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Cayley diagram

C5×C22⋊ΓL(1,𝔽8) #88308

order420
isoclinismisoclinic class of C22⋊ΓL(1,𝔽8)
automorphism?
homologyAbelianization (H1) = C3×C5 · Schur multiplier (H2) = C2

Aliases

direct productC5×C22⋊ΓL(1,𝔽8)
unambiguous semidirect productC22⋊(C5×ΓL(1,𝔽8))
unambiguous semidirect productC7⋊(C5×A4)
unambiguous semidirect product(C22×C5)⋊ΓL(1,𝔽8)
unambiguous semidirect product(C5×C7)⋊A4

Splittings

Fitting, socle, HallC22×C5×C7 ⋊ C3
derived, HallC22×C7 ⋊ C3×C5
HallC5×C7 ⋊ A4
HallC22×C5 ⋊ ΓL(1,𝔽8)
SylowC7 ⋊ C5×A4
SylowC22 ⋊ C5×ΓL(1,𝔽8)

Subgroups

subgroupG/H
center, hypercenterC5C22⋊ΓL(1,𝔽8)
commutator, cosocle, hypocenterC22×C7C3×C5
FrattiniC1G
Fitting, socleC22×C5×C7C3
LCSG ▷ C22×C7
UCSC1 ◁ C5

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 16
transitive degreeμt(G) = 140

A minimal faithful permutation representation — 2 generators on 16 points:

(2,3,4)(5,6,7,8,9)(11,12,13)(14,16,15);(1,4)(2,3)(10,11,12,14,13,15,16)

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC5×D7⋊A4, D5×C22⋊ΓL(1,𝔽8), C5⋊(D7⋊A4), …
maximal normal subgroup ofC5×D7⋊A4, D5×C22⋊ΓL(1,𝔽8), C5⋊(D7⋊A4), …
maximal quotient ofC5×C7⋊SL(2,𝔽3), C2×C5×C22⋊ΓL(1,𝔽8), C5×C22⋊(C7⋊C9), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(420, 33);