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C3⋊#6728 #10623945

order720
automorphism?
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productC3⋊#6728

Subgroups

subgroupG/H
center, Frattini, hypercenterC2ΓL(2,𝔽4)
commutator, prime radical, cosocle, hypocenterC3×SL(2,𝔽5)C2
Fitting, socleC2×C3S5
LCSG ▷ C3×SL(2,𝔽5)
UCSC1 ◁ C2

51 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 144

Permutation generators:

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

Conjugacy classes & character table

21 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#13088291, #13088292, #13088293, …
maximal normal subgroup of#13088291, #13088292, #13088293, …
maximal quotient ofC3⋊#91143, C2×C3⋊#6728

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(720, 414);