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C2×D12831 #72288809

order51324
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

dihedralD25662
direct productC2×D12831

Splittings

Fitting, socleC2×C3×C7×C13×C47 ⋊ C2
derived, HallC3×C7×C13×C47 ⋊ C22
HallC7×C13×C47 ⋊ C2×D3
HallC3×C13×C47 ⋊ C2×D7
HallC3×C7×C47 ⋊ C2×D13
HallC13×C47 ⋊ C2×D21
HallC7×C47 ⋊ C2×D39
HallC3×C7×C13 ⋊ C2×D47
HallC3×C47 ⋊ C2×D91
HallC7×C13 ⋊ C2×D141
SylowC47 ⋊ C2×D273
HallC3×C13 ⋊ C2×D329
HallC3×C7 ⋊ C2×D611
SylowC13 ⋊ C2×D987
SylowC7 ⋊ C2×D1833
SylowC3 ⋊ C2×D4277

Subgroups

subgroupG/H
center, hypercenterC2D12831
commutator, cosocle, hypocenterC3×C7×C13×C47C22
FrattiniC1G
Fitting, socleC2×C3×C7×C13×C47C2
LCSG ▷ C3×C7×C13×C47
UCSC1 ◁ C2

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 72
transitive degreeμt(G) = 25662

A minimal faithful permutation representation — 2 generators on 72 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);(4,5)(7,12)(8,11)(9,10)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)(27,72)(28,71)(29,70)(30,69)(31,68)(32,67)(33,66)(34,65)(35,64)(36,63)(37,62)(38,61)(39,60)(40,59)(41,58)(42,57)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(49,50)

Conjugacy classes & character table

12834 conjugacy classes. The character table is too large to display.

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((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), (4,5)(7,12)(8,11)(9,10)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)(27,72)(28,71)(29,70)(30,69)(31,68)(32,67)(33,66)(34,65)(35,64)(36,63)(37,62)(38,61)(39,60)(40,59)(41,58)(42,57)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(49,50));