C₃²×#1868 #11768917

order	1152
nilpotency class	3
isoclinism	isoclinic class of B(4,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃²×C₄ · Schur multiplier (H₂) = C₂³×C₃

Aliases

direct product

C₃²×#1868

Subgroups

	subgroup	G/H
center	C₃²×C₄	#120
commutator	C₂³	C₂²×C₃²×C₄
Frattini, cosocle	C₂²×C₄	C₂³×C₃²
socle	C₂×C₃²	#385
prime radical	C₂×C₃²×#41	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃²×C₄ ◁ C₂²×C₃²×C₄ ◁ G

780 conjugacy classes of subgroups; only the 324 normal subgroups are shown.

normal subgroups only

This group has 780 conjugacy classes of subgroups — too many to draw. Only its 324 normal subgroups are shown.

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

Full subgroup data →

Presentation

minimal degree	μ(G) = 38
transitive degree	μ_t(G) = 288

A minimal faithful permutation representation — 3 generators on 38 points:

(1,2,3)(7,8,11,15,12,16,22,28)(9,13,19,25,18,24,32,36)(10,14,21,27,20,26,33,37)(17,29,31,38,30,23,34,35);(4,5,6)(7,9,11,19,12,18,22,32)(8,13,15,25,16,24,28,36)(10,17,21,31,20,30,33,34)(14,23,27,35,26,29,37,38);(7,10)(8,14)(9,17)(11,21)(12,20)(13,23)(15,27)(16,26)(18,30)(19,31)(22,33)(24,29)(25,35)(28,37)(32,34)(36,38)

Conjugacy classes & character table

288 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 2928);