C₃²×#2201 #11769250

order	1152
nilpotency class	4
isoclinism	isoclinic class of #2199
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₂³×C₃

Aliases

direct product

C₃²×#2201

Subgroups

	subgroup	G/H
center, socle	C₂×C₃²	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×D₄	C₂³×C₃²
prime radical	C₃²×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃² ◁ C₂²×C₃² ◁ C₂×C₃²×D₄ ◁ G

678 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 678 conjugacy classes of subgroups — too many to draw. Only its 168 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) = 38
transitive degree	μ_t(G) = 288

A minimal faithful permutation representation — 3 generators on 38 points:

(1,2,3)(7,8)(9,17)(10,14)(11,15)(12,16)(13,19)(18,34)(20,33)(21,32)(22,28)(23,29)(24,31)(25,35)(26,36)(27,37)(30,38);(4,5,6)(7,9,12,20)(8,13,16,26)(10,21,23,18)(11,19,24,36)(14,27,29,25)(15,17,31,33)(22,37,38,35)(28,32,30,34);(7,10,12,23)(8,14,16,29)(9,18,20,21)(11,22,24,38)(13,25,26,27)(15,30,31,28)(17,32,33,34)(19,35,36,37)

Conjugacy classes & character table

153 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 3261);