C₃²×#1323 #11768372

order	1152
nilpotency class	3
isoclinism	isoclinic class of #327
automorphism	?
homology	Abelianization (H₁) = C₂×C₃²×C₈ · Schur multiplier (H₂) = C₂×C₃

Aliases

direct product

C₃²×#1323

Subgroups

	subgroup	G/H
center	C₂×C₃²×C₄	C₂²⋊C₄
commutator	C₂×C₄	C₂×C₃²×C₈
Frattini, cosocle	C₂²×C₈	C₂²×C₃²
socle	C₂²×C₃²	C₂²⋊C₈
prime radical	C₂³×C₃²	M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₃²×C₄ ◁ C₂²×C₃²×C₈ ◁ G

282 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 132 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 — 2 generators on 46 points:

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

Conjugacy classes & character table

342 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 2383);