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ΓL(1,𝔽8)×C29×A5 #49141698

order36540
automorphism?
homologyAbelianization (H1) = C3×C29 · Schur multiplier (H2) = C2

Aliases

direct productΓL(1,𝔽8)×C29×A5
unambiguous semidirect productC7⋊(C3×C29×A5)
unambiguous semidirect product(C7×C29)⋊(C3×A5)
unambiguous semidirect product(C7×A5)⋊(C3×C29)
unambiguous semidirect product(C7×C29×A5)⋊C3

Splittings

Fitting, HallC7×C29 ⋊ C3×A5
socleC7×C29×A5 ⋊ C3
derivedC7×A5 ⋊ C3×C29
SylowC7 ⋊ C3×C29×A5

Subgroups

subgroupG/H
center, hypercenterC29ΓL(1,𝔽8)×A5
commutator, hypocenterC7×A5C3×C29
FrattiniC1G
FittingC7×C29C3×A5
socleC7×C29×A5C3
cosocleC7C3×C29×A5
LCSG ▷ C7×A5
UCSC1 ◁ C29

84 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 41
transitive degreeμt(G) = 1015

A minimal faithful permutation representation — 2 generators on 41 points:

(1,2,3,4,5)(6,7,8)(9,11,10);(3,5,4)(6,12,11,10,8,9,7)(13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41)

Conjugacy classes & character table

725 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(36540, 540);