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C17⋊(ΓL(1,𝔽8)⋊D25) #31821707

order17850
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productC17⋊(ΓL(1,𝔽8)⋊D25)
unambiguous semidirect productΓL(1,𝔽8)⋊D425
unambiguous semidirect productC25⋊(C17⋊AGL(1,𝔽7))
unambiguous semidirect productD2975⋊C3

Splittings

Fitting, derived, HallC7×C17×C25 ⋊ C2×C3
HallC17×ΓL(1,𝔽8)×C25 ⋊ C2
HallD2975 ⋊ C3
HallΓL(1,𝔽8)×C25 ⋊ D17
HallC17×C25 ⋊ AGL(1,𝔽7)
HallC17×ΓL(1,𝔽8) ⋊ D25
HallC7×C25 ⋊ C3×D17
HallC7×C17 ⋊ C3×D25
SylowC25 ⋊ C17⋊AGL(1,𝔽7)
HallΓL(1,𝔽8) ⋊ D425
SylowC17 ⋊ ΓL(1,𝔽8)⋊D25
SylowC7 ⋊ C3×D425

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, Fitting, cosocle, hypocenterC7×C17×C25C2×C3
FrattiniC5C5⋊(C17⋊AGL(1,𝔽7))
socleC5×C7×C17C3×D5
LCSG ▷ C7×C17×C25
UCSC1

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 49
transitive degreeμt(G) = 2975

A minimal faithful permutation representation — 2 generators on 49 points:

(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(19,22,23,20,21,24)(26,28)(27,31)(29,34)(30,37)(32,40)(33,39)(35,36)(38,44)(41,45)(42,46)(43,48)(47,49);(1,2,4,6,8,10,12,14,16,17,15,13,11,9,7,5,3)(18,19,21,24,23,22,20)(25,39,49,42,28,31,45,43,34,37,40,38,36,35,44,32,30,29,48,41,27,26,46,47,33)

Polycyclic presentation:

2,3,5,7,17,5|c^5=f^4,c^a=c^4f,d^a=d^6,e^a=e^16,f^a=f^4,d^b=d^2

Conjugacy classes & character table

1067 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofD17×ΓL(1,𝔽8)⋊D25, AGL(1,𝔽7)×D425, D25×C17⋊AGL(1,𝔽7), …
maximal normal subgroup ofD17×ΓL(1,𝔽8)⋊D25, AGL(1,𝔽7)×D425, D25×C17⋊AGL(1,𝔽7), …
maximal quotient ofΓL(1,𝔽8)⋊Dic425, C2×C17⋊(ΓL(1,𝔽8)⋊D25)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17850, 24);