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Cayley diagram

D7×ΓL(1,𝔽8) #64420

order294
isoclinismisoclinic class of D7×ΓL(1,𝔽8)
automorphismAut(G) = AGL(1,𝔽7)2 (order 1764) · Inn(G) = G · Out(G) = C2×C3
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

direct productD7×ΓL(1,𝔽8)

Splittings

Fitting, socle, derived, SylowC72 ⋊ C2×C3
HallC7×ΓL(1,𝔽8) ⋊ C2
HallC7×D7 ⋊ C3

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC72C2×C3
LCSG ▷ C72
UCSC1

17 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 14
transitive degreeμt(G) = 42

A minimal faithful permutation representation — 2 generators on 14 points:

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

Conjugacy classes & character table

25 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD7×AGL(1,𝔽7), C2×D7×ΓL(1,𝔽8), ΓL(1,𝔽8)×AGL(1,𝔽7), …
maximal normal subgroup ofD7×AGL(1,𝔽7), C2×D7×ΓL(1,𝔽8), ΓL(1,𝔽8)×AGL(1,𝔽7), …
maximal quotient ofΓL(1,𝔽8)×Dic7, C2×D7×ΓL(1,𝔽8), D7×C7⋊C9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(294, 9);