C₃²×#2614 #11769663

order	1152
nilpotency class	2
isoclinism	isoclinic class of #2613
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃² · Schur multiplier (H₂) = C₂⁶×C₃

Aliases

direct product

C₃²×#2614

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

2346 conjugacy classes of subgroups; only the 696 normal subgroups are shown.

normal subgroups only

This group has 2346 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)(12,14)(15,16,19,22)(17,23,25,20)(18,21,27,26)(24,29,30,28);(4,5,6)(8,10)(12,14)(15,17,19,25)(16,20,22,23)(18,24,27,30)(21,28,26,29);(7,8)(9,10)(11,12)(13,14)(15,16)(17,23)(18,26)(19,22)(20,25)(21,27)(24,28)(29,30);(15,18)(16,21)(17,24)(19,27)(20,28)(22,26)(23,29)(25,30)

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, 3674);