C₇×#2031 #11730500

order	896
nilpotency class	3
isoclinism	isoclinic class of #2031
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂³

Aliases

	subgroup	G/H
center	C₄×C₇	#124
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₇
socle	C₂×C₇	#393
prime radical	C₂×C₇×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇ ◁ C₂²×C₄×C₇ ◁ G

244 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 80 normal subgroups are shown.

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

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 112

A minimal faithful permutation representation — 3 generators on 23 points:

(1,2,3,4,5,6,7)(8,9)(10,17,19,23)(11,15)(12,21)(13,16)(14,20,22,18);(10,18,19,20)(14,17,22,23);(8,10)(9,14)(11,18)(12,20)(13,19)(15,23)(16,22)(17,21)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(896, 5266);