C₉×#1533 #11766254

order	1152
nilpotency class	3
isoclinism	isoclinic class of #1510
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₉×#1533

Subgroups

	subgroup	G/H
center	C₂²×C₉	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄×C₉
Frattini, cosocle	C₃×C₄²	C₂³×C₃
socle	C₂²×C₃	C₂×C₃×C₂²⋊C₄
prime radical	C₂²×C₃×D₄	C₃×C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ C₂×C₄²×C₉ ◁ G

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

normal subgroups only

This group has 372 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 — 3 generators on 33 points:

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

Conjugacy classes & character table

234 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 265);