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C5⋊(C5⋊#11716328) #34093043

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

Aliases

unambiguous semidirect productC5⋊(C5⋊#11716328)

Splittings

Fitting, socle, derived, HallC52×C7×C19 ⋊ C2×C3
HallC52×#87705 ⋊ C2
HallC5⋊D665 ⋊ C3
HallC52×C19 ⋊ AGL(1,𝔽7)
Hall#87705 ⋊ C5⋊D5
HallC52×C7 ⋊ D19⋊C3
HallC7×C19 ⋊ C3×C5⋊D5
SylowC52 ⋊ #11716328
SylowC19 ⋊ C5⋊(C5⋊AGL(1,𝔽7))
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)(5,13,14,9,10,18)(11,19,17,16,15,12)(21,24,25,22,23,26)(28,29)(30,31)(33,34)(35,36);(1,2,5,10,15,18,17,7,6,11,16,4,8,12,14,19,13,9,3)(20,21,23,26,25,24,22)(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,e^b=e^2,f^b=f^11

Conjugacy classes & character table

592 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(19950, 82);