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A5×C71⋊C7 #44447290

order29820
automorphism?
homologyAbelianization (H1) = C7 · Schur multiplier (H2) = C2

Aliases

direct productA5×C71⋊C7
unambiguous semidirect productC71⋊(C7×A5)
unambiguous semidirect product(A5×C71)⋊C7

Splittings

Fitting, SylowC71 ⋊ C7×A5
socle, derived, HallA5×C71 ⋊ C7

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, socle, hypocenterA5×C71C7
Fitting, cosocleC71C7×A5
LCSG ▷ A5×C71
UCSC1

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 76
transitive degreeμt(G) = 355

A minimal faithful permutation representation — 2 generators on 76 points:

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

Conjugacy classes & character table

85 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(29820, 178);