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C5⋊(C17⋊(D49⋊C3)) #38735934

order24990
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productC5⋊(C17⋊(D49⋊C3))
unambiguous semidirect productC17⋊(C5⋊(D49⋊C3))
unambiguous semidirect product(C49⋊C3)⋊D85
unambiguous semidirect productD4165⋊C3

Splittings

Fitting, derived, HallC5×C17×C49 ⋊ C2×C3
HallC5×C17×C49⋊C3 ⋊ C2
HallD4165 ⋊ C3
HallC17×C49⋊C3 ⋊ D5
HallC17×C49 ⋊ C3×D5
HallC5×C49⋊C3 ⋊ D17
HallC5×C49 ⋊ C3×D17
HallC49⋊C3 ⋊ D85
HallC5×C17 ⋊ D49⋊C3
SylowC49 ⋊ C3×D85
SylowC17 ⋊ C5⋊(D49⋊C3)
SylowC5 ⋊ C17⋊(D49⋊C3)

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, Fitting, cosocle, hypocenterC5×C17×C49C2×C3
FrattiniC7C5⋊(C17⋊AGL(1,𝔽7))
socleC5×C7×C17AGL(1,𝔽7)
LCSG ▷ C5×C17×C49
UCSC1

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 71
transitive degreeμt(G) = 4165

A minimal faithful permutation representation — 2 generators on 71 points:

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

Polycyclic presentation:

2,3,5,7,17,7|d^7=f^6,c^a=c^4,d^a=d^6f,e^a=e^16,f^a=f^6,d^b=d^2f^3,f^b=f^2

Conjugacy classes & character table

812 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(24990, 24);