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C9⋊((C11⋊C5)⋊D31) #44501700

order30690
automorphism?
homologyAbelianization (H1) = C2×C5 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productC9⋊((C11⋊C5)⋊D31)
unambiguous semidirect product(C11⋊C5)⋊D279

Splittings

Fitting, derived, HallC9×C11×C31 ⋊ C2×C5
HallC9×C31×C11⋊C5 ⋊ C2
HallD3069 ⋊ C5
HallC31×C11⋊C5 ⋊ D9
HallC9×C11⋊C5 ⋊ D31
HallC11×C31 ⋊ C5×D9
HallC9×C31 ⋊ AGL(1,𝔽11)
HallC9×C11 ⋊ C5×D31
HallC11⋊C5 ⋊ D279
SylowC31 ⋊ C9⋊AGL(1,𝔽11)
SylowC11 ⋊ C5×D279
SylowC9 ⋊ (C11⋊C5)⋊D31

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, Fitting, cosocle, hypocenterC9×C11×C31C2×C5
FrattiniC3C3⋊((C11⋊C5)⋊D31)
socleC3×C11×C31C5×D3
LCSG ▷ C9×C11×C31
UCSC1

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 51
transitive degreeμt(G) = 3069

A minimal faithful permutation representation — 2 generators on 51 points:

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

Polycyclic presentation:

2,3,5,11,31,3|b^3=f^2,b^a=b^2f,d^a=d^10,e^a=e^30,f^a=f^2,d^c=d^4

Conjugacy classes & character table

984 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30690, 67);