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C5×D56⋊C3 #13132023

order1680
isoclinismisoclinic class of #1132⋊C3
automorphism?
homologyAbelianization (H1) = C22×C3×C5 · Schur multiplier (H2) = C2

Aliases

direct productC5×D56⋊C3
unambiguous semidirect productD56⋊(C3×C5)
unambiguous semidirect product(C5×D56)⋊C3

Splittings

FittingC5×C7×C8 ⋊ C2×C3
HallC5×D56 ⋊ C3
HallD56 ⋊ C3×C5
HallC5×ΓL(1,𝔽8) ⋊ D8
HallC5×C7 ⋊ C3×D8
HallΓL(1,𝔽8) ⋊ C5×D8
SylowC7 ⋊ C3×C5×D8

Subgroups

subgroupG/H
centerC2×C5C4⋊AGL(1,𝔽7)
commutator, cosocleC4×C7C22×C3×C5
FrattiniC4C2×C5×AGL(1,𝔽7)
FittingC5×C7×C8C2×C3
socleC2×C5×C7C3×D4
hypercenterC5×C8AGL(1,𝔽7)
hypocenterC7C3×C5×D8
LCSG ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C5 ◁ C4×C5 ◁ C5×C8

88 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)(6,7,12)(8,9,11)(13,14,18,20,16,19,15,17);(7,8)(9,11)(10,12)(14,17)(15,18)(19,20)

Conjugacy classes & character table

145 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD5×D56⋊C3, C5×D8×AGL(1,𝔽7), #72217934, …
maximal normal subgroup ofD5×D56⋊C3, C5×D8×AGL(1,𝔽7), #72217934, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1680, 43);