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Dic3×5−1+2 #13093723

order1500
automorphism?
homologyAbelianization (H1) = C4×C52 · Schur multiplier (H2) = C1

Aliases

direct productDic3×5−1+2
unambiguous semidirect productC3⋊(C4×5−1+2)
unambiguous semidirect product(Dic3×C25)⋊C5

Splittings

HallC3×5−1+2 ⋊ C4
SylowC3 ⋊ C4×5−1+2

Subgroups

subgroupG/H
center, FrattiniC2×C5C52×D3
commutatorC3×C5C4×C52
FittingC2×C3×5−1+2C2
socle, cosocleC2×C3×C5C2×C52
prime radicalC2×C3×C52C2×C5
hypercenterC2×5−1+2D3
hypocenterC3C4×5−1+2
LCSG ▷ C3×C5 ▷ C3
UCSC1 ◁ C2×C5 ◁ C2×5−1+2

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 32
transitive degreeμt(G) = 300

A minimal faithful permutation representation — 2 generators on 32 points:

(1,2)(3,4)(5,6,7)(8,9,11,15,22,10,13,17,24,29,14,20,26,23,30,21,19,16,25,31,28,12,18,27,32);(1,3,2,4)(6,7)(9,12,19,20,13)(11,16,17,18,26)(15,23,27,24,25)(22,29,30,31,32)

Conjugacy classes & character table

174 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1500, 24);