#72228177

order	7200
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₃

Aliases

No aliases for this group.

Splittings

derived

C₅×SL(2,𝔽₉) ⋊ C₂

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₅×A₆
commutator, hypocenter	C₅×SL(2,𝔽₉)	C₂
Fitting, socle, cosocle	C₂×C₅	C₂×A₆
LCS	G ▷ C₅×SL(2,𝔽₉)
UCS	C₁ ◁ C₂

154 conjugacy classes of subgroups; only the 8 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 8 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) = 85
transitive degree	μ_t(G) = 400

A minimal faithful permutation representation — 2 generators on 85 points:

(1,52,53,5,54,55,4,43,44,12,48,49)(2,18,6,8,10,15,3,24,9,11,7,21)(13,60,17,27,67,34,19,71,23,37,61,40)(14,62,66,16,68,70,20,72,73,22,74,59)(25,33,29,42,46,31,35,39,28,47,45,30)(26,38,36,32)(41,56,80,58,63,76,51,50,79,65,64,77)(57,78,69,75)(82,84)(83,85);(1,29,66,4,28,73)(2,44,16,3,53,22)(5,31,70,12,30,59)(6,9)(7,65,19,10,58,13)(8,49,20,11,55,14)(15,21)(17,64,25,23,63,35)(18,41,37,24,51,27)(26,71,52,36,60,43)(32,61,54,38,67,48)(33,39)(34,56,42,40,50,47)(45,46)(57,79,74,69,80,68)(62,78,76,72,75,77)(81,82,83,85,84)

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

#72228181

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((1,52,53,5,54,55,4,43,44,12,48,49)(2,18,6,8,10,15,3,24,9,11,7,21)(13,60,17,27,67,34,19,71,23,37,61,40)(14,62,66,16,68,70,20,72,73,22,74,59)(25,33,29,42,46,31,35,39,28,47,45,30)(26,38,36,32)(41,56,80,58,63,76,51,50,79,65,64,77)(57,78,69,75)(82,84)(83,85), (1,29,66,4,28,73)(2,44,16,3,53,22)(5,31,70,12,30,59)(6,9)(7,65,19,10,58,13)(8,49,20,11,55,14)(15,21)(17,64,25,23,63,35)(18,41,37,24,51,27)(26,71,52,36,60,43)(32,61,54,38,67,48)(33,39)(34,56,42,40,50,47)(45,46)(57,79,74,69,80,68)(62,78,76,72,75,77)(81,82,83,85,84));