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C11⋊C5×D61⋊C3 #34117751

order20130
automorphism?
homologyAbelianization (H1) = C2×C3×C5 · Schur multiplier (H2) = C1

Aliases

direct productC11⋊C5×D61⋊C3
unambiguous semidirect product(C11⋊C5×D61)⋊C3

Splittings

Fitting, socle, derived, HallC11×C61 ⋊ C2×C3×C5
HallC11⋊C5×C61⋊C3 ⋊ C2
HallC11⋊C5×D61 ⋊ C3
HallC11×D61⋊C3 ⋊ C5
HallC11⋊C5×C61 ⋊ C2×C3
HallC11×C61⋊C3 ⋊ C2×C5
HallC11×D61 ⋊ C3×C5
HallC61⋊C3 ⋊ C2×C11⋊C5
HallD61 ⋊ C3×C11⋊C5
SylowC61 ⋊ C2×C3×C11⋊C5
SylowC11 ⋊ C5×D61⋊C3

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC11×C61C2×C3×C5
LCSG ▷ C11×C61
UCSC1

32 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:

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

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20130, 80);