C₃²×#2204 #11769253

order	1152
nilpotency class	4
isoclinism	isoclinic class of #2204
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₂³×C₃

Aliases

	subgroup	G/H
center, socle	C₂×C₃²	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×Q₈	C₂³×C₃²
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃² ◁ C₂²×C₃² ◁ C₂×C₃²×Q₈ ◁ G

708 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 708 conjugacy classes of subgroups — too many to draw. Only its 168 normal subgroups are shown.

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

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 144

A minimal faithful permutation representation — 3 generators on 22 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 3264);