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C5×#66659 #13132562

order1680
automorphism?
homologyAbelianization (H1) = C23×C3×C5 · Schur multiplier (H2) = C22

Aliases

direct productC5×#66659

Splittings

FittingC5×C7×Q8 ⋊ C2×C3
HallC5×#1161 ⋊ C3
Hall#1161 ⋊ C3×C5
HallC5×ΓL(1,𝔽8) ⋊ #41
HallC5×C7 ⋊ C3×#41
HallΓL(1,𝔽8) ⋊ C5×#41
SylowC7 ⋊ C3×C5×#41

Subgroups

subgroupG/H
centerC2×C5C22×AGL(1,𝔽7)
commutator, cosocleC2×C7C23×C3×C5
FrattiniC2C22×C5×AGL(1,𝔽7)
FittingC5×C7×Q8C2×C3
socleC2×C5×C7C23×C3
hypercenterC5×Q8AGL(1,𝔽7)
hypocenterC7C3×C5×#41
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C5 ◁ C5×Q8

160 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 20 points:

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

Conjugacy classes & character table

175 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD5×#66659, C5×#6477⋊C3, #72218068, …
maximal normal subgroup ofD5×#66659, C5×#6477⋊C3, #72218068, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1680, 582);