C₉×#1302 #11766023

order	1152
nilpotency class	3
isoclinism	isoclinic class of #327
automorphism	?
homology	Abelianization (H₁) = C₄²×C₉ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₉×#1302

Subgroups

	subgroup	G/H
center	C₂×C₄×C₉	C₂²⋊C₄
commutator	C₂×C₄	C₄²×C₉
Frattini, cosocle	C₂×C₃×C₄²	C₂²×C₃
socle	C₂²×C₃	C₃×#95
prime radical	C₂³×C₃	C₃×C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₉ ◁ C₂×C₄²×C₉ ◁ G

216 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 102 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)(13,16)(15,17)(18,19,21,25,23,26,22,31)(20,27,29,33,28,24,30,32);(11,14,16,15)(18,20,22,30,23,28,21,29)(19,24,25,32,26,27,31,33)

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, 34);