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D11×C61⋊C3 #16860902

order4026
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

direct productD11×C61⋊C3
unambiguous semidirect productC11⋊(C2×C61⋊C3)
unambiguous semidirect product(D11×C61)⋊C3

Splittings

Fitting, socle, derived, HallC11×C61 ⋊ C2×C3
HallC11×C61⋊C3 ⋊ C2
HallD11×C61 ⋊ C3
SylowC61 ⋊ C3×D11
SylowC11 ⋊ C2×C61⋊C3

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC11×C61C2×C3
LCSG ▷ C11×C61
UCSC1

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 72
transitive degreeμt(G) = 671

A minimal faithful permutation representation — 2 generators on 72 points:

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

Conjugacy classes & character table

161 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×D11×C61⋊C3, D33×C61⋊C3, C5×D11×C61⋊C3, …
maximal normal subgroup ofC3×D11×C61⋊C3, C5×D11×C61⋊C3, D11×(C61⋊C3)⋊C5, …
maximal quotient ofD11×C61⋊C9, C3×D11×C61⋊C3, D33×C61⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4026, 11);