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C17⋊((C19⋊C3)⋊D13) #38738113

order25194
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productC17⋊((C19⋊C3)⋊D13)
unambiguous semidirect product(C19⋊C3)⋊D221

Splittings

Fitting, socle, derived, HallC13×C17×C19 ⋊ C2×C3
HallC13×C17×C19⋊C3 ⋊ C2
HallD4199 ⋊ C3
HallC17×C19⋊C3 ⋊ D13
HallC13×C19⋊C3 ⋊ D17
HallC17×C19 ⋊ C3×D13
HallC13×C19 ⋊ C3×D17
HallC13×C17 ⋊ D19⋊C3
HallC19⋊C3 ⋊ D221
SylowC19 ⋊ C3×D221
SylowC17 ⋊ (C19⋊C3)⋊D13
SylowC13 ⋊ C17⋊(D19⋊C3)

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC13×C17×C19C2×C3
LCSG ▷ C13×C17×C19
UCSC1

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 49
transitive degreeμt(G) = 4199

A minimal faithful permutation representation — 2 generators on 49 points:

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

Polycyclic presentation:

2,3,13,17,19|c^a=c^12,d^a=d^16,e^a=e^18,e^b=e^7

Conjugacy classes & character table

999 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(25194, 45);