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Cayley diagram

C2×C3⋊AGL(1,𝔽7) #6991

order252
isoclinismisoclinic class of C3⋊AGL(1,𝔽7)
automorphismAut(G) = C2×D3×AGL(1,𝔽7) (order 504) · Inn(G) = C3⋊AGL(1,𝔽7) · Out(G) = C22
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C2

Aliases

direct productC2×C3⋊AGL(1,𝔽7)
unambiguous semidirect product(C2×C3)⋊AGL(1,𝔽7)
unambiguous semidirect product(C2×ΓL(1,𝔽8))⋊D3
unambiguous semidirect product(C2×D21)⋊C3

Splittings

Fitting, socleC2×C3×C7 ⋊ C2×C3
derivedC3×C7 ⋊ C22×C3
HallC3×ΓL(1,𝔽8) ⋊ C22
SylowC7 ⋊ C2×C3×D3

Subgroups

subgroupG/H
center, hypercenterC2C3⋊AGL(1,𝔽7)
commutator, cosocle, hypocenterC3×C7C22×C3
FrattiniC1G
Fitting, socleC2×C3×C7C2×C3
LCSG ▷ C3×C7
UCSC1 ◁ C2

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 12
transitive degreeμt(G) = 42

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

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

Conjugacy classes & character table

24 conjugacy classes — view the full character table.

Relationships

maximal subgroup of(C3⋊AGL(1,𝔽7))⋊C4, #4246⋊C3, #4247⋊C3, …
maximal normal subgroup of(C3⋊AGL(1,𝔽7))⋊C4, #4246⋊C3, #4247⋊C3, …
maximal quotient ofDic42⋊C3, C4×C3⋊AGL(1,𝔽7), C4⋊(C3⋊AGL(1,𝔽7)), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(252, 30);