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C3×SL(2,𝔽9) #72228159

order2160
isoclinismisoclinic class of SL(2,𝔽9)
automorphism?
homologyAbelianization (H1) = C3 · Schur multiplier (H2) = C3

Aliases

direct productC3×SL(2,𝔽9)

Subgroups

subgroupG/H
center, Fitting, socle, hypercenterC2×C3A6
commutator, hypocenterSL(2,𝔽9)C3
Frattini, cosocleC2C3×A6
LCSG ▷ SL(2,𝔽9)
UCSC1 ◁ C2×C3

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 83
transitive degreeμt(G) = 240

A minimal faithful permutation representation — 2 generators on 83 points:

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

Conjugacy classes & character table

39 conjugacy classes — view the full character table.

Relationships

derived subgroup ofD3×SL(2,𝔽9), #72228178
maximal subgroup ofA4×SL(2,𝔽9), D3×SL(2,𝔽9), #72228176, …
maximal normal subgroup ofD3×SL(2,𝔽9), #72228178
maximal quotient ofA4×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)(5,8,42)(6,39,9)(7,36,11)(10,18,44)(12,21,51)(13,73,50)(14,52,43)(15,37,64)(16,66,71)(17,65,28)(19,61,72)(20,60,33)(22,68,77)(24,48,56)(26,62,49)(27,69,67)(29,46,59)(30,76,58)(32,74,79)(34,53,55)(35,80,40)(38,63,57)(41,82,45)(47,78,75), (4,14,80,51,78)(5,81,13,15,59)(6,77,33,40,39)(7,75,57,37,36)(8,62,66,65,17)(9,43,26,49,46)(10,83,30,16,55)(11,50,27,67,53)(12,25,35,19,24)(18,69,61,60,20)(21,74,41,63,38)(22,29,23,64,45)(28,58,32,79,56)(31,71,52,47,34)(42,48,54,72,73)(44,68,70,82,76));