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Cayley diagram

D5×ΓL(1,𝔽8) #6143

order210
isoclinismisoclinic class of D5×ΓL(1,𝔽8)
automorphismAut(G) = AGL(1,𝔽5)×AGL(1,𝔽7) (order 840) · Inn(G) = G · Out(G) = C22
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

direct productD5×ΓL(1,𝔽8)
unambiguous semidirect productC5⋊(C2×ΓL(1,𝔽8))
unambiguous semidirect product(C7×D5)⋊C3

Splittings

Fitting, socle, derived, HallC5×C7 ⋊ C2×C3
HallC5×ΓL(1,𝔽8) ⋊ C2
HallC7×D5 ⋊ C3
SylowC7 ⋊ C3×D5
SylowC5 ⋊ C2×ΓL(1,𝔽8)

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC5×C7C2×C3
LCSG ▷ C5×C7
UCSC1

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 12
transitive degreeμt(G) = 35

A minimal faithful permutation representation — 2 generators on 12 points:

(2,3)(4,5)(7,8,9)(10,12,11);(1,2,4,5,3)(6,7,8,10,9,11,12)

Conjugacy classes & character table

20 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽5)×ΓL(1,𝔽8), ΓL(1,𝔽8)⋊AGL(1,𝔽5), D5×AGL(1,𝔽7), …
maximal normal subgroup ofAGL(1,𝔽5)×ΓL(1,𝔽8), ΓL(1,𝔽8)⋊AGL(1,𝔽5), D5×AGL(1,𝔽7), …
maximal quotient ofDic5×ΓL(1,𝔽8), C2×D5×ΓL(1,𝔽8), D5×C7⋊C9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(210, 2);