C₉×#2020 #11766741

order	1152
nilpotency class	3
isoclinism	isoclinic class of #2018
automorphism	?
homology	Abelianization (H₁) = C₂³×C₉ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₉×#2020

Subgroups

	subgroup	G/H
center	C₂²×C₉	#121
commutator	C₂²×C₄	C₂³×C₉
Frattini, cosocle	C₂²×C₃×C₄	C₂³×C₃
socle	C₂²×C₃	C₃×#121
prime radical	C₂×C₃×#41	C₂²×C₃
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ C₂²×C₄×C₉ ◁ G

513 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 513 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 45
transitive degree	μ_t(G) = 576

A minimal faithful permutation representation — 3 generators on 45 points:

(1,2,3,4,5,6,7,8,9)(10,11)(12,13)(14,15,20,25)(16,26,30,37)(17,31,35,40)(18,24,19,23)(21,29,38,28)(22,34,41,33)(27,42,44,36)(32,39,43,45);(11,13)(14,16)(15,21)(17,32)(18,29)(19,28)(20,30)(22,39)(23,37)(24,26)(25,38)(27,34)(31,42)(33,44)(35,43)(36,40)(41,45);(14,17,20,35)(15,22,25,41)(16,27,30,44)(18,34,19,33)(21,36,38,42)(23,40,24,31)(26,39,37,45)(28,32,29,43)

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, 752);