C₃²×#1830 #11768879

order	1152
nilpotency class	3
isoclinism	isoclinic class of #1828
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃²×C₄ · Schur multiplier (H₂) = C₂³×C₃

Aliases

direct product

C₃²×#1830

Subgroups

	subgroup	G/H
center, socle	C₂×C₃²	#381
commutator	C₂³	C₂²×C₃²×C₄
Frattini, cosocle	C₂²×C₄	C₂³×C₃²
prime radical	C₂³×C₃²	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃² ◁ C₂²×C₃²×C₄ ◁ G

708 conjugacy classes of subgroups; only the 300 normal subgroups are shown.

normal subgroups only

This group has 708 conjugacy classes of subgroups — too many to draw. Only its 300 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)(7,8,11,16,12,15,25,30)(9,17,19,31,20,13,24,27)(10,21,23,35,22,14,28,29)(18,26,34,37,33,32,36,38);(4,5,6)(7,9,12,20)(8,13,15,17)(10,18,22,33)(11,24,25,19)(14,26,21,32)(16,31,30,27)(23,36,28,34)(29,38,35,37);(7,10)(8,14,15,21)(9,18)(11,23)(12,22)(13,26,17,32)(16,29,30,35)(19,34)(20,33)(24,36)(25,28)(27,37,31,38)

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, 2890);