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C32×C5⋊D385 #46831661

order34650
automorphism?
homologyAbelianization (H1) = C2×C32 · Schur multiplier (H2) = C3×C5

Aliases

direct productC32×C5⋊D385
unambiguous semidirect productC5⋊(C32×D385)
unambiguous semidirect productC11⋊(C32×C5⋊D35)

Splittings

Fitting, socle, HallC32×C52×C7×C11 ⋊ C2
derived, HallC52×C7×C11 ⋊ C2×C32
HallC32×C52×C11 ⋊ D7
HallC32×C52×C7 ⋊ D11
HallC32×C7×C11 ⋊ C5⋊D5
HallC52×C11 ⋊ C32×D7
HallC32×C52 ⋊ D77
HallC52×C7 ⋊ C32×D11
HallC32×C11 ⋊ C5⋊D35
HallC7×C11 ⋊ C32×C5⋊D5
HallC32×C7 ⋊ C5⋊D55
SylowC52 ⋊ C32×D77
SylowC11 ⋊ C32×C5⋊D35
SylowC7 ⋊ C32×C5⋊D55

Subgroups

subgroupG/H
center, hypercenterC32C5⋊D385
commutator, cosocle, hypocenterC52×C7×C11C2×C32
FrattiniC1G
Fitting, socleC32×C52×C7×C11C2
LCSG ▷ C52×C7×C11
UCSC1 ◁ C32

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

Conjugacy classes & character table

8676 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(34650, 304);