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C5⋊(C5⋊(ΓL(1,𝔽8)⋊D19)) #34093046

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

Aliases

unambiguous semidirect productC5⋊(C5⋊(ΓL(1,𝔽8)⋊D19))
unambiguous semidirect productΓL(1,𝔽8)⋊(C5⋊D95)

Splittings

Fitting, socle, derived, HallC52×C7×C19 ⋊ C2×C3
HallC52×C19×ΓL(1,𝔽8) ⋊ C2
HallC5⋊D665 ⋊ C3
HallC52×ΓL(1,𝔽8) ⋊ D19
HallC52×C19 ⋊ AGL(1,𝔽7)
HallC19×ΓL(1,𝔽8) ⋊ C5⋊D5
HallC52×C7 ⋊ C3×D19
HallC7×C19 ⋊ C3×C5⋊D5
SylowC52 ⋊ ΓL(1,𝔽8)⋊D19
HallΓL(1,𝔽8) ⋊ C5⋊D95
SylowC19 ⋊ C5⋊(C5⋊AGL(1,𝔽7))
SylowC7 ⋊ C3×C5⋊D95

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:

(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(21,24,25,22,23,26)(28,29)(30,31)(33,34)(35,36);(1,2,4,6,8,10,12,14,16,18,19,17,15,13,11,9,7,5,3)(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

Conjugacy classes & character table

1192 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(19950, 85);