Abstract group 50000000.qt: $C_5^6.(C_2\times C_{10}^2:\OD_{16})$

• Label: 50000000.qt
• Order: \( 2^{7} \cdot 5^{8} \)
• Exponent: \( 2^{3} \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \)
• $\card{Z(G)}$: 1
• $\card{\Aut(G)}$: \( 2^{17} \cdot 5^{8} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{10} \)
• Perm deg.: $40$
• Trans deg.: not computed
• Rank: $4$

Group information

Description:	$C_5^6.(C_2\times C_{10}^2:\OD_{16})$
Order:	\(50000000\)\(\medspace = 2^{7} \cdot 5^{8} \)
Exponent:	\(40\)\(\medspace = 2^{3} \cdot 5 \)
Automorphism group:	Group of order \(51200000000\)\(\medspace = 2^{17} \cdot 5^{8} \)
Composition factors:	$C_2$ x 7, $C_5$ x 8
Derived length:	$3$

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

Group statistics

Order	1	2	4	5	8	10
Elements	1	466975	12500000	390624	25000000	11642400	50000000
Conjugacy classes	1	19	20	3648	16	1632	5336
Divisions	1	19	12	3648	8	1632	5320

Minimal presentations

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

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

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h, i, j \mid b^{4}=c^{10}=d^{10}=e^{10}=f^{5}= \!\cdots\! \rangle}$
Permutation group:	Degree $40$ $\langle(1,6,12,38,4,9,14,36)(2,7,11,39,3,8,15,40)(5,10,13,37)(16,22,29,33)(17,21,30,34,20,23,28,32) \!\cdots\! \rangle$
Transitive group:	40T186124				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:	$(C_5^8.C_2^3)$ . $\OD_{16}$ (8)	$(C_5^8.C_2.C_2^3)$ . $D_4$ (4)	$C_5^4$ . $(C_2\times D_5^4.C_4)$ (4)	$(C_5^6.D_5:F_5)$ . $C_2^4$	all 45

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

Homology

Abelianization:	$C_{2}^{3} \times C_{4} $
Schur multiplier:	$C_{2}^{6}$
Commutator length:	not computed

Subgroups

There are 247 normal subgroups (33 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
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 32 (order at least 1562500) are shown, as well as all normal subgroups of any index.

Order 50000000: $C_5^6.(C_2\times C_{10}^2:\OD_{16})$

Order 25000000: $C_5^6.C_{10}^2:\OD_{16}$ x 4, $C_5^8.C_2.C_4.C_2^3$ x 4, $C_5^6.C_{10}^2:\OD_{16}$ x 2, $C_5^6.C_{10}^2:\OD_{16}$ x 2, $C_5^6.C_{10}^2:\OD_{16}$ x 2, $C_5^8.C_2.C_2^5$

Order 12500000: $C_5^8.C_2.C_2^4$ x 19, $C_5^8.C_4.C_2^3$ x 12, $C_5^6.C_{10}^2:C_8$ x 8, $C_5^6.C_{10}^2:C_8$ x 8, $C_5^8.C_4.C_2^3$ x 4, $C_5^6.C_{10}^2:C_8$ x 4, $C_5^7.(C_2^3\times F_5)$ x 2, $C_5^8.C_2^5$, $C_5^8.C_2^4.C_2$

Order 6250000: $C_5^8.C_4.C_2^2$ x 37, $C_5^8.C_2.C_2^3$ x 33, $C_5^8.C_2^4$ x 17, $C_5^8.\OD_{16}$ x 16, $C_5^7.(C_2^2\times F_5)$ x 12, $C_5^8.C_8.C_2$ x 12, $C_5^8.C_2^3.C_2$ x 10, $C_5^4.D_5^4$ x 2

Order 5000000: $C_5^7.C_2.C_2^5$ x 4

Order 3125000: $C_5^8.C_2^3$ x 79, $C_5^8.C_4.C_2$ x 49, $C_5^7.D_{10}.C_2$ x 19, $C_5^8.C_8$ x 8, $C_5^6.D_5:F_5$ x 8, $C_5^5.D_5^3$ x 8

Order 2500000: $C_5^7.C_2.C_2^4$ x 136, $C_5^7:(C_2^3\times C_4)$ x 8, $C_5^7.C_2^5$ x 4, $C_5^7.C_2^4.C_2$ x 4

Order 2000000: $C_5^6:(C_2^3:\OD_{16})$ x 4

Order 1562500: $C_5^8.C_2^2$ x 71, $C_5^7.D_{10}$ x 15, $C_5^8.C_4$ x 12, $C_5^2\times C_5^6.C_2^2$

Order 781250: $C_5^8.C_2$ x 7

Order 390625: $C_5^8$

Order 125000: $C_5^6.C_2^3$ x 4

Order 62500: $C_5^5.D_{10}$ x 8, $C_5^6.C_2^2$ x 4

Order 31250: $C_5^6:C_2$ x 12

Order 15625: $C_5^6$ x 4

Order 2500: $C_5^4:C_2^2$ x 2

Order 1250: $C_5^4:C_2$ x 6

Order 625: $C_5^4$ x 6

Order 25: $C_5^2$ x 4

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 50000000: $C_5^6.(C_2\times C_{10}^2:\OD_{16})$ ($C_1$)

