Abstract group 52500000.d: $C_5^7:(C_2\times \PGL(2,7))$

• Label: 52500000.d
• Order: \( 2^{5} \cdot 3 \cdot 5^{7} \cdot 7 \)
• Exponent: \( 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: 1
• $\card{\Aut(G)}$: \( 2^{6} \cdot 3 \cdot 5^{7} \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2 \)
• Perm deg.: $40$
• Trans deg.: $40$
• Rank: $2$

Group information

Description:	$C_5^7:(C_2\times \PGL(2,7))$
Order:	\(52500000\)\(\medspace = 2^{5} \cdot 3 \cdot 5^{7} \cdot 7 \)
Exponent:	\(840\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
Automorphism group:	$C_5^7:(C_4\times \PGL(2,7))$, of order \(105000000\)\(\medspace = 2^{6} \cdot 3 \cdot 5^{7} \cdot 7 \)
Composition factors:	$C_2$ x 2, $C_5$ x 7, $\PSL(2,7)$
Derived length:	$1$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	5	6	7	8	10	14	15	20	30	35	40
Elements	1	114875	35000	787500	78124	5425000	750000	7875000	7619500	3750000	4340000	5775000	7700000	3000000	5250000	52500000
Conjugacy classes	1	5	1	2	186	3	1	4	154	1	38	8	14	2	4	424
Divisions	1	5	1	2	95	3	1	2	79	1	13	4	5	1	1	214
Autjugacy classes	1	5	1	2	95	3	1	4	79	1	19	5	7	1	2	226

Minimal presentations

Permutation degree:	$40$
Transitive degree:	$40$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	16	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Permutation group:	Degree $40$ $\langle(1,26,33,24,13,8,4,29,31,22,11,6,2,27,34,25,14,9,5,30,32,23,12,7,3,28,35,21,15,10) \!\cdots\! \rangle$
Transitive group:	40T188410				more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$(C_5^7:\PGL(2,7))$ . $C_2$	$(C_5^7:\PGL(2,7))$ . $C_2$	$(C_5^7.C_2)$ . $\PGL(2,7)$	$(C_5^7:\GL(3,2))$ . $C_2^2$	all 5

Elements of the group are displayed as permutations of degree 40.

Homology

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

Subgroups

There are 8 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 $C_5^7:\GL(3,2)$
Frattini:	a subgroup isomorphic to $C_1$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times D_8$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5^7$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7$

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 1500 (order at least 35000) are shown, as well as all normal subgroups of any index.

Order 52500000: $C_5^7:(C_2\times \PGL(2,7))$

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

Order 13125000: $C_5^7:\GL(3,2)$

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

Order 3750000: $C_5^7:(C_2\times S_4)$

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

Order 2500000: $C_5^7:(C_2\times D_8)$

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

Order 1875000: $C_5^7.D_6.C_2$, $C_5^7.A_4.C_2$, $C_5^7:S_4$, $C_5^7:S_4$

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

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

Order 1250000: $C_5^7:(C_2\times D_4)$ x 2, $C_5^7:D_8$ x 2, $C_5^7:(C_2\times C_8)$, $C_5^7:D_8$, $C_5^7:D_8$

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

Order 937500: $C_5^7.D_6$ x 3, $C_5^6.C_{15}.C_2^2$ x 3, $C_5^7.C_6.C_2$, $C_5^6.(C_5\times A_4)$

Order 750000: $C_5^6:(C_2\times S_4)$

Order 656250: $C_5^6:F_7$ x 2, $C_5^6:C_7:C_6$

Order 625000: $C_5^7.C_2^3$ x 2, $C_5^7:D_4$ x 2, $C_5^7:D_4$ x 2, $C_5^7:D_4$ x 2, $C_5^6.D_{10}.C_2$, $C_5^7:C_8$, $C_5^7:C_8$

