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C7⋊(C17⋊(D31⋊C5)) #49146276

order36890
automorphism?
homologyAbelianization (H1) = C2×C5 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productC7⋊(C17⋊(D31⋊C5))
unambiguous semidirect productC17⋊(C7⋊(D31⋊C5))
unambiguous semidirect productΓL(1,𝔽32)⋊D119
unambiguous semidirect productD3689⋊C5

Splittings

Fitting, socle, derived, HallC7×C17×C31 ⋊ C2×C5
HallC7×C17×ΓL(1,𝔽32) ⋊ C2
HallD3689 ⋊ C5
HallC17×ΓL(1,𝔽32) ⋊ D7
HallC7×ΓL(1,𝔽32) ⋊ D17
HallC17×C31 ⋊ C5×D7
HallC7×C31 ⋊ C5×D17
HallΓL(1,𝔽32) ⋊ D119
HallC7×C17 ⋊ D31⋊C5
SylowC31 ⋊ C5×D119
SylowC17 ⋊ C7⋊(D31⋊C5)
SylowC7 ⋊ C17⋊(D31⋊C5)

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC7×C17×C31C2×C5
LCSG ▷ C7×C17×C31
UCSC1

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 55
transitive degreeμt(G) = 3689

A minimal faithful permutation representation — 2 generators on 55 points:

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

Polycyclic presentation:

2,5,7,17,31|c^a=c^6,d^a=d^16,e^a=e^30,e^b=e^16

Conjugacy classes & character table

662 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(36890, 24);