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C5×ΓL(1,𝔽8)⋊C16 #13131981

order1680
isoclinismisoclinic class of AGL(1,𝔽7)
automorphism?
homologyAbelianization (H1) = C3×C5×C16 · Schur multiplier (H2) = C1

Aliases

direct productC5×ΓL(1,𝔽8)⋊C16
unambiguous semidirect productΓL(1,𝔽8)⋊(C5×C16)
unambiguous semidirect product(C7⋊C16)⋊(C3×C5)
unambiguous semidirect product(C5×C7⋊C16)⋊C3

Splittings

derived, SylowC7 ⋊ C3×C5×C16
HallC5×C7⋊C16 ⋊ C3
HallC7⋊C16 ⋊ C3×C5
HallC5×ΓL(1,𝔽8) ⋊ C16
HallC5×C7 ⋊ C3×C16
HallΓL(1,𝔽8) ⋊ C5×C16

Subgroups

subgroupG/H
center, hypercenterC5×C8AGL(1,𝔽7)
commutator, hypocenterC7C3×C5×C16
FrattiniC8C5×AGL(1,𝔽7)
FittingC5×C7×C8C2×C3
socleC2×C5×C7C3×C8
prime radicalC2×C5×ΓL(1,𝔽8)C8
cosocleC7×C8C2×C3×C5
LCSG ▷ C7
UCSC1 ◁ C5×C8

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 28
transitive degreeμt(G) = 560

A minimal faithful permutation representation — 2 generators on 28 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1680, 1);