C₉×#2237 #11766958

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

Aliases

direct product

C₉×#2237

Subgroups

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

198 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 576

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

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, 969);