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C2×D14105 #72291561

order56420
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

dihedralD28210
direct productC2×D14105

Splittings

Fitting, socleC2×C5×C7×C13×C31 ⋊ C2
derived, HallC5×C7×C13×C31 ⋊ C22
HallC7×C13×C31 ⋊ C2×D5
HallC5×C13×C31 ⋊ C2×D7
HallC5×C7×C31 ⋊ C2×D13
HallC5×C7×C13 ⋊ C2×D31
HallC13×C31 ⋊ C2×D35
HallC7×C31 ⋊ C2×D65
HallC5×C31 ⋊ C2×D91
HallC7×C13 ⋊ C2×D155
HallC5×C13 ⋊ C2×D217
HallC5×C7 ⋊ C2×D403
SylowC31 ⋊ C2×D455
SylowC13 ⋊ C2×D1085
SylowC7 ⋊ C2×D2015
SylowC5 ⋊ C2×D2821

Subgroups

subgroupG/H
center, hypercenterC2D14105
commutator, cosocle, hypocenterC5×C7×C13×C31C22
FrattiniC1G
Fitting, socleC2×C5×C7×C13×C31C2
LCSG ▷ C5×C7×C13×C31
UCSC1 ◁ C2

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 58
transitive degreeμt(G) = 28210

A minimal faithful permutation representation — 2 generators on 58 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);(4,7)(5,6)(9,14)(10,13)(11,12)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)(29,58)(30,57)(31,56)(32,55)(33,54)(34,53)(35,52)(36,51)(37,50)(38,49)(39,48)(40,47)(41,46)(42,45)(43,44)

Conjugacy classes & character table

14108 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), (4,7)(5,6)(9,14)(10,13)(11,12)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)(29,58)(30,57)(31,56)(32,55)(33,54)(34,53)(35,52)(36,51)(37,50)(38,49)(39,48)(40,47)(41,46)(42,45)(43,44));