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C2×D3219 #27166177

order12876
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

dihedralD6438
direct productC2×D3219
unambiguous semidirect product(C2×C3)⋊D1073
unambiguous semidirect product(C2×C29)⋊D111
unambiguous semidirect product(C2×C37)⋊D87

Splittings

Fitting, socleC2×C3×C29×C37 ⋊ C2
derived, HallC3×C29×C37 ⋊ C22
HallC29×C37 ⋊ C2×D3
HallC3×C37 ⋊ C2×D29
HallC3×C29 ⋊ C2×D37
SylowC37 ⋊ C2×D87
SylowC29 ⋊ C2×D111
SylowC3 ⋊ C2×D1073

Subgroups

subgroupG/H
center, hypercenterC2D3219
commutator, cosocle, hypocenterC3×C29×C37C22
FrattiniC1G
Fitting, socleC2×C3×C29×C37C2
LCSG ▷ C3×C29×C37
UCSC1 ◁ C2

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 71
transitive degreeμt(G) = 6438

A minimal faithful permutation representation — 2 generators on 71 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);(4,5)(7,34)(8,33)(9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(36,71)(37,70)(38,69)(39,68)(40,67)(41,66)(42,65)(43,64)(44,63)(45,62)(46,61)(47,60)(48,59)(49,58)(50,57)(51,56)(52,55)(53,54)

Conjugacy classes & character table

3222 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofC2×D9657, C2×C3⋊(C29⋊(D37⋊C3)), D3×D3219, …
maximal normal subgroup ofC2×C3⋊(C29⋊(D37⋊C3)), C2×C3×D3219, D12876
maximal quotient ofC2×D9657, D3×D3219, C2×C3×D3219, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12876, 46);