C₇×#2225 #11730694

order	896
nilpotency class	4
isoclinism	isoclinic class of #2184
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₇×#2225

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²×C₇	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₇
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₄²×C₇ ◁ C₄×C₇×C₈ ◁ G

130 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 130 conjugacy classes of subgroups — too many to draw. Only its 68 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 31
transitive degree	μ_t(G) = 896

A minimal faithful permutation representation — 3 generators on 31 points:

(1,2,3,4,5,6,7)(8,9,11,13)(10,14,15,12)(17,21)(18,24)(19,26)(22,31)(23,30)(25,28)(27,29);(8,10,11,15)(9,12,13,14);(16,17,20,23)(18,25,27,31)(19,28,26,22)(21,29,30,24)

Conjugacy classes & character table

203 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#13267582, #13267589, #13267638, …
maximal subgroup of	C₇×#12369, C₇×#12370, C₇×#12377, …
maximal normal subgroup of	C₇×#12369, C₇×#12370, C₇×#12377, …
maximal quotient of	C₇×#12812, C₇×#12938, C₇×#13110, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(896, 5460);