C₉×#2229 #11766950

order	1152
nilpotency class	4
isoclinism	isoclinic class of #2184
automorphism	?
homology	Abelianization (H₁) = C₂³×C₉ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₉×#2229

Subgroups

	subgroup	G/H
center	C₂²×C₉	C₂×D₈
commutator	C₂×C₈	C₂³×C₉
Frattini, cosocle	C₂×C₃×C₈	C₂³×C₃
socle, prime radical	C₂²×C₃	C₂×C₃×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ C₄²×C₉ ◁ C₄×C₈×C₉ ◁ G

186 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 96 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) = 49
transitive degree	μ_t(G) = 1152

A minimal faithful permutation representation — 3 generators on 49 points:

(1,2,3,4,5,6,7,8,9)(10,11,13,15)(12,16,17,14)(18,19,21,24)(20,28,29,46)(22,34,32,47)(23,37,33,27)(25,41,39,30)(26,43,40,36)(31,48,38,44)(35,45,49,42);(10,12,13,17)(11,14,15,16)(19,24)(25,39)(26,40)(27,37)(28,46)(34,47)(42,45)(44,48);(18,20,21,29)(19,25,24,39)(22,35,32,49)(23,38,33,31)(26,44,40,48)(27,45,37,42)(28,47,46,34)(30,43,41,36)

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, 961);