C₇×#1699 #11730168

order	896
nilpotency class	3
isoclinism	isoclinic class of #1685
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂

Aliases

	subgroup	G/H
center, socle	C₂²×C₇	#121
commutator, Frattini, cosocle	C₄²	C₂³×C₇
prime radical	C₂×C₇×D₄	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₄²×C₇ ◁ G

160 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 64 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) = 448

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(896, 4934);