C₃²×#1657 #11768706

order	1152
nilpotency class	3
isoclinism	isoclinic class of #1655
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₂²×C₃

Aliases

direct product

C₃²×#1657

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃²	#120
commutator, Frattini, cosocle	C₄²	C₂³×C₃²
prime radical	C₃²×#496	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃² ◁ C₃²×C₄² ◁ G

618 conjugacy classes of subgroups; only the 180 normal subgroups are shown.

normal subgroups only

This group has 618 conjugacy classes of subgroups — too many to draw. Only its 180 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) = 30
transitive degree	μ_t(G) = 576

A minimal faithful permutation representation — 3 generators on 30 points:

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

Conjugacy classes & character table

180 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 2717);