D₅×SL(2,𝔽₉) #72228161

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

Aliases

direct product

D₅×SL(2,𝔽₉)

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

152 conjugacy classes of subgroups; only the 10 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 10 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:

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

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

A₅×SL(2,𝔽₉)

GAP

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

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