C₃²×#2203 #11769252

order	1152
nilpotency class	4
isoclinism	isoclinic class of #2202
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₂³×C₃

Aliases

direct product

C₃²×#2203

Subgroups

	subgroup	G/H
center, socle	C₂×C₃²	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×Q₈	C₂³×C₃²
prime radical	C₃²×#454	C₂
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃² ◁ C₂²×C₃² ◁ C₂×C₃²×Q₈ ◁ G

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

normal subgroups only

This group has 648 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)(9,18)(10,22)(11,14)(12,17)(13,19)(15,25)(16,27)(20,35)(21,33)(23,31)(24,32)(26,36)(28,34)(29,37)(30,38);(4,5,6)(7,9,12,21)(8,13,17,31)(10,23,25,19)(11,26,27,20)(14,29,16,30)(15,33,22,18)(24,37,28,38)(32,35,34,36);(8,14)(11,27)(13,29)(15,34)(16,17)(18,35)(20,26)(22,32)(24,28)(30,31)(33,36)(37,38)

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, 3263);