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ΓL(1,𝔽8)⋊D475 #34093007

order19950
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productΓL(1,𝔽8)⋊D475
unambiguous semidirect productC25⋊(ΓL(1,𝔽8)⋊D19)

Splittings

Fitting, derived, HallC7×C19×C25 ⋊ C2×C3
HallC19×ΓL(1,𝔽8)×C25 ⋊ C2
HallD3325 ⋊ C3
HallΓL(1,𝔽8)×C25 ⋊ D19
HallC19×C25 ⋊ AGL(1,𝔽7)
HallC19×ΓL(1,𝔽8) ⋊ D25
HallC7×C25 ⋊ C3×D19
HallC7×C19 ⋊ C3×D25
SylowC25 ⋊ ΓL(1,𝔽8)⋊D19
HallΓL(1,𝔽8) ⋊ D475
SylowC19 ⋊ ΓL(1,𝔽8)⋊D25
SylowC7 ⋊ C3×D475

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, Fitting, cosocle, hypocenterC7×C19×C25C2×C3
FrattiniC5C5⋊(ΓL(1,𝔽8)⋊D19)
socleC5×C7×C19C3×D5
LCSG ▷ C7×C19×C25
UCSC1

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 51
transitive degreeμt(G) = 3325

A minimal faithful permutation representation — 2 generators on 51 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)(28,30)(29,33)(31,36)(32,39)(34,42)(35,41)(37,38)(40,46)(43,47)(44,48)(45,50)(49,51);(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)(27,41,51,44,30,33,47,45,36,39,42,40,38,37,46,34,32,31,50,43,29,28,48,49,35)

Polycyclic presentation:

2,3,5,7,19,5|c^5=f^4,c^a=c^4f,d^a=d^6,e^a=e^18,f^a=f^4,d^b=d^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, 46);