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C2×D8585 #46805169

order34340
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

dihedralD17170
direct productC2×D8585
unambiguous semidirect product(C2×C5)⋊D1717
unambiguous semidirect product(C2×C17)⋊D505
unambiguous semidirect product(C2×C101)⋊D85

Splittings

Fitting, socleC2×C5×C17×C101 ⋊ C2
derived, HallC5×C17×C101 ⋊ C22
HallC17×C101 ⋊ C2×D5
HallC5×C101 ⋊ C2×D17
SylowC101 ⋊ C2×D85
HallC5×C17 ⋊ C2×D101
SylowC17 ⋊ C2×D505
SylowC5 ⋊ C2×D1717

Subgroups

subgroupG/H
center, hypercenterC2D8585
commutator, cosocle, hypocenterC5×C17×C101C22
FrattiniC1G
Fitting, socleC2×C5×C17×C101C2
LCSG ▷ C5×C17×C101
UCSC1 ◁ C2

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 125
transitive degreeμt(G) = 17170

A minimal faithful permutation representation — 2 generators on 125 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)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125);(4,7)(5,6)(9,24)(10,23)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)(26,125)(27,124)(28,123)(29,122)(30,121)(31,120)(32,119)(33,118)(34,117)(35,116)(36,115)(37,114)(38,113)(39,112)(40,111)(41,110)(42,109)(43,108)(44,107)(45,106)(46,105)(47,104)(48,103)(49,102)(50,101)(51,100)(52,99)(53,98)(54,97)(55,96)(56,95)(57,94)(58,93)(59,92)(60,91)(61,90)(62,89)(63,88)(64,87)(65,86)(66,85)(67,84)(68,83)(69,82)(70,81)(71,80)(72,79)(73,78)(74,77)(75,76)

Conjugacy classes & character table

8588 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(34340, 66);