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Cayley diagram

C3⋊D27 #4182

order162
isoclinismisoclinic class of C3⋊D27
automorphismAut(G) = #72221873 (order 26244) · Inn(G) = G · Out(G) = #4208
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C3

Aliases

unambiguous semidirect productC3⋊D27
unambiguous semidirect productC27⋊D3

Splittings

Fitting, derived, SylowC3×C27 ⋊ C2

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, Fitting, cosocle, hypocenterC3×C27C2
FrattiniC9C3⋊D3
socleC32D9
LCSG ▷ C3×C27
UCSC1

28 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 30
transitive degreeμt(G) = 81

A minimal faithful permutation representation — 3 generators on 30 points:

(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);(1,2,3);(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)

Polycyclic presentation:

2,3^4|b^3=d^2e,d^3=e^2,b^a=b^2d,c^a=c^2,d^a=d^2e,e^a=e^2

Conjugacy classes & character table

42 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×D27, C2×C3⋊D27, #92172, …
maximal normal subgroup ofD3×D27, C2×C3⋊D27, #92172, …
maximal quotient ofC3⋊Dic27, C2×C3⋊D27, C9⋊D27, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(162, 18);