C₉×#2245 #11766966

order	1152
nilpotency class	4
isoclinism	isoclinic class of #2218
automorphism	?
homology	Abelianization (H₁) = C₂³×C₉ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₉×#2245

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

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

normal subgroups only

This group has 198 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)(12,15)(13,14)(16,17)(18,19,25,31)(20,27,35,46)(21,26,37,44)(22,29,39,41)(23,28,32,47)(24,40,38,30)(33,45,48,43)(34,49,36,42);(11,14)(15,17)(18,20,32,48,24,34,22,21,25,35,23,33,38,36,39,37)(19,26,41,49,30,43,28,27,31,44,29,42,40,45,47,46);(10,12,13,16)(11,15,14,17)(18,21,22,36,38,48,32,35,25,37,39,34,24,33,23,20)(19,27,28,45,40,49,41,44,31,46,47,43,30,42,29,26)

Conjugacy classes & character table

234 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 977);