C₃²×#3027 #11770076

order	1152
nilpotency class	3
isoclinism	isoclinic class of #3026
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃² · Schur multiplier (H₂) = C₂⁵×C₃×C₄

Aliases

direct product

C₃²×#3027

Subgroups

	subgroup	G/H
center, socle	C₂×C₃²	C₂×#120
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃² ◁ C₂²×C₃²×C₄ ◁ G

2316 conjugacy classes of subgroups; only the 636 normal subgroups are shown.

normal subgroups only

This group has 2316 conjugacy classes of subgroups — too many to draw. Only its 636 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) = 144

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

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

Conjugacy classes & character table

261 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 4087);