C₃²×#1467 #11768516

order	1152
nilpotency class	3
isoclinism	isoclinic class of #1462
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃²×C₄ · Schur multiplier (H₂) = C₂³×C₃

Aliases

direct product

C₃²×#1467

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃²	C₂×C₂²⋊C₄
commutator	C₂³	C₂²×C₃²×C₄
Frattini, cosocle	C₂²×C₄	C₂³×C₃²
prime radical	C₂³×C₃²	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃² ◁ C₂×C₃²×C₄² ◁ G

708 conjugacy classes of subgroups; only the 300 normal subgroups are shown.

normal subgroups only

This group has 708 conjugacy classes of subgroups — too many to draw. Only its 300 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)(7,8,10,12,11,13,9,14)(15,16)(17,23,25,20)(18,21)(19,22)(24,29,30,27)(26,28);(4,5,6)(15,17,19,25)(16,20,22,23)(18,24,26,30)(21,27,28,29);(7,9,11,10)(8,12,13,14)(15,18,19,26)(16,21,22,28)(17,24,25,30)(20,27,23,29)

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, 2527);