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C25⋊((C19⋊C3)⋊D7) #34093006

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

Aliases

unambiguous semidirect productC25⋊((C19⋊C3)⋊D7)
unambiguous semidirect product(C19⋊C3)⋊D175

Splittings

Fitting, derived, HallC7×C19×C25 ⋊ C2×C3
HallC7×C25×C19⋊C3 ⋊ C2
HallD3325 ⋊ C3
HallC25×C19⋊C3 ⋊ D7
HallC19×C25 ⋊ C3×D7
HallC7×C19⋊C3 ⋊ D25
HallC7×C25 ⋊ D19⋊C3
HallC7×C19 ⋊ C3×D25
HallC19⋊C3 ⋊ D175
SylowC25 ⋊ (C19⋊C3)⋊D7
SylowC19 ⋊ C3×D175
SylowC7 ⋊ C25⋊(D19⋊C3)

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, Fitting, cosocle, hypocenterC7×C19×C25C2×C3
FrattiniC5C5⋊((C19⋊C3)⋊D7)
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:

(2,6,7,3,4,8)(5,14,11,9,10,16)(12,17,19,18,15,13)(21,22)(23,24)(25,26)(28,30)(29,33)(31,36)(32,39)(34,42)(35,41)(37,38)(40,46)(43,47)(44,48)(45,50)(49,51);(1,2,5,11,17,14,18,4,8,13,19,7,6,12,10,15,16,9,3)(20,21,23,25,26,24,22)(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,e^b=e^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, 45);