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C3⋊(C7⋊(C31⋊AGL(1,𝔽5))) #27189239

order13020
automorphism?
homologyAbelianization (H1) = C4 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productC3⋊(C7⋊(C31⋊AGL(1,𝔽5)))
unambiguous semidirect productC7⋊(C3⋊(C31⋊AGL(1,𝔽5)))
unambiguous semidirect productC31⋊(C3⋊(C7⋊AGL(1,𝔽5)))

Splittings

Fitting, socle, derived, HallC3×C5×C7×C31 ⋊ C4
HallC5×C7×C31 ⋊ Dic3
HallC3×C7×C31 ⋊ AGL(1,𝔽5)
HallC3×C5×C31 ⋊ Dic7
HallC7×C31 ⋊ ΓL(1,𝔽16)
HallC5×C31 ⋊ Dic21
HallC3×C5×C7 ⋊ Dic31
HallC3×C31 ⋊ C7⋊AGL(1,𝔽5)
HallC5×C7 ⋊ Dic93
SylowC31 ⋊ C3⋊(C7⋊AGL(1,𝔽5))
HallC3×C7 ⋊ C31⋊AGL(1,𝔽5)
HallC3×C5 ⋊ Dic217
SylowC7 ⋊ C3⋊(C31⋊AGL(1,𝔽5))
SylowC5 ⋊ Dic651
SylowC3 ⋊ C7⋊(C31⋊AGL(1,𝔽5))

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, hypocenterC3×C5×C7×C31C4
prime radical, cosocleC3×C7×D5×C31C2
LCSG ▷ C3×C5×C7×C31
UCSC1

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 46
transitive degreeμt(G) = 3255

A minimal faithful permutation representation — 2 generators on 46 points:

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

Polycyclic presentation:

2^2,3,5,7,31|a^2=b,c^a=c^2,d^a=d^2,e^a=e^6,f^a=f^30,d^b=d^4

Conjugacy classes & character table

1305 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13020, 206);