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C3×ΓL(1,𝔽8)×C41⋊C5 #27166551

order12915
automorphism?
homologyAbelianization (H1) = C32×C5 · Schur multiplier (H2) = C3

Aliases

direct productC3×ΓL(1,𝔽8)×C41⋊C5
unambiguous semidirect productC7⋊(C32×C41⋊C5)
unambiguous semidirect product(C3×C7)⋊(C3×C41⋊C5)
unambiguous semidirect productC41⋊(C3×C5×ΓL(1,𝔽8))
unambiguous semidirect product(C3×C41)⋊(C5×ΓL(1,𝔽8))
unambiguous semidirect product(ΓL(1,𝔽8)×C41)⋊(C3×C5)
unambiguous semidirect product(C7×C41⋊C5)⋊C32
unambiguous semidirect product(C3×ΓL(1,𝔽8)×C41)⋊C5
unambiguous semidirect product(C3×C7×C41⋊C5)⋊C3

Splittings

Fitting, socleC3×C7×C41 ⋊ C3×C5
derived, HallC7×C41 ⋊ C32×C5
HallC3×ΓL(1,𝔽8)×C41 ⋊ C5
HallC7×C41⋊C5 ⋊ C32
SylowC41 ⋊ C3×C5×ΓL(1,𝔽8)
SylowC7 ⋊ C32×C41⋊C5

Subgroups

subgroupG/H
center, hypercenterC3ΓL(1,𝔽8)×C41⋊C5
commutator, cosocle, hypocenterC7×C41C32×C5
FrattiniC1G
Fitting, socleC3×C7×C41C3×C5
LCSG ▷ C7×C41
UCSC1 ◁ C3

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 51
transitive degreeμt(G) = 861

A minimal faithful permutation representation — 2 generators on 51 points:

(2,3,5,9,17)(4,7,13,24,35)(6,11,20,12,22)(8,15,23,34,21)(10,19,30,18,29)(14,25,36,40,33)(16,28,27,38,41)(26,37,31,39,32)(43,44,45)(46,48,47);(1,2,4,8,16,25,29,34,40,36,3,6,12,23,30,37,9,18,5,10,7,14,26,38,39,41,19,31,11,21,15,27,24,35,20,32,13,17,22,33,28)(42,43,44,46,45,47,48)(49,50,51)

Conjugacy classes & character table

195 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×ΓL(1,𝔽8)⋊(D41⋊C5), C3×AGL(1,𝔽7)×C41⋊C5, C3×ΓL(1,𝔽8)×D41⋊C5, …
maximal normal subgroup ofC3×ΓL(1,𝔽8)⋊(D41⋊C5), C3×AGL(1,𝔽7)×C41⋊C5, C3×ΓL(1,𝔽8)×D41⋊C5, …
maximal quotient ofC2×C3×ΓL(1,𝔽8)×C41⋊C5

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12915, 6);