C₇×#2042 #11730511

order	896
nilpotency class	3
isoclinism	isoclinic class of #1999
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂³

Aliases

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

318 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 100 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)(10,11)(13,14)(16,17,21,23)(18,19,22,20);(12,13)(14,15)(16,18)(17,19)(20,23)(21,22);(9,11)(12,14,15,13)(17,22)(18,23)(19,20)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(896, 5277);