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C3×C22⋊#13131975 #34117649

order20124
automorphism?
homologyAbelianization (H1) = C32 · Schur multiplier (H2) = C2×C3

Aliases

direct productC3×C22⋊#13131975
unambiguous semidirect product(C22×C3)⋊#13131975

Splittings

Fitting, socleC22×C3×C13×C43 ⋊ C3
derived, HallC22×C13×C43 ⋊ C32
HallC13×C43 ⋊ C3×A4
HallC22×C43 ⋊ C3×C13⋊C3
HallC22×C13 ⋊ C3×C43⋊C3
SylowC43 ⋊ C3×C22⋊(C13⋊C3)
SylowC13 ⋊ C3×C22⋊(C43⋊C3)
SylowC22 ⋊ C3×#13131975

Subgroups

subgroupG/H
center, hypercenterC3C22⋊#13131975
commutator, cosocle, hypocenterC22×C13×C43C32
FrattiniC1G
Fitting, socleC22×C3×C13×C43C3
LCSG ▷ C22×C13×C43
UCSC1 ◁ C3

56 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 63
transitive degreeμt(G) = 6708

A minimal faithful permutation representation — 2 generators on 63 points:

(2,3,5)(4,7,12)(6,10,9)(8,14,22)(11,17,19)(13,20,16)(15,24,30)(18,26,31)(21,29,28)(23,32,25)(27,36,38)(33,39,37)(34,40,35)(41,43,42)(45,46,48)(47,50,53)(49,52,51)(54,56,55)(58,59,60);(1,2,4,8,15,17,3,6,11,18,27,26,7,13,21,30,38,36,14,23,33,40,22,31,24,34,41,43,32,12,19,28,37,42,39,20,5,9,16,25,35,29,10)(44,45,47,46,49,53,50,54,56,48,51,55,52)(57,60)(58,59)(61,62,63)

Conjugacy classes & character table

2244 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20124, 252);