C₉×#1331 #11766052

order	1152
nilpotency class	4
isoclinism	isoclinic class of D₁₆
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₉×#1331

Subgroups

	subgroup	G/H
center	C₂×C₄×C₉	D₈
commutator	C₈	C₂×C₈×C₉
Frattini, cosocle	C₃×C₄×C₈	C₂²×C₃
socle	C₂²×C₃	C₃×#102
prime radical	C₂×C₃×D₈	C₃×C₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₉ ◁ C₄²×C₉ ◁ C₄×C₈×C₉ ◁ G

165 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 165 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 576

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

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

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, 63);