C₃²×#1676 #11768725

order	1152
nilpotency class	3
isoclinism	isoclinic class of #1665
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₂×C₃

Aliases

direct product

C₃²×#1676

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²×C₃²	#121
commutator, Frattini, cosocle	C₄²	C₂³×C₃²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃² ◁ C₃²×C₄² ◁ G

432 conjugacy classes of subgroups; only the 216 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 216 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) = 1152

A minimal faithful permutation representation — 3 generators on 38 points:

(1,2,3)(7,8,11,14)(9,15,18,22)(10,19,16,13)(12,20,21,17)(23,24,27,30)(25,31,33,38)(26,34,36,29)(28,35,37,32);(4,5,6)(7,9,16,20,11,18,10,17)(8,12,19,22,14,21,13,15);(23,25,27,33)(24,28,30,37)(26,35,36,32)(29,38,34,31)

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, 2736);