Finite Group Databasebeta

← Prev (#67396) Next (#67398) →

Cayley diagram

C3⋊D3×ΓL(1,𝔽8) #67397

order378
isoclinismisoclinic class of C3⋊D3×ΓL(1,𝔽8)
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C3

Aliases

direct productC3⋊D3×ΓL(1,𝔽8)
unambiguous semidirect productC3⋊(D3×ΓL(1,𝔽8))

Splittings

Fitting, socle, derivedC32×C7 ⋊ C2×C3
HallC32×ΓL(1,𝔽8) ⋊ C2
SylowC7 ⋊ C3×C3⋊D3

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC32×C7C2×C3
LCSG ▷ C32×C7
UCSC1

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 13
transitive degreeμt(G) = 63

A minimal faithful permutation representation — 3 generators on 13 points:

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

Conjugacy classes & character table

30 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofΓL(1,𝔽8)×#157, ΓL(1,𝔽8)⋊#157, C3⋊D3×AGL(1,𝔽7), …
maximal normal subgroup ofΓL(1,𝔽8)×#157, ΓL(1,𝔽8)⋊#157, C3⋊D3×AGL(1,𝔽7), …
maximal quotient ofΓL(1,𝔽8)×C3⋊Dic3, C2×C3⋊D3×ΓL(1,𝔽8), ΓL(1,𝔽8)×C3⋊D9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(378, 50);