C₉×#1849 #11766570

order	1152
nilpotency class	3
isoclinism	isoclinic class of #468
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₉ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₉×#1849

Subgroups

	subgroup	G/H
center	C₂×C₄×C₉	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₉
Frattini, cosocle	C₂²×C₃×C₄	C₂³×C₃
socle	C₂²×C₃	C₂×C₃×C₄⋊C₄
prime radical	C₂³×C₃	C₂×C₃×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₉ ◁ C₂×C₄²×C₉ ◁ G

318 conjugacy classes of subgroups; only the 204 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 204 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) = 576

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

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

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, 581);