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C22×C3×#13131976 #34117652

order20124
automorphism?
homologyAbelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C3

Aliases

direct productC22×C3×#13131976

Splittings

Fitting, socleC22×C3×C13×C43 ⋊ C3
derived, HallC13×C43 ⋊ C22×C32
HallC22×C13×C43 ⋊ C32
HallC22×C43 ⋊ C3×C13⋊C3
HallC22×C13 ⋊ C3×C43⋊C3
SylowC43 ⋊ C22×C3×C13⋊C3
SylowC13 ⋊ C22×C3×C43⋊C3

Subgroups

subgroupG/H
center, hypercenterC22×C3#13131976
commutator, cosocle, hypocenterC13×C43C22×C32
FrattiniC1G
Fitting, socleC22×C3×C13×C43C3
LCSG ▷ C13×C43
UCSC1 ◁ C22×C3

120 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)(47,50,54)(49,52,51)(53,55,56)(62,63);(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)(44,45,47,48,51,50,54,56,55,46,49,53,52)(57,58,59)(60,61)

Conjugacy classes & character table

2268 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20124, 255);