12}$ x 2, $C_5^7.C_3.D_4$ x 2, $C_5^5.C_{10}^2.C_6$ x 2, $C_5^5.C_3.C_{10}^2.C_2$ x 2, $C_5^5.A_4.C_5.C_{10}$ x 2, $C_5^6.C_{15}.C_2^3$, $C_5^6.(C_2\times A_5)$, $C_5^5.C_{10}.C_{30}.C_2$, $C_5^4.C_{15}.C_{10}^2.C_2$, $C_5^4.C_{10}^2.C_{30}$, $C_5^3.C_{10}^2.C_{15}.C_{10}$, $C_5^3.C_{10}^2.C_5.D_{15}$, $C_5^3.A_4.C_5^3.C_{10}$, $C_5^3.(C_5\times A_4).C_5^3.C_2$, $C_5\times C_5^5.A_4.C_{10}$

Order 1640625: $C_5\wr (C_7:C_3)$

Order 1562500: $C_5^7.D_{10}$ x 4, $C_5^4.C_5^3.C_{20}$ x 2, $C_5^7.C_{10}.C_2$, $C_5^5.C_5^3.C_4$, $C_5^5.C_5^3.C_2^2$, $C_5^5.C_5.C_{10}^2$, $C_5^4.C_5^3.C_5.C_4$

Order 1500000: $C_5^3.C_{10}^2.C_5.D_6.C_2$ x 32, $C_5^6.D_6.C_2^3$ x 7, $C_5^4.D_{10}.S_5$ x 7, $C_5^6:(C_2^2\times S_4)$ x 6

Order 1406250: $C_5^7.C_3.C_6$ x 2, $C_5^7.C_3.S_3$, $C_5^6.C_{15}.C_6$, $C_5^5.C_{15}.C_{30}$, $C_5^4.C_{15}^2.C_{10}$, $C_5^4.C_3.C_5^3.C_6$

Order 1312500: $C_5^6:(C_2\times F_7)$

Order 1250000: $C_5^7.C_2^3.C_2$ x 10, $C_5^7.C_2^4$ x 2, $C_5^6.D_5.C_2^3$ x 2, $C_5^6.C_{10}.C_2^3$ x 2, $C_5^5.D_5^2.C_2^2$ x 2, $C_5^4.C_5^3.C_2^3.C_2$ x 2, $C_5^4.C_{10}.C_{10}^2.C_2$, $C_5^3.C_5^3.D_{10}.C_2^2$

Order 1125000: $C_5^6.S_3^2.C_2$ x 19, $C_5\wr (S_3\wr C_2)$ x 12, $C_5^6:\SOPlus(4,2)$ x 12, $C_5^3.C_{10}^2.C_{15}.C_6$ x 7, $C_5^6.(S_3\times A_4)$ x 6, $C_5^6.(C_6\times A_4)$ x 6, $C_5^5.A_4:D_{15}$ x 6, $C_5^5.(C_5\times C_3:S_4)$ x 6, $C_5^6:(C_2\times C_3^2:C_4)$ x 6, $C_5^6:(S_3\times D_6)$ x 6, $C_5^5.(C_5\times S_3\times A_4)$ x 5, $C_5^3:C_2^2.C_5^3.C_3.C_6$ x 4, $C_5^4.C_{15}^2.D_4$ x 3, $C_5^6.C_3:S_3.C_2^2$, $C_5^6.C_3.D_6.C_2$, $C_5^4:A_4\times C_5^2:S_3$, $C_5^4.C_3:S_3.C_{10}^2$, $C_5^3.A_4.C_{10}\times C_5^2:C_3$, $C_5^5:A_6$

Order 1093750: $C_5^7:D_7$, $C_5\wr D_7$, $C_5^7:C_{14}$

Order 937500: $C_5^6.A_5$ x 19, $C_5^7.D_6$ x 13, $C_5^7.C_6.C_2$ x 6, $C_5\wr A_5$ x 6, $C_5^6.C_{30}.C_2$ x 4, $C_5^6.C_{15}.C_2^2$ x 4, $C_5^5.C_3.C_{10}^2$ x 3, $C_5^4:C_2^2\times C_5\times C_5^2:C_3$ x 2, $C_5^4.C_{15}.C_{10}^2$ x 2, $C_5^7.C_{12}$, $C_5^5.C_{15}.C_{20}$, $C_5^5.C_{10}^2.C_3$, $C_5^4:C_4\times C_5\times C_5^2:C_3$, $C_5^3:S_3\times C_5^4:C_2$, $C_5^3.C_{15}.C_5.C_{10}^2$, $C_5^3.C_{10}^2.C_5.C_{15}$, $C_5\times C_5^5.(C_5\times A_4)$

Order 900000: $C_5^5.(S_3\times S_4).C_2$, $C_5^3.C_{10}^2.S_3^2.C_2$

Order 781250: $C_5^7.C_{10}$ x 3, $C_5^7.D_5$, $C_5^6.C_5.C_{10}$, $C_5^5.C_5^3.C_2$, $C_5^4.C_5^3.C_{10}$

Order 156250: $C_5^7.C_2$

Order 78125: $C_5^7$

Order 15625: $C_5^6$

Order 5: $C_5$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 787500000: $C_5^6.(D_5\times S_7)$ ($C_1$)

Order 393750000: $C_5^7.S_7$ ($C_2$), $C_5^7.S_7$ ($C_2$), $C_5^6.(D_5\times A_7)$ ($C_2$)

Order 196875000: $C_5^7.A_7$ ($C_2^2$)

Order 78750000: $C_5^6.S_7$ ($D_5$)

Order 39375000: $C_5^6.A_7$ ($D_{10}$)

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

Order 78125: $C_5^7$ ($C_2\times S_7$)

Order 15625: $C_5^6$ ($D_5\times S_7$)

Order 5: $C_5$ ($C_5^6.(C_2\times S_7)$)

Order 1: $C_1$ ($C_5^6.(D_5\times S_7)$)

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 $1576 \times 1576$ character table is not available for this group.

Rational character table

The $806 \times 806$ rational character table is not available for this group.