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C2×D13167 #72289688

order52668
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

dihedralD26334
direct productC2×D13167

Splittings

FittingC2×C7×C9×C11×C19 ⋊ C2
derived, HallC7×C9×C11×C19 ⋊ C22
HallC9×C11×C19 ⋊ C2×D7
HallC7×C11×C19 ⋊ C2×D9
HallC7×C9×C19 ⋊ C2×D11
HallC7×C9×C11 ⋊ C2×D19
HallC11×C19 ⋊ C2×D63
HallC9×C19 ⋊ C2×D77
HallC7×C19 ⋊ C2×D99
HallC9×C11 ⋊ C2×D133
HallC7×C11 ⋊ C2×D171
HallC7×C9 ⋊ C2×D209
SylowC19 ⋊ C2×D693
SylowC11 ⋊ C2×D1197
SylowC9 ⋊ C2×D1463
SylowC7 ⋊ C2×D1881

Subgroups

subgroupG/H
center, hypercenterC2D13167
commutator, cosocle, hypocenterC7×C9×C11×C19C22
FrattiniC3C2×D4389
FittingC2×C7×C9×C11×C19C2
socleC2×C3×C7×C11×C19D3
LCSG ▷ C7×C9×C11×C19
UCSC1 ◁ C2

120 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 48
transitive degreeμt(G) = 26334

A minimal faithful permutation representation — 2 generators on 48 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);(4,11)(5,10)(6,9)(7,8)(13,18)(14,17)(15,16)(20,29)(21,28)(22,27)(23,26)(24,25)(31,48)(32,47)(33,46)(34,45)(35,44)(36,43)(37,42)(38,41)(39,40)

Conjugacy classes & character table

13170 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), (4,11)(5,10)(6,9)(7,8)(13,18)(14,17)(15,16)(20,29)(21,28)(22,27)(23,26)(24,25)(31,48)(32,47)(33,46)(34,45)(35,44)(36,43)(37,42)(38,41)(39,40));