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C2×D3009 #27130213

order12036
automorphismAut(G) order 11169408
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

dihedralD6018
direct productC2×D3009
unambiguous semidirect product(C2×C3)⋊D1003
unambiguous semidirect product(C2×C17)⋊D177
unambiguous semidirect product(C2×C59)⋊D51

Splittings

Fitting, socleC2×C3×C17×C59 ⋊ C2
derived, HallC3×C17×C59 ⋊ C22
HallC17×C59 ⋊ C2×D3
HallC3×C59 ⋊ C2×D17
SylowC59 ⋊ C2×D51
HallC3×C17 ⋊ C2×D59
SylowC17 ⋊ C2×D177
SylowC3 ⋊ C2×D1003

Subgroups

subgroupG/H
center, hypercenterC2D3009
commutator, cosocle, hypocenterC3×C17×C59C22
FrattiniC1G
Fitting, socleC2×C3×C17×C59C2
LCSG ▷ C3×C17×C59
UCSC1 ◁ C2

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 81
transitive degreeμt(G) = 6018

A minimal faithful permutation representation — 2 generators on 81 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);(4,5)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(24,81)(25,80)(26,79)(27,78)(28,77)(29,76)(30,75)(31,74)(32,73)(33,72)(34,71)(35,70)(36,69)(37,68)(38,67)(39,66)(40,65)(41,64)(42,63)(43,62)(44,61)(45,60)(46,59)(47,58)(48,57)(49,56)(50,55)(51,54)(52,53)

Conjugacy classes & character table

3012 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofC2×D9027, D3×D3009, C2×C3×D3009, …
maximal normal subgroup ofC2×C3×D3009, D12036
maximal quotient ofC2×D9027, D3×D3009, C2×C3×D3009, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12036, 22);