C₉×#1338 #11766059

order	1152
nilpotency class	3
isoclinism	isoclinic class of #327
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₉ · Schur multiplier (H₂) = C₁

Aliases

direct product

C₉×#1338

Subgroups

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

129 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 129 conjugacy classes of subgroups — too many to draw. Only its 72 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) = 33
transitive degree	μ_t(G) = 1152

A minimal faithful permutation representation — 2 generators on 33 points:

(1,2,3,4,5,6,7,8,9)(11,14)(12,16)(15,17)(18,19,21,26,23,24,29,31,22,25,30,32,20,27,28,33);(10,11,13,15)(12,17,16,14)(18,20,22,23)(19,24,25,27)(21,29,30,28)(26,33,32,31)

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, 70);