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C5×D3451 #46828508

order34510
automorphism?
homologyAbelianization (H1) = C2×C5 · Schur multiplier (H2) = C1

Aliases

direct productC5×D3451
unambiguous semidirect productC7⋊(C5×D493)
unambiguous semidirect productC17⋊(C5×D203)
unambiguous semidirect productC29⋊(C5×D119)

Splittings

Fitting, socle, HallC5×C7×C17×C29 ⋊ C2
derived, HallC7×C17×C29 ⋊ C2×C5
HallC5×C17×C29 ⋊ D7
HallC5×C7×C29 ⋊ D17
HallC5×C7×C17 ⋊ D29
HallC17×C29 ⋊ C5×D7
HallC7×C29 ⋊ C5×D17
HallC5×C29 ⋊ D119
HallC7×C17 ⋊ C5×D29
HallC5×C17 ⋊ D203
HallC5×C7 ⋊ D493
SylowC29 ⋊ C5×D119
SylowC17 ⋊ C5×D203
SylowC7 ⋊ C5×D493

Subgroups

subgroupG/H
center, hypercenterC5D3451
commutator, cosocle, hypocenterC7×C17×C29C2×C5
FrattiniC1G
Fitting, socleC5×C7×C17×C29C2
LCSG ▷ C7×C17×C29
UCSC1 ◁ C5

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 58
transitive degreeμt(G) = 17255

A minimal faithful permutation representation — 2 generators on 58 points:

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

Conjugacy classes & character table

8635 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(34510, 9);