Abstract group 250000.ck: $C_5^6:\OD_{16}$

• Label: 250000.ck
• Order: \( 2^{4} \cdot 5^{6} \)
• Exponent: \( 2^{3} \cdot 5 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 5 \)
• $\card{Z(G)}$: 5
• $\card{\Aut(G)}$: \( 2^{12} \cdot 5^{5} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{8} \)
• Perm deg.: $30$
• Trans deg.: $40$
• Rank: $2$

Group information

Description:	$C_5^6:\OD_{16}$
Order:	\(250000\)\(\medspace = 2^{4} \cdot 5^{6} \)
Exponent:	\(40\)\(\medspace = 2^{3} \cdot 5 \)
Automorphism group:	Group of order \(12800000\)\(\medspace = 2^{12} \cdot 5^{5} \)
Composition factors:	$C_2$ x 4, $C_5$ x 6
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	20	40
Elements	1	675	2500	15624	25000	46200	60000	100000	250000
Conjugacy classes	1	2	3	1064	4	188	52	16	1330
Divisions	1	2	2	280	2	50	8	2	347
Autjugacy classes	1	2	2	35	1	14	6	1	62

Minimal presentations

Permutation degree:	$30$
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	8	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f \mid b^{20}=c^{5}=d^{5}=e^{5}=f^{5}=[c,d]=[c,e]= \!\cdots\! \rangle}$
Permutation group:	Degree $30$ $\langle(1,2,4,7,5,8,10,3)(6,9)(11,12,14,13,15)(16,17,19,21,24,23,25,22)(18,20) \!\cdots\! \rangle$
Transitive group:	40T106969				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:	$(C_5^6:C_8)$ . $C_2$ (2)	$(C_5^5:C_8)$ . $C_{10}$ (2)	$C_5^4$ . $(C_{20}.C_{20})$	$(C_5^5:\OD_{16})$ . $C_5$	all 29

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

Homology

Abelianization:	$C_{2} \times C_{20} \simeq C_{2} \times C_{4} \times C_{5}$
Schur multiplier:	$C_{5}$
Commutator length:	$1$

Subgroups

There are 42 normal subgroups (30 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 $C_5$
Commutator:	a subgroup isomorphic to 6250.278
Frattini:	a subgroup isomorphic to $C_1$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
2-Sylow subgroup:	$P_{ 2 } \simeq$ $\OD_{16}$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5^6$

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 125 (order at least 2000) are shown, as well as all normal subgroups of any index.

Order 250000: $C_5^6:\OD_{16}$

Order 125000: $C_5^6:C_8$ x 2, $C_5^2\times C_5^4:(C_2\times C_4)$

Order 62500: $C_5^2\times C_5^4:C_4$ x 2, $C_5^2\times C_5^4:C_2^2$

Order 50000: $C_5^5:\OD_{16}$, $C_5^5.\OD_{16}$

Order 31250: $C_5^2\times C_5^4:C_2$ x 2

Order 25000: $C_5^4:(C_2\times C_{20})$ x 4, $C_5^2\times C_5^3:(C_2\times C_4)$ x 2, $C_5^5:C_8$ x 2, $C_5^4:C_{40}$ x 2

Order 15625: $C_5^6$

Order 12500: $C_5^2\times C_5:(C_5:F_5)$ x 8, $C_5\times C_5^4:C_4$ x 8, $C_5^2\times D_5\times C_5:D_5$ x 4, $C_5\times C_5^4:C_2^2$ x 4

Order 10000: $C_5^4:\OD_{16}$ x 2, $C_5^4.\OD_{16}$

Order 6250: $C_5^5:C_2$ x 46, $C_5^5:C_2$ x 24, $C_5^4\times D_5$ x 4, $C_5^5:C_2$ x 4

