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C2×C25⋊(D43⋊C3) #27166349

order12900
automorphism?
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C2

Aliases

direct productC2×C25⋊(D43⋊C3)
unambiguous semidirect product(C2×C25)⋊(D43⋊C3)
unambiguous semidirect product(C2×C43⋊C3)⋊D25
unambiguous semidirect product(C2×D1075)⋊C3

Splittings

FittingC2×C25×C43 ⋊ C2×C3
derived, HallC25×C43 ⋊ C22×C3
HallC2×D1075 ⋊ C3
HallC25×C43⋊C3 ⋊ C22
HallC43⋊C3 ⋊ C2×D25
SylowC43 ⋊ C2×C3×D25
SylowC25 ⋊ C2×D43⋊C3

Subgroups

subgroupG/H
center, hypercenterC2C25⋊(D43⋊C3)
commutator, cosocle, hypocenterC25×C43C22×C3
FrattiniC5C2×C5⋊(D43⋊C3)
FittingC2×C25×C43C2×C3
socleC2×C5×C43C3×D5
LCSG ▷ C25×C43
UCSC1 ◁ C2

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 70
transitive degreeμt(G) = 2150

A minimal faithful permutation representation — 2 generators on 70 points:

(2,6,7,3,4,8)(5,14,15,9,10,24)(11,27,28,17,18,37)(12,26,22,20,16,13)(19,29,34,21,30,40)(23,25,39,31,32,35)(33,38,42,36,41,43)(45,47)(46,50)(48,53)(49,56)(51,59)(52,58)(54,55)(57,63)(60,64)(61,65)(62,67)(66,68);(1,2,5,11,19,20,4,8,13,23,33,32,10,16,24,34,42,41,18,29,35,37,28,39,30,27,38,43,40,15,26,14,25,36,31,22,7,6,12,21,17,9,3)(44,58,68,61,47,50,64,62,53,56,59,57,55,54,63,51,49,48,67,60,46,45,65,66,52)(69,70)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12900, 18);