C₃²×#2229 #11769278

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×#2229

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²×C₃²	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₃²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃² ◁ C₃²×C₄² ◁ C₃²×C₄×C₈ ◁ G

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

normal subgroups only

This group has 372 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.

Full subgroup data →

Presentation

minimal degree	μ(G) = 46
transitive degree	μ_t(G) = 1152

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

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

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, 3289);