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Cayley diagram

C3⋊D3×AGL(1,𝔽5) #67197

order360
isoclinismisoclinic class of C3⋊D3×AGL(1,𝔽5)
automorphism?
homologyAbelianization (H1) = C2×C4 · Schur multiplier (H2) = C2×C3

Aliases

direct productC3⋊D3×AGL(1,𝔽5)
unambiguous semidirect product(C3×AGL(1,𝔽5))⋊D3

Splittings

Fitting, socle, derived, HallC32×C5 ⋊ C2×C4
SylowC32 ⋊ C2×AGL(1,𝔽5)
SylowC5 ⋊ C4×C3⋊D3

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, hypocenterC32×C5C2×C4
prime radicalD5×C3⋊D3C2
cosocleC32×D5C22
LCSG ▷ C32×C5
UCSC1

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 11
transitive degreeμt(G) = 45

A minimal faithful permutation representation — 3 generators on 11 points:

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

Conjugacy classes & character table

30 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽5)×#157, D32×AGL(1,𝔽5), C2×C3⋊D3×AGL(1,𝔽5), …
maximal normal subgroup ofAGL(1,𝔽5)×#157, D32×AGL(1,𝔽5), C2×C3⋊D3×AGL(1,𝔽5), …
maximal quotient ofAGL(1,𝔽5)×C3⋊Dic3, (C2×AGL(1,𝔽5))⋊(C3⋊D3), #10623965, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(360, 127);