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C3×C61⋊C5 #11745016

order915
isoclinismisoclinic class of C61⋊C5
automorphism?
homologyAbelianization (H1) = C3×C5 · Schur multiplier (H2) = C1

Aliases

direct productC3×C61⋊C5
unambiguous semidirect product(C3×C61)⋊C5

Splittings

Fitting, socle, HallC3×C61 ⋊ C5
derived, SylowC61 ⋊ C3×C5

Subgroups

subgroupG/H
center, hypercenterC3C61⋊C5
commutator, cosocle, hypocenterC61C3×C5
FrattiniC1G
Fitting, socleC3×C61C5
LCSG ▷ C61
UCSC1 ◁ C3

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 64
transitive degreeμt(G) = 183

A minimal faithful permutation representation — 2 generators on 64 points:

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

Conjugacy classes & character table

51 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×D61⋊C5, D3×C61⋊C5, C3⋊(D61⋊C5), …
maximal normal subgroup ofC3×D61⋊C5, D3×C61⋊C5, C3⋊(D61⋊C5), …
maximal quotient ofC2×C3×C61⋊C5, C9×C61⋊C5, C32×C61⋊C5, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(915, 2);