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C22×SL(2,𝔽9) #72228158

order2880
isoclinismisoclinic class of SL(2,𝔽9)
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2×C3

Aliases

direct productC22×SL(2,𝔽9)

Subgroups

subgroupG/H
center, Fitting, socle, hypercenterC23A6
commutator, hypocenterSL(2,𝔽9)C22
Frattini, cosocleC2C22×A6
LCSG ▷ SL(2,𝔽9)
UCSC1 ◁ C23

257 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 84
transitive degreeμt(G) = 320

A minimal faithful permutation representation — 2 generators on 84 points:

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

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

derived subgroup ofA4×SL(2,𝔽9)
maximal subgroup ofA4×SL(2,𝔽9)
maximal normal subgroup 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)(5,30,73,43,77)(6,76,52,9,65)(7,83,27,80,60)(8,12,18,28,62)(10,58,71,38,37)(11,53,68,34,33)(13,16,66,61,78)(14,82,51,24,54)(15,32,63,84,79)(17,46,57,20,36)(19,41,35,59,29)(21,74,72,22,56)(23,39,44,64,67)(25,50,70,45,31)(26,69,75,49,48)(40,42,55,81,47), (3,4)(5,41,42,24,7,45)(6,59,8,36,27,53)(9,74,38,57,15,16)(10,12,37,61,68,66)(11,70,33,62,40,28)(13,48,67,71,64,26)(14,22,23,73,63,69)(17,76)(18,83,46,34,35,52)(19,60)(20,78,56,65,58,84)(21,79)(25,39,31,81,75,55)(29,30,47,80,54,50)(32,77,49,72,51,44)(43,82));