C₃²×#1609 #11768658

order	1152
nilpotency class	3
isoclinism	isoclinic class of #1595
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₂²×C₃

Aliases

direct product

C₃²×#1609

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃²	#120
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₃²
prime radical	C₃²×#121	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃² ◁ C₂²×C₃²×C₄ ◁ G

648 conjugacy classes of subgroups; only the 192 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 192 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) = 288

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

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

Conjugacy classes & character table

207 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 2669);