C₇×#1995 #11730464

order	896
nilpotency class	3
isoclinism	isoclinic class of #468
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₇×#1995

Subgroups

	subgroup	G/H
center	C₂×C₄×C₇	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂³×C₇
socle	C₂²×C₇	C₂×C₂²⋊C₄
prime radical	C₂³×C₇	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂×C₄²×C₇ ◁ G

244 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 112 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)(10,14)(11,12)(13,15)(16,17,20,23,21,24,19,29)(18,25,28,30,26,31,27,22);(9,12)(13,14)(16,18)(17,22,24,30)(19,27)(20,28)(21,26)(23,25,29,31);(8,10,11,15)(9,13,12,14)(18,26)(22,30)(25,31)(27,28)

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#13240610, #13241191, #13246166, …
maximal subgroup of	C₇×#7129, C₇×#7360, C₇×#8494, …
maximal normal subgroup of	C₇×#7129, C₇×#7360, C₇×#8494, …
maximal quotient of	C₇×#9872, C₇×#9873, C₇×#9874, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(896, 5230);