Finite Group Databasebeta

← Prev (#34093052) Next (#34093054) →

C5⋊(C5⋊((C19⋊C3)⋊D7)) #34093053

order19950
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C5

Aliases

unambiguous semidirect productC5⋊(C5⋊((C19⋊C3)⋊D7))
unambiguous semidirect product(C19⋊C3)⋊(C5⋊D35)

Splittings

Fitting, socle, derived, HallC52×C7×C19 ⋊ C2×C3
HallC52×C7×C19⋊C3 ⋊ C2
HallC5⋊D665 ⋊ C3
HallC52×C19⋊C3 ⋊ D7
HallC52×C19 ⋊ C3×D7
HallC7×C19⋊C3 ⋊ C5⋊D5
HallC52×C7 ⋊ D19⋊C3
HallC7×C19 ⋊ C3×C5⋊D5
HallC19⋊C3 ⋊ C5⋊D35
SylowC52 ⋊ (C19⋊C3)⋊D7
SylowC19 ⋊ C3×C5⋊D35
SylowC7 ⋊ C5⋊(C5⋊(D19⋊C3))

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC52×C7×C19C2×C3
LCSG ▷ C52×C7×C19
UCSC1

128 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 36
transitive degreeμt(G) = 3325

A minimal faithful permutation representation — 3 generators on 36 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)(28,29)(30,31)(33,34)(35,36);(1,2,5,11,17,14,18,4,8,13,19,7,6,12,10,15,16,9,3)(20,21,23,25,26,24,22)(32,33,35,36,34);(27,28,30,31,29)

Polycyclic presentation:

2,3,5^2,7,19|c^a=c^4,d^a=d^4,e^a=e^6,f^a=f^18,f^b=f^7

Conjugacy classes & character table

792 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(19950, 92);