C₉×#1274 #11765995

order	1152
nilpotency class	2
isoclinism	isoclinic class of H(3,ℤ/4)
automorphism	?
homology	Abelianization (H₁) = C₄×C₈×C₉ · Schur multiplier (H₂) = C₄²

Aliases

direct product

C₉×#1274

Subgroups

	subgroup	G/H
center	C₂×C₄×C₉	C₄²
commutator	C₄	C₄×C₈×C₉
Frattini, cosocle	C₂×C₃×C₄²	C₂²×C₃
socle	C₂²×C₃	C₃×#95
prime radical	C₂³×C₃	C₃×C₂²⋊C₄
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₉ ◁ G

252 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 132 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) = 288

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)(20,26,28,32)(21,24)(22,25)(23,27,30,33)(29,31);(18,20)(19,23,25,30)(21,27)(22,28)(24,26,31,32)(29,33)

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, 6);