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C9⋊(ΓL(1,𝔽32)⋊Dic5) #42124349

order27900
automorphism?
homologyAbelianization (H1) = C4×C5 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productC9⋊(ΓL(1,𝔽32)⋊Dic5)
unambiguous semidirect productΓL(1,𝔽32)⋊Dic45
unambiguous semidirect productDic1395⋊C5

Splittings

derivedC5×C9×C31 ⋊ C4×C5
HallC5×C9×ΓL(1,𝔽32) ⋊ C4
HallC5×ΓL(1,𝔽32) ⋊ Dic9
HallC9×C31 ⋊ C5×Dic5
SylowC31 ⋊ C5×Dic45
SylowC9 ⋊ ΓL(1,𝔽32)⋊Dic5

Subgroups

subgroupG/H
center, hypercenterC2C5⋊(C9⋊(D31⋊C5))
commutator, hypocenterC5×C9×C31C4×C5
FrattiniC2×C3C3⋊(C5⋊(D31⋊C5))
Fitting, cosocleC2×C5×C9×C31C2×C5
socleC2×C3×C5×C31C5×D3
prime radicalC2×C3×C5×ΓL(1,𝔽32)D3
LCSG ▷ C5×C9×C31
UCSC1 ◁ C2

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2,5,3,5,31,2,3|a^2=f,c^3=g^2,c^a=c^2g,d^a=d^4,e^a=e^30,g^a=g^2,e^b=e^16

Conjugacy classes & character table

510 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(27900, 96);