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ΓL(1,𝔽8)⋊D725 #44478548

order30450
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productΓL(1,𝔽8)⋊D725
unambiguous semidirect productC25⋊(ΓL(1,𝔽8)⋊D29)
unambiguous semidirect productC29⋊(ΓL(1,𝔽8)⋊D25)
unambiguous semidirect productD5075⋊C3

Splittings

Fitting, derived, HallC7×C25×C29 ⋊ C2×C3
HallΓL(1,𝔽8)×C25×C29 ⋊ C2
HallD5075 ⋊ C3
HallC25×C29 ⋊ AGL(1,𝔽7)
HallΓL(1,𝔽8)×C29 ⋊ D25
HallΓL(1,𝔽8)×C25 ⋊ D29
HallC7×C29 ⋊ C3×D25
HallC7×C25 ⋊ C3×D29
SylowC29 ⋊ ΓL(1,𝔽8)⋊D25
SylowC25 ⋊ ΓL(1,𝔽8)⋊D29
HallΓL(1,𝔽8) ⋊ D725
SylowC7 ⋊ C3×D725

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, Fitting, cosocle, hypocenterC7×C25×C29C2×C3
FrattiniC5C5⋊(ΓL(1,𝔽8)⋊D29)
socleC5×C7×C29C3×D5
LCSG ▷ C7×C25×C29
UCSC1

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 61
transitive degreeμt(G) = 5075

A minimal faithful permutation representation — 2 generators on 61 points:

(2,5,6,3,4,7)(9,11)(10,14)(12,17)(13,20)(15,23)(16,22)(18,19)(21,27)(24,28)(25,29)(26,31)(30,32)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(54,55)(56,57)(58,59)(60,61);(1,2,4,7,6,5,3)(8,22,32,25,11,14,28,26,17,20,23,21,19,18,27,15,13,12,31,24,10,9,29,30,16)(33,34,36,38,40,42,44,46,48,50,52,54,56,58,60,61,59,57,55,53,51,49,47,45,43,41,39,37,35)

Polycyclic presentation:

2,3,5,7,29,5|c^5=f^4,c^a=c^4f,d^a=d^6,e^a=e^28,f^a=f^4,d^b=d^2

Conjugacy classes & character table

1817 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30450, 32);