Order 5000: $C_5^3:(C_2\times C_{20})$ x 12, $C_5^3:C_{40}$ x 4, $C_5^2\times D_5:F_5$ x 3, $C_5^4:C_8$ x 2, $C_{10}\times C_5^2:C_{20}$, $C_5^4:(C_2\times C_4)$

Order 3125: $C_5^5$ x 280

Order 2500: $C_5\times C_5:(C_5:F_5)$ x 48, $C_5\times D_5\times C_5:D_5$ x 24, $C_5^3:C_{20}$ x 21, $C_5^2\times D_5^2$ x 11, $C_5^2\times C_5^2:C_4$ x 3, $C_5^4:C_4$ x 2, $C_5^4:C_2^2$, $C_5^2:C_{10}^2$

Order 2000: $C_5^3:\OD_{16}$ x 2, $C_5^3:\OD_{16}$ x 2

Order 1250: $C_5^4:C_2$

Order 625: $C_5^4$ x 3

Order 125: $C_5^3$ x 4

Order 25: $C_5^2$ x 3

Order 16: $\OD_{16}$

Order 5: $C_5$ x 2

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 250000: $C_5^6:\OD_{16}$ ($C_1$)

Order 125000: $C_5^6:C_8$ ($C_2$) x 2, $C_5^2\times C_5^4:(C_2\times C_4)$ ($C_2$)

Order 62500: $C_5^2\times C_5^4:C_4$ ($C_2^2$), $C_5^2\times C_5^4:C_4$ ($C_4$), $C_5^2\times C_5^4:C_2^2$ ($C_4$)

Order 50000: $C_5^5:\OD_{16}$ ($C_5$)

Order 31250: $C_5^2\times C_5^4:C_2$ ($C_2\times C_4$)

Order 25000: $C_5^5:C_8$ ($C_{10}$) x 2, $C_5^4:(C_2\times C_{20})$ ($C_{10}$), $C_5^4:(C_2\times C_{20})$ ($D_5$)

Order 15625: $C_5^6$ ($\OD_{16}$)

Order 12500: $C_5\times C_5^4:C_4$ ($C_2\times C_{10}$), $C_5\times C_5^4:C_4$ ($D_{10}$), $C_5\times C_5^4:C_4$ ($C_{20}$), $C_5\times C_5^4:C_4$ ($C_5:C_4$), $C_5\times C_5^4:C_2^2$ ($C_{20}$), $C_5\times C_5^4:C_2^2$ ($C_5:C_4$)

Order 6250: $C_5^5:C_2$ ($C_2\times C_{20}$), $C_5^5:C_2$ ($C_{10}:C_4$)

Order 5000: $C_5^4:(C_2\times C_4)$ ($C_5\times D_5$)

Order 3125: $C_5^5$ ($C_5\times \OD_{16}$), $C_5^5$ ($C_5:\OD_{16}$)

Order 2500: $C_5^4:C_4$ ($C_5:C_{20}$), $C_5^4:C_4$ ($C_5\times D_{10}$), $C_5^4:C_2^2$ ($C_5:C_{20}$)

Order 1250: $C_5^4:C_2$ ($C_{10}:C_{20}$)

Order 625: $C_5^4$ ($C_5^2:\OD_{16}$) x 2, $C_5^4$ ($C_{20}.C_{20}$)

Order 125: $C_5^3$ ($C_5^3:\OD_{16}$) x 2, $C_5^3$ ($C_5^3:\OD_{16}$) x 2

Order 25: $C_5^2$ ($C_5^4:\OD_{16}$) x 2, $C_5^2$ ($C_5^4.\OD_{16}$)

Order 5: $C_5$ ($C_5^5:\OD_{16}$), $C_5$ ($C_5^5.\OD_{16}$)

Order 1: $C_1$ ($C_5^6:\OD_{16}$)

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 8 larger groups in the database.

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

Character theory

Complex character table

The $1330 \times 1330$ character table is not available for this group.

Rational character table

The $347 \times 347$ rational character table is not available for this group.