C₉×#1991 #11766712

order	1152
nilpotency class	3
isoclinism	isoclinic class of #465
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₉ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₉×#1991

Subgroups

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

378 conjugacy classes of subgroups; only the 192 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 192 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) = 37
transitive degree	μ_t(G) = 1152

A minimal faithful permutation representation — 3 generators on 37 points:

(1,2,3,4,5,6,7,8,9)(10,11,12,13)(14,15)(16,20)(17,18)(19,21)(22,23)(24,30)(25,34)(26,28)(27,32)(29,31)(33,36)(35,37);(11,13)(15,18)(19,20)(22,24,31,37,26,33,25,32)(23,27,34,36,28,35,29,30);(14,16,17,21)(15,19,18,20)(23,28)(27,35)(29,34)(30,36)

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, 723);