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Cayley diagram

D3×D27 #66346

order324
isoclinismisoclinic class of D3×D27
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

direct productD3×D27
unambiguous semidirect product(C3×D27)⋊C2
unambiguous semidirect product(D3×C27)⋊C2
unambiguous semidirect product(C3⋊D27)⋊C2

Splittings

Fitting, derived, SylowC3×C27 ⋊ C22

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, Fitting, cosocle, hypocenterC3×C27C22
FrattiniC9D32
socleC32C2×D9
LCSG ▷ C3×C27
UCSC1

46 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 30
transitive degreeμt(G) = 54

A minimal faithful permutation representation — 2 generators on 30 points:

(1,2,3)(5,8)(6,12)(7,15)(9,10)(11,21)(13,14)(16,23)(17,22)(18,28)(19,27)(20,26)(24,25)(29,30);(2,3)(4,23,17,24,28,26,6,5,27,7,30,21,13,10,9,14,11,29,15,19,8,12,20,18,25,22,16)

Conjugacy classes & character table

45 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×D3×D27, D9×D27, ((C3⋊D3)⋊C27)⋊C2, …
maximal normal subgroup ofC2×D3×D27, C3×D3×D27, D3×D27⋊C3, …
maximal quotient of#10619539, Dic3×D27, (C3⋊D27)⋊C4, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(324, 38);