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(#13049955⋊C2)⋊C5 #27189108

order13020
automorphism?
homologyAbelianization (H1) = C22×C3×C5 · Schur multiplier (H2) = C2

Aliases

unambiguous semidirect product(#13049955⋊C2)⋊C5
unambiguous semidirect product(#13049955⋊C5)⋊C2
unambiguous semidirect product(#13049959⋊C5)⋊C2
unambiguous semidirect product(#13049961⋊C5)⋊C2

Splittings

Fitting, socle, derived, HallC7×C31 ⋊ C22×C3×C5
HallD7×D31⋊C5 ⋊ C3
Hall#10620253⋊C5 ⋊ C22
Hall#13049955⋊C2 ⋊ C5
HallC7×ΓL(1,𝔽32) ⋊ C22×C3
HallD7×D31 ⋊ C3×C5
Hall#10620253 ⋊ C22×C5
HallΓL(1,𝔽32) ⋊ C2×AGL(1,𝔽7)
SylowC31 ⋊ C2×C5×AGL(1,𝔽7)
SylowC7 ⋊ C2×AGL(1,𝔽31)

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC7×C31C22×C3×C5
LCSG ▷ C7×C31
UCSC1

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 38
transitive degreeμt(G) = 217

A minimal faithful permutation representation — 2 generators on 38 points:

(1,2,14,17,30,3,23,29,16,18,4,15,7,20,31)(5,8,19,24,9,13,27,22,21,26,25,10,28,6,11)(33,36,37,34,35,38);(2,3)(4,7)(5,8)(6,14)(9,11)(10,16)(12,18)(13,20)(15,27)(17,26)(19,21)(22,23)(24,25)(28,31)(29,30)(32,33,35,38,37,36,34)

Polycyclic presentation:

2^2,3,5,7,31|f^a=f^30,e^b=e^6,e^c=e^2,f^c=f^25,f^d=f^16

Conjugacy classes & character table

75 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13020, 75);