Abstract group 656250.a: $C_5^6:C_7:C_6$

• Label: 656250.a
• Order: \( 2 \cdot 3 \cdot 5^{6} \cdot 7 \)
• Exponent: \( 2 \cdot 3 \cdot 5 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2 \cdot 3 \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{4} \cdot 3^{2} \cdot 5^{6} \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 3 \)
• Perm deg.: $35$
• Trans deg.: $35$
• Rank: $2$

Group information

Description:	$C_5^6:C_7:C_6$
Order:	\(656250\)\(\medspace = 2 \cdot 3 \cdot 5^{6} \cdot 7 \)
Exponent:	\(210\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 7 \)
Automorphism group:	$C_5^6.C_{42}.C_6.C_2^2$, of order \(15750000\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5^{6} \cdot 7 \)
Composition factors:	$C_2$, $C_3$, $C_5$ x 6, $C_7$
Derived length:	$3$

This group is nonabelian, solvable, and an A-group. Whether it is monomial has not been computed.

Group statistics

Order	1	2	3	5	6	7	14	15
Elements	1	15625	8750	15624	218750	93750	93750	210000	656250
Conjugacy classes	1	1	2	380	2	2	2	24	414
Divisions	1	1	1	190	1	1	1	6	202
Autjugacy classes	1	1	2	22	2	1	1	2	32

Dimension	1	2	3	6	14	28	42	56	84
Irr. complex chars.	6	0	4	0	36	0	368	0	0	414
Irr. rational chars.	2	2	0	2	0	6	0	6	184	202

Minimal presentations

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

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	14	14	28
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h \mid a^{6}=b^{7}=c^{5}=d^{5}=e^{5}=f^{5}= \!\cdots\! \rangle}$
Permutation group:	Degree $35$ $\langle(1,13,18,2,12,19)(3,11,20,5,14,17)(4,15,16)(6,33,27,9,35,30)(7,32,28,8,31,29) \!\cdots\! \rangle$
Transitive group:	35T91				more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_5^6:C_7)$ $\,\rtimes\,$ $C_6$	$C_5^6$ $\,\rtimes\,$ $(C_7:C_6)$	$(C_5^6:C_{14})$ $\,\rtimes\,$ $C_3$	$(C_5^6:C_7:C_3)$ $\,\rtimes\,$ $C_2$	more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

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

Homology

Abelianization:	$C_{6} \simeq C_{2} \times C_{3}$
Schur multiplier:	$C_{5}$
Commutator length:	$1$

Subgroups

There are 7 normal subgroups, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $C_5^6:C_7:C_6$
Commutator:	$G' \simeq$ $C_5^6:C_7$	$G/G' \simeq$ $C_6$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_5^6:C_7:C_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_5^6$	$G/\operatorname{Fit} \simeq$ $C_7:C_6$
Radical:	$R \simeq$ $C_5^6:C_7:C_6$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_5^6$	$G/\operatorname{soc} \simeq$ $C_7:C_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5^6$
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 21 (order at least 31250) are shown, as well as all normal subgroups of any index.

Order 656250: $C_5^6:C_7:C_6$

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

Order 218750: $C_5^6:C_{14}$

Order 109375: $C_5^6:C_7$

Order 93750: $C_5^6.C_6$

Order 46875: $C_5^6:C_3$

Order 31250: $C_5^6:C_2$

Order 15625: $C_5^6$

Order 42: $C_7:C_6$

Order 7: $C_7$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 21 (order at least 31250) are shown, as well as all normal subgroups of any index.

Order 656250: $C_5^6:C_7:C_6$

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

Order 218750: $C_5^6:C_{14}$

Order 109375: $C_5^6:C_7$

Order 93750: $C_5^6.C_6$

Order 46875: $C_5^6:C_3$

Order 31250: $C_5^6:C_2$

Order 15625: $C_5^6$

Order 42: $C_7:C_6$

Order 7: $C_7$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 656250: $C_5^6:C_7:C_6$ ($C_1$)

Order 328125: $C_5^6:C_7:C_3$ ($C_2$)

Order 218750: $C_5^6:C_{14}$ ($C_3$)

Order 109375: $C_5^6:C_7$ ($C_6$)

Order 31250: $C_5^6:C_2$ ($C_7:C_3$)

Order 15625: $C_5^6$ ($C_7:C_6$)

Order 1: $C_1$ ($C_5^6:C_7:C_6$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 656250: $C_5^6:C_7:C_6$ ($C_1$)

Order 328125: $C_5^6:C_7:C_3$ ($C_2$)

Order 218750: $C_5^6:C_{14}$ ($C_3$)

Order 109375: $C_5^6:C_7$ ($C_6$)

Order 31250: $C_5^6:C_2$ ($C_7:C_3$)

Order 15625: $C_5^6$ ($C_7:C_6$)

Order 1: $C_1$ ($C_5^6:C_7:C_6$)

Series

Derived series	$C_5^6:C_7:C_6$	$\rhd$	$C_5^6:C_7$	$\rhd$	$C_5^6$	$\rhd$	$C_1$
Chief series	$C_5^6:C_7:C_6$	$\rhd$	$C_5^6:C_7:C_3$	$\rhd$	$C_5^6:C_7$	$\rhd$	$C_5^6$	$\rhd$	$C_1$
Lower central series	$C_5^6:C_7:C_6$	$\rhd$	$C_5^6:C_7$
Upper central series	$C_1$

Supergroups

This group is a maximal subgroup of 5 larger groups in the database.

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

Character theory

Complex character table

See the $414 \times 414$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $202 \times 202$ rational character table (warning: may be slow to load).