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C3×C5×(C43⋊C3)⋊C7 #27214403

order13545
automorphism?
homologyAbelianization (H1) = C32×C5×C7 · Schur multiplier (H2) = C3

Aliases

direct productC3×C5×(C43⋊C3)⋊C7
unambiguous semidirect product(C43⋊C3)⋊(C3×C5×C7)
unambiguous semidirect product(C43⋊C7)⋊(C32×C5)
unambiguous semidirect product(C3×C43⋊C3)⋊(C5×C7)
unambiguous semidirect product(C5×C43⋊C3)⋊(C3×C7)
unambiguous semidirect product(C3×C43⋊C7)⋊(C3×C5)
unambiguous semidirect product(C5×C43⋊C7)⋊C32
unambiguous semidirect product(C3×C5×C43⋊C3)⋊C7
unambiguous semidirect product(C3×C5×C43⋊C7)⋊C3

Splittings

Fitting, socleC3×C5×C43 ⋊ C3×C7
derived, SylowC43 ⋊ C32×C5×C7
HallC3×C5×C43⋊C3 ⋊ C7
HallC5×C43⋊C7 ⋊ C32
HallC3×C43⋊C3 ⋊ C5×C7
HallC43⋊C7 ⋊ C32×C5
HallC5×C43 ⋊ C32×C7

Subgroups

subgroupG/H
center, hypercenterC3×C5(C43⋊C3)⋊C7
commutator, cosocle, hypocenterC43C32×C5×C7
FrattiniC1G
Fitting, socleC3×C5×C43C3×C7
LCSG ▷ C43
UCSC1 ◁ C3×C5

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 51
transitive degreeμt(G) = 645

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

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

Conjugacy classes & character table

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

Relationships

maximal subgroup ofC3×C5×AGL(1,𝔽43), C3×C5⋊AGL(1,𝔽43), C3×D5×(C43⋊C3)⋊C7, …
maximal normal subgroup ofC3×C5×AGL(1,𝔽43), C3×C5⋊AGL(1,𝔽43), C3×D5×(C43⋊C3)⋊C7, …
maximal quotient ofC2×C3×C5×(C43⋊C3)⋊C7

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13545, 9);