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C13⋊C3×(C43⋊C3)⋊C4 #34117477

order20124
automorphism?
homologyAbelianization (H1) = C32×C4 · Schur multiplier (H2) = C3

Aliases

direct productC13⋊C3×(C43⋊C3)⋊C4
unambiguous semidirect product(C43⋊C3)⋊(C4×C13⋊C3)
unambiguous semidirect product(C13×(C43⋊C3)⋊C4)⋊C3
unambiguous semidirect product(C13⋊C3×Dic43)⋊C3
unambiguous semidirect product#20269435⋊C3
unambiguous semidirect product#20269436⋊C3

Splittings

derived, HallC13×C43 ⋊ C32×C4
HallC13⋊C3×C43⋊C3 ⋊ C4
HallC13×Dic43 ⋊ C32
HallDic43 ⋊ C3×C13⋊C3
SylowC43 ⋊ C3×C4×C13⋊C3
SylowC13 ⋊ C3×(C43⋊C3)⋊C4

Subgroups

subgroupG/H
center, Frattini, hypercenterC2C13⋊C3×D43⋊C3
commutator, hypocenterC13×C43C32×C4
Fitting, socle, cosocleC2×C13×C43C2×C32
prime radicalC2×C13⋊C3×C43⋊C3C2
LCSG ▷ C13×C43
UCSC1 ◁ C2

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 60
transitive degreeμt(G) = 2236

A minimal faithful permutation representation — 2 generators on 60 points:

(2,6,7,3,4,8)(5,14,15,9,10,24)(11,27,28,17,18,36)(12,26,22,20,16,13)(19,32,34,23,25,37)(21,38,39,30,29,33)(31,41,42,40,35,43)(44,45,47,46,49,53,50,54,56,48,51,55,52)(57,60,59,58);(1,2,5,11,19,22,7,6,12,21,31,39,15,26,14,25,35,42,28,37,38,27,18,29,34,36,43,41,32,10,16,24,33,40,30,20,4,8,13,23,17,9,3)(45,46,48)(47,50,53)(49,52,51)(54,56,55)

Conjugacy classes & character table

182 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20124, 80);