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C2×#20269523 #27211867

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

Aliases

direct productC2×#20269523

Splittings

Fitting, socleC2×C11×C61 ⋊ C2×C5
derived, HallC11×C61 ⋊ C22×C5
Hall#16809821 ⋊ C22
HallC2×D11×C61 ⋊ C5
SylowC61 ⋊ C2×AGL(1,𝔽11)
HallC2×D11 ⋊ C61⋊C5
SylowC11 ⋊ C22×C61⋊C5

Subgroups

subgroupG/H
center, hypercenterC2#20269523
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)(4,7,13,24,37)(6,11,21,36,31)(8,15,27,20,35)(10,19,34,18,32)(12,16,29,45,53)(14,26,23,40,48)(22,38,54,52,49)(25,41,46,33,50)(28,44,42,57,60)(30,47,59,56,61)(39,55,51,58,43)(63,67,68,66,71,64,65,72,69,70);(1,2,4,8,16,30,34,7,14,17,31,48,60,41,29,46,50,21,37,53,9,18,33,51,55,44,54,35,26,36,52,11,22,39,3,6,12,23,40,56,24,15,28,32,49,61,59,57,47,45,58,38,27,43,19,13,25,42,5,10,20)(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, 59);