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C5×C22⋊C4×ΓL(1,𝔽8) #13132248

order1680
automorphism?
homologyAbelianization (H1) = C2×C3×C4×C5 · Schur multiplier (H2) = C22

Aliases

direct productC5×C22⋊C4×ΓL(1,𝔽8)
unambiguous semidirect product(C7×C22⋊C4)⋊(C3×C5)
unambiguous semidirect product(C5×C7×C22⋊C4)⋊C3

Splittings

Fitting, HallC5×C7×C22⋊C4 ⋊ C3
HallC7×C22⋊C4 ⋊ C3×C5
HallC5×C7 ⋊ C3×C22⋊C4
SylowC7 ⋊ C3×C5×C22⋊C4

Subgroups

subgroupG/H
centerC22×C5C22×ΓL(1,𝔽8)
commutatorC2×C7C2×C3×C4×C5
FrattiniC22C22×C5×ΓL(1,𝔽8)
FittingC5×C7×C22⋊C4C3
socleC22×C5×C7C22×C3
prime radicalC23×C5×ΓL(1,𝔽8)C2
cosocleC22×C7C22×C3×C5
hypercenterC5×C22⋊C4ΓL(1,𝔽8)
hypocenterC7C3×C5×C22⋊C4
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C22×C5 ◁ C5×C22⋊C4

136 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 20
transitive degreeμt(G) = 280

A minimal faithful permutation representation — 2 generators on 20 points:

(1,2,3,4,5)(6,7,9,11)(8,10,12,13)(15,16,17)(18,20,19);(7,10)(11,13)(14,15,16,18,17,19,20)

Conjugacy classes & character table

250 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD5×C22⋊C4×ΓL(1,𝔽8), C5×C22⋊C4×AGL(1,𝔽7), C22⋊C4×C5⋊AGL(1,𝔽7), …
maximal normal subgroup ofD5×C22⋊C4×ΓL(1,𝔽8), C5×C22⋊C4×AGL(1,𝔽7), C22⋊C4×C5⋊AGL(1,𝔽7), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1680, 268);