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C3⋊#16864281 #27166166

order12876
automorphism?
homologyAbelianization (H1) = C4 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productC3⋊#16864281

Splittings

Fitting, socle, derived, HallC3×C29×C37 ⋊ C4
HallC29×C37 ⋊ Dic3
HallC3×C37 ⋊ #1181
HallC3×C29 ⋊ #3866
SylowC37 ⋊ C3⋊#1181
SylowC29 ⋊ C3⋊#3866
SylowC3 ⋊ #16864281

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, hypocenterC3×C29×C37C4
prime radical, cosocleC3×D1073C2
LCSG ▷ C3×C29×C37
UCSC1

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 69
transitive degreeμt(G) = 3219

A minimal faithful permutation representation — 2 generators on 69 points:

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

Polycyclic presentation:

2^2,3,29,37|a^2=b,c^a=c^2,d^a=d^17,e^a=e^6,d^b=d^28,e^b=e^36

Conjugacy classes & character table

810 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC9⋊#16864281, C3⋊(#16864281⋊C3), C3×C3⋊#16864281, …
maximal normal subgroup ofC3⋊(#16864281⋊C3), C3×C3⋊#16864281
maximal quotient ofC9⋊#16864281, C3×C3⋊#16864281, C3⋊(C3⋊#16864281)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12876, 35);