C₇×#1631 #11730100

order	896
nilpotency class	3
isoclinism	isoclinic class of #1619
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂²

Aliases

	subgroup	G/H
center, socle	C₂²×C₇	#120
commutator, Frattini, cosocle	C₄²	C₂³×C₇
prime radical	C₇×#550	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₄²×C₇ ◁ G

256 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 72 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)(8,9,11,14)(10,12,15,13)(17,20)(18,22)(21,23);(9,12)(10,15)(13,14)(18,22)(20,23);(16,17,19,21)(18,23,22,20)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(896, 4866);