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C25⋊#13093366 #49150221

order37050
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productC25⋊#13093366

Splittings

Fitting, derived, HallC13×C19×C25 ⋊ C2×C3
HallC25×#10625202 ⋊ C2
HallD6175 ⋊ C3
Hall#10625202 ⋊ D25
HallC19×C25 ⋊ D13⋊C3
HallC13×C25 ⋊ D19⋊C3
HallC13×C19 ⋊ C3×D25
SylowC25 ⋊ #13093366
SylowC19 ⋊ C25⋊(D13⋊C3)
SylowC13 ⋊ C25⋊(D19⋊C3)

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, Fitting, cosocle, hypocenterC13×C19×C25C2×C3
FrattiniC5C5⋊#13093366
socleC5×C13×C19C3×D5
LCSG ▷ C13×C19×C25
UCSC1

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 57
transitive degreeμt(G) = 6175

A minimal faithful permutation representation — 2 generators on 57 points:

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

Polycyclic presentation:

2,3,5,13,19,5|c^5=f^4,c^a=c^4f,d^a=d^12,e^a=e^18,f^a=f^4,d^b=d^3,e^b=e^7

Conjugacy classes & character table

1067 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(37050, 47);