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#135×C13⋊C3 #11928740

order1248
automorphism?
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C22

Aliases

direct product#135×C13⋊C3
unambiguous semidirect product(C13×#135)⋊C3

Splittings

Fitting, HallC13×#135 ⋊ C3
SylowC13 ⋊ C3×#135

Subgroups

subgroupG/H
centerC4D4×C13⋊C3
commutator, cosocleC4×C13C23×C3
FrattiniC4C23×C13⋊C3
FittingC13×#135C3
socleC2×C13C2×C3×D4
hypercenter#135C13⋊C3
hypocenterC13C3×#135
LCSG ▷ C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C4 ◁ C2×C4 ◁ #135

124 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 29
transitive degreeμt(G) = 208

A minimal faithful permutation representation — 3 generators on 29 points:

(1,2)(3,10)(4,8)(5,13)(6,9)(7,11)(12,15)(14,16)(18,19,21)(20,23,26)(22,25,24)(27,29,28);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,18,20,19,22,26,23,27,29,21,24,28,25);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1248, 852);