C₇×#2039 #11730508

order	896
nilpotency class	3
isoclinism	isoclinic class of #2002
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂³

Aliases

	subgroup	G/H
center, socle	C₂²×C₇	#120
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₇
prime radical	C₇×#537	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ G

306 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 84 normal subgroups are shown.

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

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 3 generators on 19 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(896, 5274);