Abstract group 6250000.n: $C_5^4.D_5^4$

• Label: 6250000.n
• Order: \( 2^{4} \cdot 5^{8} \)
• Exponent: \( 2 \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: 1
• $\card{\Aut(G)}$: \( 2^{22} \cdot 3 \cdot 5^{8} \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{18} \cdot 3 \cdot 7 \)
• Perm deg.: $40$
• Trans deg.: $40$
• Rank: $4$

Group information

Description:	$C_5^4.D_5^4$
Order:	\(6250000\)\(\medspace = 2^{4} \cdot 5^{8} \)
Exponent:	\(10\)\(\medspace = 2 \cdot 5 \)
Automorphism group:	Group of order \(34406400000000\)\(\medspace = 2^{22} \cdot 3 \cdot 5^{8} \cdot 7 \)
Composition factors:	$C_2$ x 4, $C_5$ x 8
Derived length:	$2$

This group is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.

Group statistics

Order	1	2	5	10
Elements	1	399375	390624	5460000	6250000
Conjugacy classes	1	15	24960	1344	26320
Divisions	1	15	12480	672	13168

Minimal presentations

Permutation degree:	$40$
Transitive degree:	$40$
Rank:	$4$
Inequivalent generating quadruples:	not computed

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

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h, i \mid b^{10}=c^{10}=d^{10}=e^{5}=f^{5}= \!\cdots\! \rangle}$
Permutation group:	Degree $40$ $\langle(1,13,3,11,5,14,2,12,4,15)(6,40,8,38,10,36,7,39,9,37)(16,29,18,27,20,30,17,28,19,26) \!\cdots\! \rangle$
Transitive group:	40T171396				more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	not computed
Trans. wreath product:	not computed
Possibly split product:	$C_5^4$ . $D_5^4$ (56)	$C_5^6$ . $D_{10}^2$ (28)	$(C_5^5.D_5^3)$ . $C_2$ (8)	$(C_5^7:C_2^3)$ . $D_5$ (8)	all 19

Elements of the group are displayed as words in the presentation generators from the presentation above.

Homology

Abelianization:	$C_{2}^{4} $
Schur multiplier:	not computed
Commutator length:	not computed

Subgroups

There are 624 normal subgroups (4 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:	not computed
Frattini:	a subgroup isomorphic to $C_1$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^4$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5^8$

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 16 (order at least 390625) are shown, as well as all normal subgroups of any index.

Order 6250000: $C_5^4.D_5^4$

Order 3125000: $C_5^5.D_5^3$ x 8, $C_5^8.C_2^3$ x 7

Order 1562500: $C_5\times C_5^7.C_2^2$ x 16, $C_5^8.C_2^2$ x 10, $C_5^7.D_{10}$ x 6, $C_5^2\times C_5^5.D_{10}$ x 2, $(C_5^4:C_2)^2$

Order 1250000: $C_5^7:C_2^4$ x 8

Order 781250: $C_5^2\times C_5^6:C_2$ x 5, $C_5\times C_5^7.C_2$ x 5, $C_5^3\times C_5^5:C_2$ x 3, $C_5^8.C_2$ x 2

Order 625000: $C_5^7.C_2^3$ x 112, $C_5^7:C_2^3$ x 56, $C_5^7:C_2^3$ x 8

Order 390625: $C_5^8$

Order 312500: $C_5^7.C_2^2$ x 47, $C_5\times C_5^5.D_{10}$ x 9

Order 156250: $C_5\times C_5^6:C_2$ x 22, $C_5^7.C_2$ x 20, $C_5^2\times C_5^5:C_2$ x 13, $C_5^3\times C_5^4:C_2$

Order 78125: $C_5^7$ x 8

Order 62500: $C_5^5.D_{10}$ x 28

Order 31250: $C_5^6:C_2$ x 54, $C_5\times C_5^5:C_2$ x 26, $C_5^2\times C_5^4:C_2$ x 4

Order 15625: $C_5^6$ x 28

Order 6250: $C_5^5:C_2$ x 56

Order 3125: $C_5^5$ x 56

Order 1250: $C_5^4:C_2$ x 14

Order 625: $C_5^4$ x 70

Order 125: $C_5^3$ x 56

Order 25: $C_5^2$ x 28

Order 16: $C_2^4$

Order 5: $C_5$ x 8

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 6250000: $C_5^4.D_5^4$ ($C_1$)

Order 3125000: $C_5^5.D_5^3$ ($C_2$) x 8, $C_5^8.C_2^3$ ($C_2$) x 7

Order 1562500: $C_5\times C_5^7.C_2^2$ ($C_2^2$) x 16, $C_5^8.C_2^2$ ($C_2^2$) x 10, $C_5^7.D_{10}$ ($C_2^2$) x 6, $C_5^2\times C_5^5.D_{10}$ ($C_2^2$) x 2, $(C_5^4:C_2)^2$ ($C_2^2$)

Order 781250: $C_5^2\times C_5^6:C_2$ ($C_2^3$) x 5, $C_5\times C_5^7.C_2$ ($C_2^3$) x 5, $C_5^3\times C_5^5:C_2$ ($C_2^3$) x 3, $C_5^8.C_2$ ($C_2^3$) x 2

Order 625000: $C_5^7:C_2^3$ ($D_5$) x 8

Order 390625: $C_5^8$ ($C_2^4$)

Order 312500: $C_5^7.C_2^2$ ($D_{10}$) x 47, $C_5\times C_5^5.D_{10}$ ($D_{10}$) x 9

Order 156250: $C_5\times C_5^6:C_2$ ($C_2\times D_{10}$) x 22, $C_5^7.C_2$ ($C_2\times D_{10}$) x 20, $C_5^2\times C_5^5:C_2$ ($C_2\times D_{10}$) x 13, $C_5^3\times C_5^4:C_2$ ($C_2\times D_{10}$)

Order 78125: $C_5^7$ ($C_2^2\times D_{10}$) x 8

Order 62500: $C_5^5.D_{10}$ ($D_5^2$) x 28

Order 31250: $C_5^6:C_2$ ($D_5\times D_{10}$) x 54, $C_5\times C_5^5:C_2$ ($D_5\times D_{10}$) x 26, $C_5^2\times C_5^4:C_2$ ($D_5\times D_{10}$) x 4

Order 15625: $C_5^6$ ($D_{10}^2$) x 28

Order 6250: $C_5^5:C_2$ ($D_5^3$) x 56

Order 3125: $C_5^5$ ($C_2\times D_5^3$) x 56

Order 1250: $C_5^4:C_2$ ($C_5:D_5^3$) x 14

Order 625: $C_5^4$ ($D_5^4$) x 56, $C_5^4$ ($C_{10}:D_5^3$) x 14

Order 125: $C_5^3$ ($C_5:D_5^4$) x 56

Order 25: $C_5^2$ ($C_5^6:C_2^4$) x 28

Order 5: $C_5$ ($C_5^7:C_2^4$) x 8

Order 1: $C_1$ ($C_5^4.D_5^4$)

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 7 larger groups in the database.

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

Character theory

Complex character table

The $26320 \times 26320$ character table is not available for this group.

Rational character table

The $13168 \times 13168$ rational character table is not available for this group.