C₃²×#2658 #11769707

order	1152
nilpotency class	2
isoclinism	isoclinic class of #2613
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃² · Schur multiplier (H₂) = C₂⁴×C₃×C₄

Aliases

direct product

C₃²×#2658

Subgroups

	subgroup	G/H
center, socle	C₂³×C₃²	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃²
prime radical	C₂²×C₃²×D₄	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₃² ◁ G

1836 conjugacy classes of subgroups; only the 696 normal subgroups are shown.

normal subgroups only

This group has 1836 conjugacy classes of subgroups — too many to draw. Only its 696 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 — 4 generators on 30 points:

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

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, 3718);