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C17×ΓL(1,𝔽8)×A5 #36339595

order21420
automorphism?
homologyAbelianization (H1) = C3×C17 · Schur multiplier (H2) = C2

Aliases

direct productC17×ΓL(1,𝔽8)×A5
unambiguous semidirect productC7⋊(C3×C17×A5)
unambiguous semidirect product(C7×C17)⋊(C3×A5)
unambiguous semidirect product(C7×A5)⋊(C3×C17)
unambiguous semidirect product(C7×C17×A5)⋊C3

Splittings

Fitting, HallC7×C17 ⋊ C3×A5
socleC7×C17×A5 ⋊ C3
derivedC7×A5 ⋊ C3×C17
SylowC7 ⋊ C3×C17×A5

Subgroups

subgroupG/H
center, hypercenterC17ΓL(1,𝔽8)×A5
commutator, hypocenterC7×A5C3×C17
FrattiniC1G
FittingC7×C17C3×A5
socleC7×C17×A5C3
cosocleC7C3×C17×A5
LCSG ▷ C7×A5
UCSC1 ◁ C17

84 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 29
transitive degreeμt(G) = 595

A minimal faithful permutation representation — 2 generators on 29 points:

(18,19,20)(21,23,24)(25,28,29,26,27);(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,20,22,23,24,19,21)(25,26,27,28,29)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(21420, 427);