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C2×C3⋊(C17⋊(D31⋊C5)) #44539325

order31620
automorphism?
homologyAbelianization (H1) = C22×C5 · Schur multiplier (H2) = C2

Aliases

direct productC2×C3⋊(C17⋊(D31⋊C5))
unambiguous semidirect product(C2×C3)⋊(C17⋊(D31⋊C5))
unambiguous semidirect product(C2×C17)⋊(C3⋊(D31⋊C5))
unambiguous semidirect product(C2×ΓL(1,𝔽32))⋊D51
unambiguous semidirect product(C2×D1581)⋊C5

Splittings

Fitting, socleC2×C3×C17×C31 ⋊ C2×C5
derived, HallC3×C17×C31 ⋊ C22×C5
HallC3×C17×ΓL(1,𝔽32) ⋊ C22
HallC2×D1581 ⋊ C5
HallC17×ΓL(1,𝔽32) ⋊ C2×D3
HallC17×C31 ⋊ C2×C5×D3
HallC3×ΓL(1,𝔽32) ⋊ C2×D17
HallΓL(1,𝔽32) ⋊ C2×D51
HallC3×C31 ⋊ C2×C5×D17
HallC3×C17 ⋊ C2×D31⋊C5
SylowC31 ⋊ C2×C5×D51
SylowC17 ⋊ C2×C3⋊(D31⋊C5)
SylowC3 ⋊ C2×C17⋊(D31⋊C5)

Subgroups

subgroupG/H
center, hypercenterC2C3⋊(C17⋊(D31⋊C5))
commutator, cosocle, hypocenterC3×C17×C31C22×C5
FrattiniC1G
Fitting, socleC2×C3×C17×C31C2×C5
LCSG ▷ C3×C17×C31
UCSC1 ◁ C2

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 53
transitive degreeμt(G) = 3162

A minimal faithful permutation representation — 2 generators on 53 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(31620, 82);