C₃²×#2642 #11769691

order	1152
nilpotency class	2
isoclinism	isoclinic class of #2613
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃² · Schur multiplier (H₂) = C₂³×C₃×C₄

Aliases

direct product

C₃²×#2642

Subgroups

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

1248 conjugacy classes of subgroups; only the 600 normal subgroups are shown.

normal subgroups only

This group has 1248 conjugacy classes of subgroups — too many to draw. Only its 600 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) = 1152

A minimal faithful permutation representation — 4 generators on 34 points:

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

Conjugacy classes & character table

288 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 3702);