C₉×#2203 #11766924

order	1152
nilpotency class	4
isoclinism	isoclinic class of #2202
automorphism	?
homology	Abelianization (H₁) = C₂³×C₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₉×#2203

Subgroups

	subgroup	G/H
center	C₂×C₉	B(4,𝔽₂)
commutator	C₂×Q₈	C₂³×C₉
Frattini, cosocle	C₂×C₃×Q₈	C₂³×C₃
socle	C₂×C₃	C₃×B(4,𝔽₂)
prime radical	C₃×#454	C₂×C₃
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₉ ◁ C₂²×C₉ ◁ C₂×Q₈×C₉ ◁ G

324 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 84 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) = 41
transitive degree	μ_t(G) = 288

A minimal faithful permutation representation — 3 generators on 41 points:

(1,2,3,4,5,6,7,8,9)(10,11)(12,21)(13,25)(14,17)(15,20)(16,22)(18,28)(19,30)(23,38)(24,36)(26,34)(27,35)(29,39)(31,37)(32,40)(33,41);(10,12,15,24)(11,16,20,34)(13,26,28,22)(14,29,30,23)(17,32,19,33)(18,36,25,21)(27,40,31,41)(35,38,37,39);(11,17)(14,30)(16,32)(18,37)(19,20)(21,38)(23,29)(25,35)(27,31)(33,34)(36,39)(40,41)

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, 935);