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Cayley diagram

C5×ΓL(1,𝔽8)⋊C4 #88276

order420
isoclinismisoclinic class of AGL(1,𝔽7)
automorphismAut(G) = C2×C4×AGL(1,𝔽7) (order 336) · Inn(G) = AGL(1,𝔽7) · Out(G) = C2×C4
homologyAbelianization (H1) = C3×C4×C5 · Schur multiplier (H2) = C1

Aliases

direct productC5×ΓL(1,𝔽8)⋊C4
unambiguous semidirect productΓL(1,𝔽8)⋊(C4×C5)
unambiguous semidirect productDic7⋊(C3×C5)
unambiguous semidirect product(C5×Dic7)⋊C3

Splittings

derived, SylowC7 ⋊ C3×C4×C5
HallC5×Dic7 ⋊ C3
HallC5×ΓL(1,𝔽8) ⋊ C4
HallC5×C7 ⋊ C3×C4
HallDic7 ⋊ C3×C5
HallΓL(1,𝔽8) ⋊ C4×C5

Subgroups

subgroupG/H
center, hypercenterC2×C5AGL(1,𝔽7)
commutator, hypocenterC7C3×C4×C5
FrattiniC2C5×AGL(1,𝔽7)
Fitting, socleC2×C5×C7C2×C3
prime radicalC2×C5×ΓL(1,𝔽8)C2
cosocleC2×C7C2×C3×C5
LCSG ▷ C7
UCSC1 ◁ C2×C5

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:

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

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD5×ΓL(1,𝔽8)⋊C4, #63304⋊C3, #63307⋊C3, …
maximal normal subgroup ofD5×ΓL(1,𝔽8)⋊C4, #63304⋊C3, #63307⋊C3, …
maximal quotient ofC5×ΓL(1,𝔽8)⋊C8, C2×C5×ΓL(1,𝔽8)⋊C4, C5×Dic7⋊C9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(420, 1);