C₃²×#1597 #11768646

order	1152
nilpotency class	3
isoclinism	isoclinic class of #1595
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₂³×C₃

Aliases

	subgroup	G/H
center, socle	C₂²×C₃²	#120
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₃²
prime radical	C₃²×#447	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃² ◁ C₂²×C₃²×C₄ ◁ G

816 conjugacy classes of subgroups; only the 192 normal subgroups are shown.

normal subgroups only

This group has 816 conjugacy classes of subgroups — too many to draw. Only its 192 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) = 288

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 2657);