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Cayley diagram

C2×AGL(1,𝔽13) #64629

order312
isoclinismisoclinic class of AGL(1,𝔽13)
automorphismAut(G) = C2×AGL(1,𝔽13) (order 312) · Inn(G) = AGL(1,𝔽13) · Out(G) = C2
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2

Aliases

direct productC2×AGL(1,𝔽13)
unambiguous semidirect product#261⋊(C2×C3)
unambiguous semidirect product(D13⋊C3)⋊C4
unambiguous semidirect product(C2×#261)⋊C3

Splittings

Fitting, socleC2×C13 ⋊ C3×C4
derived, SylowC13 ⋊ C2×C3×C4
HallC2×#261 ⋊ C3
HallC13⋊C3 ⋊ C2×C4

Subgroups

subgroupG/H
center, hypercenterC2AGL(1,𝔽13)
commutator, hypocenterC13C2×C3×C4
FrattiniC1G
Fitting, socleC2×C13C3×C4
prime radicalC2×D13⋊C3C2
cosocleD13C22×C3
LCSG ▷ C13
UCSC1 ◁ C2

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 15
transitive degreeμt(G) = 26

A minimal faithful permutation representation — 2 generators on 15 points:

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

Conjugacy classes & character table

26 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC4×AGL(1,𝔽13), C4⋊AGL(1,𝔽13), (C22⋊#261)⋊C3, …
maximal normal subgroup ofC4×AGL(1,𝔽13), C4⋊AGL(1,𝔽13), (C22⋊#261)⋊C3, …
maximal quotient of(D13⋊C3)⋊C8, #6118⋊C3, C4×AGL(1,𝔽13), …
automorphism group ofDic13, C2×D13, C3×D13, …
holomorph ofC2×C13

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(312, 45);