C₃²×#2237 #11769286

order	1152
nilpotency class	4
isoclinism	isoclinic class of #2184
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₂×C₃

Aliases

direct product

C₃²×#2237

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃²	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₃²
prime radical	C₂³×C₃²	D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃² ◁ C₂²×C₃²×C₄ ◁ C₂²×C₃²×C₈ ◁ G

396 conjugacy classes of subgroups; only the 180 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 180 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) = 46
transitive degree	μ_t(G) = 576

A minimal faithful permutation representation — 3 generators on 46 points:

(1,2,3)(7,8)(9,12)(10,11)(13,14)(15,16,20,25)(17,21,28,36)(18,22,31,38)(19,32,30,24)(23,39,37,29)(26,33,42,45)(27,43,41,35)(34,46,44,40);(4,5,6)(8,11)(12,14)(15,17,20,28)(16,21,25,36)(18,26,31,42)(19,27,30,41)(22,33,38,45)(23,34,37,44)(24,35,32,43)(29,40,39,46);(7,9,10,13)(8,12,11,14)(16,22)(18,29)(19,30)(21,33)(23,24)(25,38)(26,40)(27,41)(31,39)(32,37)(34,35)(36,45)(42,46)(43,44)

Conjugacy classes & character table

261 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 3297);