C₃²×#2211 #11769260

order	1152
nilpotency class	4
isoclinism	isoclinic class of #2184
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₂²×C₃

Aliases

direct product

C₃²×#2211

Subgroups

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

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

normal subgroups only

This group has 474 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) = 42
transitive degree	μ_t(G) = 576

A minimal faithful permutation representation — 3 generators on 42 points:

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

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, 3271);