C₃²×#2381 #11769430

order	1152
nilpotency class	2
isoclinism	isoclinic class of #560
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃²×C₄ · Schur multiplier (H₂) = C₂³×C₃×C₄

Aliases

direct product

C₃²×#2381

Subgroups

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

1272 conjugacy classes of subgroups; only the 792 normal subgroups are shown.

normal subgroups only

This group has 1272 conjugacy classes of subgroups — too many to draw. Only its 792 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)(12,15)(13,17)(20,24)(21,27)(26,29)(28,33);(4,5,6)(12,15)(16,18)(19,20,23,24)(21,25,27,31)(22,29,30,26)(28,34,33,32);(11,12,14,15)(13,16,17,18)(19,21)(20,25)(22,28)(23,27)(24,31)(26,32)(29,34)(30,33);(11,13)(12,16)(14,17)(15,18)(19,22,23,30)(20,26,24,29)(21,28,27,33)(25,32,31,34)

Conjugacy classes & character table

342 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 3441);