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C7×C3⋊D705 #44444125

order29610
automorphism?
homologyAbelianization (H1) = C2×C7 · Schur multiplier (H2) = C3

Aliases

direct productC7×C3⋊D705
unambiguous semidirect productC3⋊(C7×D705)
unambiguous semidirect productC5⋊(C7×C3⋊D141)
unambiguous semidirect productC47⋊(C7×C3⋊D15)

Splittings

Fitting, socle, HallC32×C5×C7×C47 ⋊ C2
derived, HallC32×C5×C47 ⋊ C2×C7
HallC32×C7×C47 ⋊ D5
HallC5×C7×C47 ⋊ C3⋊D3
HallC32×C47 ⋊ C7×D5
HallC7×C47 ⋊ C3⋊D15
HallC32×C5×C7 ⋊ D47
HallC5×C47 ⋊ C7×C3⋊D3
HallC32×C7 ⋊ D235
SylowC47 ⋊ C7×C3⋊D15
HallC32×C5 ⋊ C7×D47
HallC5×C7 ⋊ C3⋊D141
SylowC32 ⋊ C7×D235
SylowC5 ⋊ C7×C3⋊D141

Subgroups

subgroupG/H
center, hypercenterC7C3⋊D705
commutator, cosocle, hypocenterC32×C5×C47C2×C7
FrattiniC1G
Fitting, socleC32×C5×C7×C47C2
LCSG ▷ C32×C5×C47
UCSC1 ◁ C7

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 65
transitive degreeμt(G) = 14805

A minimal faithful permutation representation — 3 generators on 65 points:

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

Conjugacy classes & character table

7413 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(29610, 62);