Order 546875: $C_5^7:C_7$

Order 500000: $C_5^6:(C_2\times D_8)$

Order 468750: $C_5^7.C_6$ x 3, $C_5^6.C_{15}.C_2$ x 2, $C_5^5.C_{15}.C_{10}$ x 2

Order 437500: $C_5^6:D_{14}$

Order 375000: $C_5^6:(C_2\times D_6)$ x 2, $C_5^6:S_4$ x 2, $C_5^6.D_6.C_2$, $C_5^6:(C_2\times A_4)$

Order 328125: $C_5^6:C_7:C_3$

Order 312500: $C_5^7.C_2^2$ x 5, $C_5^7.C_4$ x 2, $C_5^6.D_{10}$ x 2

Order 250000: $C_5^6:D_8$ x 4, $C_5^6:(C_2\times D_4)$ x 2, $C_5^6:(C_2\times D_4)$ x 2, $C_5^6:(C_2\times C_8)$, $C_5^6:(C_2\times C_8)$

Order 234375: $C_5\times C_5^6:C_3$

Order 218750: $C_5^6:D_7$ x 2, $C_5^6:C_{14}$

Order 187500: $C_5^6:D_6$ x 8, $C_5^6.D_6$ x 4, $C_5\wr D_6$ x 4, $C_5^6:(C_2\times C_6)$ x 4, $C_5^6:D_6$ x 4, $C_5^5.C_{15}.C_2^2$ x 2, $C_5^6.C_6.C_2$, $C_5^6:A_4$

Order 156250: $C_5^7.C_2$ x 5

Order 125000: $C_5^6.C_2^3$ x 16, $C_5^5.D_{10}.C_2$ x 7, $C_5^6:D_4$ x 4, $C_5^6:D_4$ x 4, $C_5^6:D_4$ x 4, $C_5^6:D_4$ x 2, $C_5^6:C_8$ x 2, $C_5^6:C_8$, $C_5^6:C_8$

Order 109375: $C_5^6:C_7$

Order 100000: $D_5\times C_5^4:D_8$

Order 93750: $C_5\wr S_3$ x 16, $C_5^6.C_6$ x 15, $C_5\wr C_6$ x 8, $C_5^6.S_3$ x 2, $C_5^4.C_{15}.C_{10}$ x 2

Order 78125: $C_5^7$

Order 75000: $C_5^4:(S_3\times D_{10})$ x 3, $C_5^5.D_6.C_2$ x 2

Order 62500: $C_5^5.D_{10}$ x 81, $C_5^6.C_2^2$ x 55, $C_5^6.C_4$ x 14

Order 50000: $C_5\wr C_4:D_{10}$ x 6, $C_5^4:D_4\times D_5$ x 4, $C_5^5:D_8$ x 2, $C_5^4:D_{40}$, $C_5^5:D_8$, $C_5^4:(C_8\times D_5)$, $C_5^4:C_8\times D_5$, $(C_5^4\times D_5):C_8$

Order 46875: $C_5^6:C_3$ x 9, $C_5\times C_5^5:C_3$ x 4

Order 37500: $C_5^5.D_6$ x 10, $C_5^5:D_6$ x 10, $C_5^4.C_{15}.C_2^2$ x 6, $C_5^5:D_6$ x 6, $C_5^5:(C_2\times C_6)$ x 5, $C_5^5:D_6$ x 3, $C_5^4:D_{30}$ x 3, $C_5^5.C_6.C_2$ x 2

Order 672: $C_2\times \PGL(2,7)$

Order 32: $C_2\times D_8$

Order 7: $C_7$

Order 3: $C_3$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 52500000: $C_5^7:(C_2\times \PGL(2,7))$ ($C_1$)

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

Order 13125000: $C_5^7:\GL(3,2)$ ($C_2^2$)

Order 156250: $C_5^7.C_2$ ($\PGL(2,7)$)

Order 78125: $C_5^7$ ($C_2\times \PGL(2,7)$)

Order 1: $C_1$ ($C_5^7:(C_2\times \PGL(2,7))$)

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 1 larger groups in the database.

Character theory

Complex character table

The $424 \times 424$ character table is not available for this group.

Rational character table

The $214 \times 214$ rational character table is not available for this group.