C₇×#2200 #11730669

order	896
nilpotency class	4
isoclinism	isoclinic class of #2199
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂³

Aliases

	subgroup	G/H
center, socle	C₂×C₇	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×D₄	C₂³×C₇
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂²×C₇ ◁ C₂×C₇×D₄ ◁ G

260 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 56 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)(10,16)(11,14)(12,15)(13,18)(17,22)(19,21)(20,23);(8,10,12,17)(9,13,15,20)(11,18,19,23)(14,16,21,22);(10,17)(11,19)(13,20)(16,22)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(896, 5435);