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C3×C7×C3⋊(C5⋊D55) #46831671

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

Aliases

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

Splittings

Fitting, socle, HallC32×C52×C7×C11 ⋊ C2
derivedC3×C52×C11 ⋊ C2×C3×C7
HallC32×C52×C11 ⋊ C2×C7
HallC52×C7×C11 ⋊ C3×D3
HallC32×C52×C7 ⋊ D11
HallC32×C7×C11 ⋊ C5⋊D5
HallC52×C11 ⋊ C3×D3×C7
HallC32×C52 ⋊ C7×D11
HallC52×C7 ⋊ C3×D33
HallC32×C11 ⋊ C7×C5⋊D5
HallC7×C11 ⋊ C3×C3⋊(C5⋊D5)
HallC32×C7 ⋊ C5⋊D55
SylowC52 ⋊ C3×C7×D33
SylowC11 ⋊ C3×C7×C3⋊(C5⋊D5)
SylowC32 ⋊ C7×C5⋊D55

Subgroups

subgroupG/H
center, hypercenterC3×C7C3⋊(C5⋊D55)
commutator, cosocle, hypocenterC3×C52×C11C2×C3×C7
FrattiniC1G
Fitting, socleC32×C52×C7×C11C2
LCSG ▷ C3×C52×C11
UCSC1 ◁ C3×C7

288 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 132 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degreeμ(G) = 34
transitive degreeμt(G) = 11550

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

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

Conjugacy classes & character table

8694 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(34650, 314);