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Cayley diagram

Dic5×ΓL(1,𝔽8) #88277

order420
isoclinismisoclinic class of D5×ΓL(1,𝔽8)
automorphismAut(G) = C2×AGL(1,𝔽5)×AGL(1,𝔽7) (order 1680) · Inn(G) = D5×ΓL(1,𝔽8) · Out(G) = C23
homologyAbelianization (H1) = C3×C4 · Schur multiplier (H2) = C1

Aliases

direct productDic5×ΓL(1,𝔽8)
unambiguous semidirect product(C7×Dic5)⋊C3

Splittings

derived, HallC5×C7 ⋊ C3×C4
HallC7×Dic5 ⋊ C3
HallC5×ΓL(1,𝔽8) ⋊ C4
SylowC7 ⋊ C3×Dic5
SylowC5 ⋊ C4×ΓL(1,𝔽8)

Subgroups

subgroupG/H
center, Frattini, hypercenterC2D5×ΓL(1,𝔽8)
commutator, hypocenterC5×C7C3×C4
Fitting, socle, cosocleC2×C5×C7C2×C3
prime radicalC2×C5×ΓL(1,𝔽8)C2
LCSG ▷ C5×C7
UCSC1 ◁ C2

24 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)(5,7,6)(9,10)(11,12)(13,14,15,16);(1,2,3,5,4,6,7)(8,9,11,12,10)

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofΓL(1,𝔽8)×#170, ΓL(1,𝔽8)⋊#170, Dic5×AGL(1,𝔽7), …
maximal normal subgroup ofΓL(1,𝔽8)×#170, ΓL(1,𝔽8)⋊#170, Dic5×AGL(1,𝔽7), …
maximal quotient ofΓL(1,𝔽8)×#168, C2×Dic5×ΓL(1,𝔽8), Dic5×C7⋊C9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(420, 2);