Order 25000000: $C_5^6.C_{10}^2:\OD_{16}$ ($C_2$) x 4, $C_5^8.C_2.C_4.C_2^3$ ($C_2$) x 4, $C_5^6.C_{10}^2:\OD_{16}$ ($C_2$) x 2, $C_5^6.C_{10}^2:\OD_{16}$ ($C_2$) x 2, $C_5^6.C_{10}^2:\OD_{16}$ ($C_2$) x 2, $C_5^8.C_2.C_2^5$ ($C_2$)

Order 12500000: $C_5^8.C_4.C_2^3$ ($C_2^2$) x 8, $C_5^6.C_{10}^2:C_8$ ($C_2^2$) x 8, $C_5^6.C_{10}^2:C_8$ ($C_2^2$) x 8, $C_5^8.C_2.C_2^4$ ($C_4$) x 6, $C_5^8.C_2.C_2^4$ ($C_2^2$) x 5, $C_5^6.C_{10}^2:C_8$ ($C_2^2$) x 4, $C_5^8.C_2^5$ ($C_4$), $C_5^8.C_2^4.C_2$ ($C_2^2$), $C_5^7.(C_2^3\times F_5)$ ($C_2^2$), $C_5^7.(C_2^3\times F_5)$ ($C_4$)

Order 6250000: $C_5^8.C_4.C_2^2$ ($C_2\times C_4$) x 9, $C_5^8.C_8.C_2$ ($C_2^3$) x 8, $C_5^8.C_2.C_2^3$ ($C_2\times C_4$) x 6, $C_5^8.C_2^4$ ($C_2\times C_4$) x 5, $C_5^7.(C_2^2\times F_5)$ ($D_4$) x 4, $C_5^7.(C_2^2\times F_5)$ ($C_2\times C_4$) x 4, $C_5^8.C_4.C_2^2$ ($C_2^3$) x 3, $C_5^8.C_4.C_2^2$ ($D_4$) x 2, $C_5^8.C_2^3.C_2$ ($C_2^3$) x 2, $C_5^8.C_2^3.C_2$ ($C_2\times C_4$) x 2, $C_5^8.C_2.C_2^3$ ($D_4$) x 2, $C_5^7.(C_2^2\times F_5)$ ($C_2^3$) x 2, $C_5^4.D_5^4$ ($C_2\times C_4$) x 2

Order 3125000: $C_5^8.C_4.C_2$ ($C_2^2:C_4$) x 9, $C_5^8.C_4.C_2$ ($C_2\times D_4$) x 8, $C_5^8.C_2^3$ ($\OD_{16}$) x 8, $C_5^8.C_2^3$ ($C_2^2\times C_4$) x 7, $C_5^8.C_4.C_2$ ($C_2^2\times C_4$) x 4, $C_5^8.C_2^3$ ($C_2^2:C_4$) x 4, $C_5^7.D_{10}.C_2$ ($C_2^2:C_4$) x 2, $C_5^7.D_{10}.C_2$ ($C_2\times D_4$) x 2, $C_5^7.D_{10}.C_2$ ($C_2^2\times C_4$) x 2, $C_5^6.D_5:F_5$ ($C_2\times D_4$) x 2, $C_5^6.D_5:F_5$ ($C_2^2:C_4$), $C_5^6.D_5:F_5$ ($C_2^4$), $C_5^6.D_5:F_5$ ($C_2^2\times C_4$)

Order 1562500: $C_5^8.C_2^2$ ($C_2\times \OD_{16}$) x 10, $C_5^8.C_4$ ($C_2^3:C_4$) x 6, $C_5^7.D_{10}$ ($C_2^3:C_4$) x 6, $C_5^8.C_4$ ($C_2^2\times D_4$) x 2, $C_5^7.D_{10}$ ($C_2\times \OD_{16}$) x 2, $C_5^7.D_{10}$ ($C_2^3\times C_4$)

Order 781250: $C_5^8.C_2$ ($C_2^2:\OD_{16}$) x 4, $C_5^8.C_2$ ($C_2^2\times \OD_{16}$) x 2, $C_5^8.C_2$ ($C_2^4:C_4$)

Order 390625: $C_5^8$ ($C_2^3:\OD_{16}$)

Order 125000: $C_5^6.C_2^3$ ($C_5^2:\OD_{16}$) x 4

Order 62500: $C_5^5.D_{10}$ ($C_2\times C_5^2:\OD_{16}$) x 8, $C_5^6.C_2^2$ ($C_2\times C_5^2:\OD_{16}$) x 4

Order 31250: $C_5^6:C_2$ ($C_{10}^2:\OD_{16}$) x 8, $C_5^6:C_2$ ($C_{10}^2:\OD_{16}$) x 4

Order 15625: $C_5^6$ ($C_2\times C_{10}^2:\OD_{16}$) x 4

Order 2500: $C_5^4:C_2^2$ ($C_5^2\wr C_2:\OD_{16}$) x 2

Order 1250: $C_5^4:C_2$ ($D_5^4.C_4$) x 4, $C_5^4:C_2$ ($C_5^4:(C_2^2\times \OD_{16})$) x 2

Order 625: $C_5^4$ ($C_2\times D_5^4.C_4$) x 4, $C_5^4$ ($C_5:D_5^3:\OD_{16}$) x 2

Order 25: $C_5^2$ ($C_5^6:(C_2^3:\OD_{16})$) x 4

Order 1: $C_1$ ($C_5^6.(C_2\times C_{10}^2:\OD_{16})$)

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 33 larger groups in the database.

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

Character theory

Complex character table

The $5336 \times 5336$ character table is not available for this group.

Rational character table

The $5320 \times 5320$ rational character table is not available for this group.