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C3⋊(C7⋊(C7⋊(Dic11⋊C5))) #44568469

order32340
automorphism?
homologyAbelianization (H1) = C4×C5 · Schur multiplier (H2) = C7

Aliases

unambiguous semidirect productC3⋊(C7⋊(C7⋊(Dic11⋊C5)))
unambiguous semidirect productC7⋊(C3⋊(C7⋊(Dic11⋊C5)))
unambiguous semidirect product(C11⋊C5)⋊(C3⋊(C7⋊Dic7))
unambiguous semidirect product(C3⋊(C7⋊Dic77))⋊C5

Splittings

derived, HallC3×C72×C11 ⋊ C4×C5
HallC3×C72×C11⋊C5 ⋊ C4
HallC3⋊(C7⋊Dic77) ⋊ C5
HallC72×C11⋊C5 ⋊ Dic3
HallC72×C11 ⋊ C5×Dic3
HallC3×C11⋊C5 ⋊ C7⋊Dic7
HallC3×C72 ⋊ Dic11⋊C5
HallC11⋊C5 ⋊ C3⋊(C7⋊Dic7)
SylowC72 ⋊ C3⋊(Dic11⋊C5)
HallC3×C11 ⋊ C5×C7⋊Dic7
SylowC11 ⋊ C5×C3⋊(C7⋊Dic7)
SylowC3 ⋊ C7⋊(C7⋊(Dic11⋊C5))

Subgroups

subgroupG/H
center, Frattini, hypercenterC2C3⋊(C7⋊(C7⋊AGL(1,𝔽11)))
commutator, hypocenterC3×C72×C11C4×C5
Fitting, socle, cosocleC2×C3×C72×C11C2×C5
prime radicalC2×C3×C72×C11⋊C5C2
LCSG ▷ C3×C72×C11
UCSC1 ◁ C2

240 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 32
transitive degreeμt(G) = 6468

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2,5,3,7^2,11,2|a^2=g,c^a=c^2,d^a=d^6,e^a=e^6,f^a=f^10,f^b=f^4

Conjugacy classes & character table

1044 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32340, 171);