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Cayley diagram

C2×C7⋊H(3,𝔽3) #67375

order378
isoclinismisoclinic class of #4427
automorphism?
homologyAbelianization (H1) = C2×C32 · Schur multiplier (H2) = C32

Aliases

direct productC2×C7⋊H(3,𝔽3)
unambiguous semidirect product(C2×C7)⋊H(3,𝔽3)
unambiguous semidirect product(C2×C32)⋊ΓL(1,𝔽8)
unambiguous semidirect product(C2×C3×ΓL(1,𝔽8))⋊C3

Splittings

FittingC2×C32×C7 ⋊ C3
HallC2×C7 ⋊ H(3,𝔽3)
SylowC7 ⋊ C2×H(3,𝔽3)

Subgroups

subgroupG/H
centerC2×C3C3×ΓL(1,𝔽8)
commutator, cosocleC3×C7C2×C32
FrattiniC3C2×C3×ΓL(1,𝔽8)
FittingC2×C32×C7C3
socleC2×C3×C7C32
hypercenterC2×C32ΓL(1,𝔽8)
hypocenterC7C2×H(3,𝔽3)
LCSG ▷ C3×C7 ▷ C7
UCSC1 ◁ C2×C3 ◁ C2×C32

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 18
transitive degreeμt(G) = 126

A minimal faithful permutation representation — 2 generators on 18 points:

(1,2)(3,4,5,7,6,8,9)(11,14,15)(13,17,18);(4,5,6)(7,9,8)(10,11,13)(12,15,17)(14,18,16)

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofDic7⋊H(3,𝔽3), (C3⋊Dic3)⋊ΓL(1,𝔽8), #10625400, …
maximal normal subgroup ofDic7⋊H(3,𝔽3), (C3⋊Dic3)⋊ΓL(1,𝔽8), #10625400, …
maximal quotient ofC4×C7⋊H(3,𝔽3), C22×C7⋊H(3,𝔽3), C2×#10588037, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(378, 28);