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C7×C3⋊(C3⋊(C5⋊D55)) #46831683

order34650
automorphism?
homologyAbelianization (H1) = C2×C7 · Schur multiplier (H2) = C3×C5

Aliases

direct productC7×C3⋊(C3⋊(C5⋊D55))
unambiguous semidirect productC3⋊(C7×C3⋊(C5⋊D55))
unambiguous semidirect productC5⋊(C7×C3⋊D165)
unambiguous semidirect productC11⋊(C7×C3⋊(C3⋊(C5⋊D5)))

Splittings

Fitting, socle, HallC32×C52×C7×C11 ⋊ C2
derived, HallC32×C52×C11 ⋊ C2×C7
HallC52×C7×C11 ⋊ C3⋊D3
HallC32×C52×C7 ⋊ D11
HallC32×C7×C11 ⋊ C5⋊D5
HallC52×C11 ⋊ C7×C3⋊D3
HallC32×C52 ⋊ C7×D11
HallC52×C7 ⋊ C3⋊D33
HallC32×C11 ⋊ C7×C5⋊D5
HallC7×C11 ⋊ C3⋊(C3⋊(C5⋊D5))
HallC32×C7 ⋊ C5⋊D55
SylowC52 ⋊ C7×C3⋊D33
SylowC11 ⋊ C7×C3⋊(C3⋊(C5⋊D5))
SylowC32 ⋊ C7×C5⋊D55

Subgroups

subgroupG/H
center, hypercenterC7C3⋊(C3⋊(C5⋊D55))
commutator, cosocle, hypocenterC32×C52×C11C2×C7
FrattiniC1G
Fitting, socleC32×C52×C7×C11C2
LCSG ▷ C32×C52×C11
UCSC1 ◁ C7

384 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 34
transitive degreeμt(G) = 17325

A minimal faithful permutation representation — 3 generators on 34 points:

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

Conjugacy classes & character table

8673 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(34650, 326);