C₉×#1380 #11766101

order	1152
nilpotency class	4
isoclinism	isoclinic class of D₁₆
automorphism	?
homology	Abelianization (H₁) = C₄²×C₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₉×#1380

Subgroups

	subgroup	G/H
center	C₂³×C₉	D₈
commutator	C₈	C₄²×C₉
Frattini, cosocle	C₂²×C₃×C₈	C₂²×C₃
socle, prime radical	C₂³×C₃	C₃×D₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₉ ◁ C₂²×C₄×C₉ ◁ C₂²×C₈×C₉ ◁ G

228 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 144 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)(10,11)(12,13)(14,15,16,17)(19,24)(20,28)(21,30)(22,29)(25,26)(27,33)(31,32);(11,13)(18,19,21,25,22,26,20,24,23,27,28,32,29,31,30,33)

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, 112);