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C5⋊((C19⋊C3)⋊D61) #46851950

order34770
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productC5⋊((C19⋊C3)⋊D61)
unambiguous semidirect product(C19⋊C3)⋊D305

Splittings

Fitting, socle, derived, HallC5×C19×C61 ⋊ C2×C3
HallC5×C19⋊C3×C61 ⋊ C2
HallD5795 ⋊ C3
HallC19⋊C3×C61 ⋊ D5
HallC19×C61 ⋊ C3×D5
HallC5×C61 ⋊ D19⋊C3
HallC5×C19⋊C3 ⋊ D61
HallC5×C19 ⋊ C3×D61
SylowC61 ⋊ C5⋊(D19⋊C3)
HallC19⋊C3 ⋊ D305
SylowC19 ⋊ C3×D305
SylowC5 ⋊ (C19⋊C3)⋊D61

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC5×C19×C61C2×C3
LCSG ▷ C5×C19×C61
UCSC1

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 85
transitive degreeμt(G) = 5795

A minimal faithful permutation representation — 2 generators on 85 points:

(2,3)(4,5)(7,11,12,8,9,13)(10,19,16,14,15,21)(17,22,24,23,20,18)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(54,55)(56,57)(58,59)(60,61)(62,63)(64,65)(66,67)(68,69)(70,71)(72,73)(74,75)(76,77)(78,79)(80,81)(82,83)(84,85);(1,2,4,5,3)(6,7,10,16,22,19,23,9,13,18,24,12,11,17,15,20,21,14,8)(25,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,85,83,81,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27)

Polycyclic presentation:

2,3,5,19,61|c^a=c^4,d^a=d^18,e^a=e^60,d^b=d^7

Conjugacy classes & character table

1377 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(34770, 58);