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C2×D8533 #46802438

order34132
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

dihedralD17066
direct productC2×D8533
unambiguous semidirect product(C2×C7)⋊D1219
unambiguous semidirect product(C2×C23)⋊D371
unambiguous semidirect product(C2×C53)⋊D161

Splittings

Fitting, socleC2×C7×C23×C53 ⋊ C2
derived, HallC7×C23×C53 ⋊ C22
HallC23×C53 ⋊ C2×D7
HallC7×C53 ⋊ C2×D23
HallC7×C23 ⋊ C2×D53
SylowC53 ⋊ C2×D161
SylowC23 ⋊ C2×D371
SylowC7 ⋊ C2×D1219

Subgroups

subgroupG/H
center, hypercenterC2D8533
commutator, cosocle, hypocenterC7×C23×C53C22
FrattiniC1G
Fitting, socleC2×C7×C23×C53C2
LCSG ▷ C7×C23×C53
UCSC1 ◁ C2

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 85
transitive degreeμt(G) = 17066

A minimal faithful permutation representation — 2 generators on 85 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);(4,9)(5,8)(6,7)(11,32)(12,31)(13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)(34,85)(35,84)(36,83)(37,82)(38,81)(39,80)(40,79)(41,78)(42,77)(43,76)(44,75)(45,74)(46,73)(47,72)(48,71)(49,70)(50,69)(51,68)(52,67)(53,66)(54,65)(55,64)(56,63)(57,62)(58,61)(59,60)

Conjugacy classes & character table

8536 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(34132, 27);