C₉×#2249 #11766970

order	1152
nilpotency class	4
isoclinism	isoclinic class of #2215
automorphism	?
homology	Abelianization (H₁) = C₂³×C₉ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₉×#2249

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	C₂×C₃×D₁₆	C₂×C₃
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ C₄²×C₉ ◁ C₄×C₈×C₉ ◁ G

276 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 108 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) = 45
transitive degree	μ_t(G) = 576

A minimal faithful permutation representation — 3 generators on 45 points:

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

Conjugacy classes & character table

342 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 981);