Finite Group Databasebeta

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Cayley diagram

C3×C7×C3⋊D3 #67403

order378
isoclinismisoclinic class of C3⋊D3
automorphism?
homologyAbelianization (H1) = C2×C3×C7 · Schur multiplier (H2) = C3

Aliases

direct productC3×C7×C3⋊D3
unambiguous semidirect productC3⋊(C3×D3×C7)

Splittings

Fitting, socle, HallC33×C7 ⋊ C2
derivedC32 ⋊ C2×C3×C7
SylowC33 ⋊ C2×C7

Subgroups

subgroupG/H
center, hypercenterC3×C7C3⋊D3
commutator, cosocle, hypocenterC32C2×C3×C7
FrattiniC1G
Fitting, socleC33×C7C2
LCSG ▷ C32
UCSC1 ◁ C3×C7

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 16
transitive degreeμt(G) = 126

A minimal faithful permutation representation — 3 generators on 16 points:

(1,2,3)(5,6)(8,9)(10,11,12,13,14,15,16);(7,8,9);(4,5,6)

Conjugacy classes & character table

126 conjugacy classes — view the full character table.

Relationships

derived subgroup ofD32⋊D21, #13094927, #13094928, …
maximal subgroup ofC3×C7×#157, C3×C7⋊#157, C7×C3⋊#157, …
maximal normal subgroup ofC3×C7×#157, C3×C7⋊#157, C7×C3⋊#157, …
maximal quotient ofC3×C7×C3⋊Dic3, C2×C3×C7×C3⋊D3, C3×C7×C3⋊D9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(378, 56);