C₉×#2178 #11766899

order	1152
nilpotency class	4
isoclinism	isoclinic class of D₁₆
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₉×#2178

Subgroups

	subgroup	G/H
center	C₈×C₉	D₈
commutator	C₈	C₂²×C₄×C₉
Frattini, cosocle	C₂×C₃×C₈	C₂³×C₃
socle	C₂×C₃	C₃×C₄×D₈
prime radical	C₃×#508	C₂×C₃
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₉ ◁ C₂×C₈×C₉ ◁ C₄×C₈×C₉ ◁ G

219 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 219 conjugacy classes of subgroups — too many to draw. Only its 114 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)(10,11,15,19,16,21,31,20)(12,22,24,33,26,39,25,41)(13,18,27,36,28,37,14,29)(17,32,34,40,35,30,38,23);(10,12)(11,17)(13,23)(14,30)(15,24)(16,26)(18,33)(19,34)(20,38)(21,35)(22,29)(25,31)(27,32)(28,40)(36,39)(37,41);(10,13,15,27,16,28,31,14)(11,18,19,36,21,37,20,29)(12,23,24,32,26,40,25,30)(17,33,34,39,35,41,38,22)

Conjugacy classes & character table

396 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 910);