C₇×#2061 #11730530

order	896
nilpotency class	3
isoclinism	isoclinic class of #2002
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#2061

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#120
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₇
prime radical	C₂³×C₇	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ G

230 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 224

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

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

Conjugacy classes & character table

182 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#13241423, #13266711, #13266725, …
maximal subgroup of	C₇×#7442, C₇×#12235, C₇×#12237, …
maximal normal subgroup of	C₇×#7442, C₇×#12235, C₇×#12237, …
maximal quotient of	C₇×#10568, C₇×#10569, C₇×#10571, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(896, 5296);