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C5×D5×D129 #27166457

order12900
automorphism?
homologyAbelianization (H1) = C22×C5 · Schur multiplier (H2) = C2

Aliases

direct productC5×D5×D129
unambiguous semidirect product(C3×D5)⋊(C5×D43)
unambiguous semidirect product(D5×C43)⋊(C5×D3)

Splittings

Fitting, socle, HallC3×C52×C43 ⋊ C22
derivedC3×C5×C43 ⋊ C22×C5
HallC52×C43 ⋊ C2×D3
HallC3×C43 ⋊ C2×C5×D5
HallC3×C52 ⋊ C2×D43
SylowC43 ⋊ C5×D3×D5
SylowC52 ⋊ C2×D129
SylowC3 ⋊ C5×D5×D43

Subgroups

subgroupG/H
center, hypercenterC5D5×D129
commutator, cosocle, hypocenterC3×C5×C43C22×C5
FrattiniC1G
Fitting, socleC3×C52×C43C22
LCSG ▷ C3×C5×C43
UCSC1 ◁ C5

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 56
transitive degreeμt(G) = 1290

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

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

Conjugacy classes & character table

1320 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12900, 126);