C₃²×#2657 #11769706

order	1152
nilpotency class	2
isoclinism	isoclinic class of #2613
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃² · Schur multiplier (H₂) = C₂²×C₃×C₄³

Aliases

direct product

C₃²×#2657

Subgroups

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

2532 conjugacy classes of subgroups; only the 696 normal subgroups are shown.

normal subgroups only

This group has 2532 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) = 22
transitive degree	μ_t(G) = 576

A minimal faithful permutation representation — 4 generators on 22 points:

(1,2,3)(7,8,9,10)(15,16)(17,19)(18,20)(21,22);(4,5,6)(11,12,13,14)(15,16,18,20)(17,19,22,21);(7,8)(9,10)(11,12)(13,14)(15,16)(17,21)(18,20)(19,22);(15,17,18,22)(16,19,20,21)

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, 3717);