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Cayley diagram

C2×C3⋊ΓL(1,𝔽16) #67217

order360
isoclinismisoclinic class of C3⋊ΓL(1,𝔽16)
automorphism?
homologyAbelianization (H1) = C2×C4 · Schur multiplier (H2) = C2×C3

Aliases

direct productC2×C3⋊ΓL(1,𝔽16)
unambiguous semidirect product(C2×C3)⋊ΓL(1,𝔽16)
unambiguous semidirect productD5⋊(C3⋊Dic3)

Splittings

Fitting, socleC2×C32×C5 ⋊ C4
derived, HallC32×C5 ⋊ C2×C4
SylowC32 ⋊ C2×AGL(1,𝔽5)
SylowC5 ⋊ C2×C3⋊Dic3

Subgroups

subgroupG/H
center, hypercenterC2C3⋊ΓL(1,𝔽16)
commutator, hypocenterC32×C5C2×C4
FrattiniC1G
Fitting, socleC2×C32×C5C4
prime radicalC2×C32×D5C2
cosocleC32×D5C22
LCSG ▷ C32×C5
UCSC1 ◁ C2

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 13
transitive degreeμt(G) = 90

A minimal faithful permutation representation — 3 generators on 13 points:

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

Conjugacy classes & character table

42 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽5)×C3⋊Dic3, (C2×AGL(1,𝔽5))⋊(C3⋊D3), #10623965, …
maximal normal subgroup ofAGL(1,𝔽5)×C3⋊Dic3, (C2×AGL(1,𝔽5))⋊(C3⋊D3), #10623965, …
maximal quotient ofD5⋊(C3⋊(C3⋊C8)), #10624093, C4×C3⋊ΓL(1,𝔽16), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(360, 147);