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C2×#20269515 #27211874

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

Aliases

direct productC2×#20269515

Splittings

Fitting, socleC2×C11×C61 ⋊ C2×C5
derived, HallC11×C61 ⋊ C22×C5
Hall#16809822 ⋊ C22
HallC2×C11×D61 ⋊ C5
HallC2×D61 ⋊ C11⋊C5
SylowC61 ⋊ C22×C11⋊C5
SylowC11 ⋊ C2×D61⋊C5

Subgroups

subgroupG/H
center, hypercenterC2#20269515
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:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13420, 66);