C₃²×#2098 #11769147

order	1152
nilpotency class	3
isoclinism	isoclinic class of #2010
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₂²×C₃

Aliases

direct product

C₃²×#2098

Subgroups

	subgroup	G/H
center	C₃²×C₄	#127
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₃²
socle	C₂×C₃²	#399
prime radical	C₂×C₃²×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃²×C₄ ◁ C₂²×C₃²×C₄ ◁ G

630 conjugacy classes of subgroups; only the 240 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 240 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)(7,8,11,16,12,17,13,27)(9,18,20,30,22,34,21,14)(10,23,25,33,26,15,28,32)(19,35,24,38,36,31,37,29);(4,5,6)(7,9,13,20,12,22,11,21)(8,14,27,18,17,30,16,34)(10,24,28,36,26,37,25,19)(15,31,33,29,23,35,32,38);(7,10)(8,15)(9,19)(11,25)(12,26)(13,28)(14,29)(16,32)(17,23)(18,35)(20,24)(21,37)(22,36)(27,33)(30,38)(31,34)

Conjugacy classes & character table

234 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 3158);