C₉×#1885 #11766606

order	1152
nilpotency class	3
isoclinism	isoclinic class of #453
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₉ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₉×#1885

Subgroups

	subgroup	G/H
center	C₂²×C₉	#120
commutator	C₂×C₄	C₂²×C₄×C₉
Frattini, cosocle	C₂²×C₃×C₄	C₂³×C₃
socle	C₂²×C₃	C₃×#120
prime radical	C₂²×C₃×D₄	C₂²×C₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ C₂²×C₄×C₉ ◁ G

540 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 168 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) = 29
transitive degree	μ_t(G) = 288

A minimal faithful permutation representation — 3 generators on 29 points:

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

Conjugacy classes & character table

288 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 617);