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C2×D3243 #27188448

order12972
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

dihedralD6486
direct productC2×D3243
unambiguous semidirect product(C2×C3)⋊D1081
unambiguous semidirect product(C2×C23)⋊D141
unambiguous semidirect product(C2×C47)⋊D69

Splittings

Fitting, socleC2×C3×C23×C47 ⋊ C2
derived, HallC3×C23×C47 ⋊ C22
HallC23×C47 ⋊ C2×D3
HallC3×C47 ⋊ C2×D23
HallC3×C23 ⋊ C2×D47
SylowC47 ⋊ C2×D69
SylowC23 ⋊ C2×D141
SylowC3 ⋊ C2×D1081

Subgroups

subgroupG/H
center, hypercenterC2D3243
commutator, cosocle, hypocenterC3×C23×C47C22
FrattiniC1G
Fitting, socleC2×C3×C23×C47C2
LCSG ▷ C3×C23×C47
UCSC1 ◁ C2

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 75
transitive degreeμt(G) = 6486

A minimal faithful permutation representation — 2 generators on 75 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);(4,5)(7,28)(8,27)(9,26)(10,25)(11,24)(12,23)(13,22)(14,21)(15,20)(16,19)(17,18)(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

3246 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofD12972, C2×D16215
maximal normal subgroup ofD12972
maximal quotient ofD12972, Dic6486, C2×D16215

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12972, 28);