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C3⋊#6729 #10623946

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

Aliases

unambiguous semidirect productC3⋊#6729

Splittings

derivedC3×SL(2,𝔽5) ⋊ C2

Subgroups

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

59 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 120

Permutation generators:

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

Conjugacy classes & character table

21 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#13088289, D3×#6729, #13088295, …
maximal normal subgroup of#13088289, D3×#6729, #13088295, …
maximal quotient ofC3⋊#91143, C2×C3⋊#6729

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(720, 415);