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C2×D6699 #40999699

order26796
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

dihedralD13398
direct productC2×D6699
unambiguous semidirect product(C2×C3)⋊D2233
unambiguous semidirect product(C2×C7)⋊D957
unambiguous semidirect product(C2×C11)⋊D609
unambiguous semidirect product(C2×C29)⋊D231

Splittings

Fitting, socleC2×C3×C7×C11×C29 ⋊ C2
derived, HallC3×C7×C11×C29 ⋊ C22
HallC7×C11×C29 ⋊ C2×D3
HallC3×C11×C29 ⋊ C2×D7
HallC3×C7×C29 ⋊ C2×D11
HallC11×C29 ⋊ C2×D21
HallC3×C7×C11 ⋊ C2×D29
HallC7×C29 ⋊ C2×D33
HallC3×C29 ⋊ C2×D77
HallC7×C11 ⋊ C2×D87
HallC3×C11 ⋊ C2×D203
SylowC29 ⋊ C2×D231
HallC3×C7 ⋊ C2×D319
SylowC11 ⋊ C2×D609
SylowC7 ⋊ C2×D957
SylowC3 ⋊ C2×D2233

Subgroups

subgroupG/H
center, hypercenterC2D6699
commutator, cosocle, hypocenterC3×C7×C11×C29C22
FrattiniC1G
Fitting, socleC2×C3×C7×C11×C29C2
LCSG ▷ C3×C7×C11×C29
UCSC1 ◁ C2

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 52
transitive degreeμt(G) = 13398

A minimal faithful permutation representation — 2 generators on 52 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);(4,5)(7,12)(8,11)(9,10)(14,23)(15,22)(16,21)(17,20)(18,19)(25,52)(26,51)(27,50)(28,49)(29,48)(30,47)(31,46)(32,45)(33,44)(34,43)(35,42)(36,41)(37,40)(38,39)

Conjugacy classes & character table

6702 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofD26796
maximal normal subgroup ofD26796
maximal quotient ofDic13398, D26796

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(26796, 132);