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Cayley diagram

C22×C5×ΓL(1,𝔽8) #88306

order420
isoclinismisoclinic class of ΓL(1,𝔽8)
automorphismAut(G) = C4×D3×AGL(1,𝔽7) (order 1008) · Inn(G) = ΓL(1,𝔽8) · Out(G) = C2×C4×D3
homologyAbelianization (H1) = C22×C3×C5 · Schur multiplier (H2) = C2

Aliases

direct productC22×C5×ΓL(1,𝔽8)

Splittings

Fitting, socle, HallC22×C5×C7 ⋊ C3
derived, SylowC7 ⋊ C22×C3×C5
HallC5×C7 ⋊ C22×C3
HallC22×C7 ⋊ C3×C5

Subgroups

subgroupG/H
center, hypercenterC22×C5ΓL(1,𝔽8)
commutator, cosocle, hypocenterC7C22×C3×C5
FrattiniC1G
Fitting, socleC22×C5×C7C3
LCSG ▷ C7
UCSC1 ◁ C22×C5

40 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:

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

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×C5×ΓL(1,𝔽8)⋊C4, C5×(C22⋊D7)⋊C3, C2×Dic5×ΓL(1,𝔽8), …
maximal normal subgroup ofC2×C5×ΓL(1,𝔽8)⋊C4, C5×(C22⋊D7)⋊C3, C2×Dic5×ΓL(1,𝔽8), …
maximal quotient ofC2×C4×C5×ΓL(1,𝔽8), C5×D4×ΓL(1,𝔽8), C5×Q8×ΓL(1,𝔽8), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(420, 31);