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C2×#20269521 #27211869

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

Aliases

direct productC2×#20269521

Splittings

Fitting, socleC2×C11×C61 ⋊ C2×C5
derived, HallC11×C61 ⋊ C22×C5
Hall#16809822 ⋊ C22
HallC2×D11×C61 ⋊ C5
SylowC61 ⋊ C2×AGL(1,𝔽11)
HallC2×D11 ⋊ C61⋊C5
SylowC11 ⋊ C22×C61⋊C5

Subgroups

subgroupG/H
center, hypercenterC2#20269521
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:

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

Conjugacy classes & character table

190 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13420, 61);