Abstract group 625000.bp: $C_5^7:D_4$

• Label: 625000.bp
• Order: \( 2^{3} \cdot 5^{7} \)
• Exponent: \( 2^{2} \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: 1
• $\card{\Aut(G)}$: \( 2^{14} \cdot 3 \cdot 5^{8} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{11} \cdot 3 \cdot 5 \)
• Perm deg.: $35$
• Trans deg.: $40$
• Rank: $2$

Group information

Description:	$C_5^7:D_4$
Order:	\(625000\)\(\medspace = 2^{3} \cdot 5^{7} \)
Exponent:	\(20\)\(\medspace = 2^{2} \cdot 5 \)
Automorphism group:	Group of order \(19200000000\)\(\medspace = 2^{14} \cdot 3 \cdot 5^{8} \)
Composition factors:	$C_2$ x 3, $C_5$ x 7
Derived length:	$3$

This group is nonabelian and solvable. Whether it is monomial has not been computed.

Group statistics

Order	1	2	4	5	10	20
Elements	1	3125	31250	78124	387500	125000	625000
Conjugacy classes	1	3	1	9844	162	4	10015
Divisions	1	3	1	2563	51	1	2620

Minimal presentations

Permutation degree:	$35$
Transitive degree:	$40$
Rank:	$2$
Inequivalent generating pairs:	not computed

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

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h \mid b^{20}=c^{5}=d^{5}=e^{5}=f^{5}=g^{5}= \!\cdots\! \rangle}$
Permutation group:	Degree $35$ $\langle(1,2,4,7)(3,5,9,12)(6,10,15,20)(8,13,19,16)(11,17)(14,18)(21,22,24,23,25,27,29,30,26,28) \!\cdots\! \rangle$
Transitive group:	40T139820				more information
Direct product:	not computed
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$C_5^6$ . $D_{20}$	$C_5^6$ . $(C_5:D_4)$ (2)	$C_5$ . $(C_5^6:D_4)$ (2)	$C_5$ . $(C_5^6:D_4)$	all 22

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

Homology

Abelianization:	$C_{2}^{2} $
Schur multiplier:	$C_{10}$
Commutator length:	not computed

Subgroups

There are 79 normal subgroups (15 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:	$P_{ 2 } \simeq$ $D_4$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5^7$

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 25 (order at least 25000) are shown, as well as all normal subgroups of any index.

Order 625000: $C_5^7:D_4$

Order 312500: $C_5^7.C_2^2$ x 2, $C_5\times C_5^6.C_4$

Order 156250: $C_5^7.C_2$ x 2, $C_5\times C_5^6:C_2$

Order 125000: $C_5^6:D_4$ x 2, $C_5^6:D_4$

Order 78125: $C_5^7$

Order 62500: $C_5^6.C_2^2$ x 20, $C_5\times C_5^5:C_4$ x 10, $C_5^5.D_{10}$ x 2, $C_5^6.C_4$

Order 31250: $C_5^6:C_2$ x 174, $C_5\times C_5^5:C_2$ x 93

Order 25000: $C_5^4:D_{20}$ x 6, $C_5^5:D_4$ x 2, $C_5^4:D_{20}$

Order 15625: $C_5^6$ x 3

Order 6250: $C_5^5:C_2$ x 3

Order 3125: $C_5^5$ x 9

Order 1250: $C_5^4:C_2$

Order 625: $C_5^4$ x 19

Order 125: $C_5^3$ x 19

Order 25: $C_5^2$ x 9

Order 8: $D_4$

Order 5: $C_5$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 625000: $C_5^7:D_4$ ($C_1$)

Order 312500: $C_5^7.C_2^2$ ($C_2$) x 2, $C_5\times C_5^6.C_4$ ($C_2$)

Order 156250: $C_5\times C_5^6:C_2$ ($C_2^2$)

Order 78125: $C_5^7$ ($D_4$)

Order 62500: $C_5^5.D_{10}$ ($D_5$) x 2, $C_5^6.C_4$ ($D_5$)

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

Order 15625: $C_5^6$ ($C_5:D_4$) x 2, $C_5^6$ ($D_{20}$)

Order 6250: $C_5^5:C_2$ ($D_5^2$) x 3

Order 3125: $C_5^5$ ($D_5\wr C_2$) x 6, $C_5^5$ ($C_5:D_{20}$) x 2, $C_5^5$ ($D_{10}:D_5$)

Order 1250: $C_5^4:C_2$ ($C_5:D_5^2$)

Order 625: $C_5^4$ ($C_5^3:D_4$) x 12, $C_5^4$ ($C_5^2:D_{20}$) x 6, $C_5^4$ ($C_5^2:D_{20}$)

Order 125: $C_5^3$ ($C_5^3:D_{20}$) x 12, $C_5^3$ ($C_5^4:D_4$) x 6, $C_5^3$ ($C_5^4:D_4$)

Order 25: $C_5^2$ ($C_5^4:D_{20}$) x 6, $C_5^2$ ($C_5^5:D_4$) x 2, $C_5^2$ ($C_5^4:D_{20}$)

Order 5: $C_5$ ($C_5^6:D_4$) x 2, $C_5$ ($C_5^6:D_4$)

Order 1: $C_1$ ($C_5^7:D_4$)

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 11 larger groups in the database.

This group is a maximal quotient of 1 larger groups in the database.

Character theory

Complex character table

The $10015 \times 10015$ character table is not available for this group.

Rational character table

The $2620 \times 2620$ rational character table is not available for this group.