C₃²×#1818 #11768867

order	1152
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃²×C₄ · Schur multiplier (H₂) = C₂³×C₃

Aliases

direct product

C₃²×#1818

Subgroups

	subgroup	G/H
center	C₂²×C₃²×C₄	C₂³
commutator	C₂³	C₂²×C₃²×C₄
Frattini, cosocle	C₂²×C₄	C₂³×C₃²
socle	C₂³×C₃²	C₂²×C₄
prime radical	C₂⁴×C₃²	C₂×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₃²×C₄ ◁ G

876 conjugacy classes of subgroups; only the 360 normal subgroups are shown.

normal subgroups only

This group has 876 conjugacy classes of subgroups — too many to draw. Only its 360 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) = 34
transitive degree	μ_t(G) = 576

A minimal faithful permutation representation — 3 generators on 34 points:

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

Conjugacy classes & character table

396 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 2878);