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C2×#20269528 #27211864

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

Aliases

direct productC2×#20269528

Splittings

Fitting, socleC2×C11×C61 ⋊ C2×C5
derived, HallC11×C61 ⋊ C22×C5
Hall#16809823 ⋊ C22
HallC2×D671 ⋊ C5
SylowC61 ⋊ C2×AGL(1,𝔽11)
SylowC11 ⋊ C2×D61⋊C5

Subgroups

subgroupG/H
center, hypercenterC2#20269528
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:

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

Conjugacy classes & character table

154 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13420, 56);