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Cayley diagram

C2×C3⋊D27 #66384

order324
isoclinismisoclinic class of C3⋊D27
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2×C3

Aliases

direct productC2×C3⋊D27
unambiguous semidirect product(C2×C3)⋊D27
unambiguous semidirect product(C2×C27)⋊D3

Splittings

FittingC2×C3×C27 ⋊ C2
derived, SylowC3×C27 ⋊ C22

Subgroups

subgroupG/H
center, hypercenterC2C3⋊D27
commutator, cosocle, hypocenterC3×C27C22
FrattiniC9C2×C3⋊D3
FittingC2×C3×C27C2
socleC2×C32D9
LCSG ▷ C3×C27
UCSC1 ◁ C2

70 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 32
transitive degreeμt(G) = 162

A minimal faithful permutation representation — 3 generators on 32 points:

(1,2)(4,5)(7,10)(8,14)(9,17)(11,12)(13,23)(15,16)(18,25)(19,24)(20,30)(21,29)(22,28)(26,27)(31,32);(1,2)(3,4,5);(6,25,19,26,30,28,8,7,29,9,32,23,15,12,11,16,13,31,17,21,10,14,22,20,27,24,18)

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

maximal subgroup of(C3⋊D27)⋊C4, #10619542, #10619545, …
maximal normal subgroup of(C3⋊D27)⋊C4, #10619542, #10619545, …
maximal quotient of#10619657, C4×C3⋊D27, C4⋊(C3⋊D27), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(324, 76);