C₃²×#2212 #11769261

order	1152
nilpotency class	4
isoclinism	isoclinic class of #2212
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₂²×C₃

Aliases

direct product

C₃²×#2212

Subgroups

	subgroup	G/H
center, socle	C₂×C₃²	#459
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₃²
prime radical	C₃²×#509	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃² ◁ C₂×C₃²×C₄ ◁ C₃²×C₄×C₈ ◁ G

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

normal subgroups only

This group has 462 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) = 38
transitive degree	μ_t(G) = 288

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

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

Conjugacy classes & character table

207 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 3272);