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C3×C5×ΓL(1,𝔽8)×C41 #27166553

order12915
isoclinismisoclinic class of ΓL(1,𝔽8)
automorphism?
homologyAbelianization (H1) = C32×C5×C41 · Schur multiplier (H2) = C3

Aliases

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

Splittings

Fitting, socleC3×C5×C7×C41 ⋊ C3
derived, SylowC7 ⋊ C32×C5×C41
HallC5×C7×C41 ⋊ C32
HallC7×C41 ⋊ C32×C5
HallC5×C7 ⋊ C32×C41

Subgroups

subgroupG/H
center, hypercenterC3×C5×C41ΓL(1,𝔽8)
commutator, cosocle, hypocenterC7C32×C5×C41
FrattiniC1G
Fitting, socleC3×C5×C7×C41C3
LCSG ▷ C7
UCSC1 ◁ C3×C5×C41

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 56
transitive degreeμt(G) = 4305

A minimal faithful permutation representation — 2 generators on 56 points:

(2,3,4)(5,7,6)(8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53);(1,2,3,5,4,6,7)(54,55,56)

Conjugacy classes & character table

3075 conjugacy classes. The character table is too large to display.

Relationships

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12915, 8);