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C2×(C61⋊C5)⋊D11 #27211876

order13420
automorphism?
homologyAbelianization (H1) = C22×C5 · Schur multiplier (H2) = C2

Aliases

direct productC2×(C61⋊C5)⋊D11
unambiguous semidirect product(C2×C61⋊C5)⋊D11

Splittings

Fitting, socleC2×C11×C61 ⋊ C2×C5
derived, HallC11×C61 ⋊ C22×C5
HallC11×C61⋊C5 ⋊ C22
HallC2×D671 ⋊ C5
HallC61⋊C5 ⋊ C2×D11
SylowC61 ⋊ C2×C5×D11
SylowC11 ⋊ C2×D61⋊C5

Subgroups

subgroupG/H
center, hypercenterC2(C61⋊C5)⋊D11
commutator, cosocle, hypocenterC11×C61C22×C5
FrattiniC1G
Fitting, socleC2×C11×C61C2×C5
LCSG ▷ C11×C61
UCSC1 ◁ C2

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 74
transitive degreeμt(G) = 1342

A minimal faithful permutation representation — 2 generators on 74 points:

(2,6,7,11,28,3,4,12,14,22)(5,19,20,29,44,9,10,15,37,27)(8,31,32,33,45,16,17,36,49,50)(13,18,43,51,48,25,26,34,58,39)(21,35,56,40,41,24,38,52,23,57)(30,59,53,42,60,46,47,61,55,54)(63,64)(65,66)(67,68)(69,70)(71,72);(1,2,5,11,23,42,29,44,25,4,8,18,35,53,56,47,60,32,10,21,7,15,30,43,58,57,50,22,39,17,33,49,31,48,28,45,41,51,34,46,20,12,24,19,36,54,59,52,61,38,26,16,6,13,27,37,55,40,14,9,3)(62,63,65,67,69,71,72,70,68,66,64)(73,74)

Conjugacy classes & character table

202 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13420, 68);