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Cayley diagram

C7×D27 #67350

order378
isoclinismisoclinic class of D27
automorphism?
homologyAbelianization (H1) = C2×C7 · Schur multiplier (H2) = C1

Aliases

direct productC7×D27
unambiguous semidirect productC27⋊(C2×C7)

Splittings

Fitting, HallC7×C27 ⋊ C2
derived, SylowC27 ⋊ C2×C7

Subgroups

subgroupG/H
center, hypercenterC7D27
commutator, cosocle, hypocenterC27C2×C7
FrattiniC9D3×C7
FittingC7×C27C2
socleC3×C7D9
LCSG ▷ C27
UCSC1 ◁ C7

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 34
transitive degreeμt(G) = 189

A minimal faithful permutation representation — 2 generators on 34 points:

(1,2,3,4,5,6,7)(9,12)(10,16)(11,19)(13,14)(15,25)(17,18)(20,27)(21,26)(22,32)(23,31)(24,30)(28,29)(33,34);(8,27,21,28,32,30,10,9,31,11,34,25,17,14,13,18,15,33,19,23,12,16,24,22,29,26,20)

Conjugacy classes & character table

105 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD7×D27, C2×C7×D27, C7×D81, …
maximal normal subgroup ofD7×D27, C2×C7×D27, ΓL(1,𝔽8)×D27, …
maximal quotient ofC7×Dic27, C2×C7×D27, C7×D81, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(378, 3);