Finite Group Databasebeta

← Prev (#72228171) Next (#72228173) →

C3⋊D3×SL(2,𝔽5) #72228172

order2160
automorphism?
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C3

Aliases

direct productC3⋊D3×SL(2,𝔽5)

Splittings

derivedC32×SL(2,𝔽5) ⋊ C2

Subgroups

subgroupG/H
center, Frattini, hypercenterC2C3⋊D3×A5
commutator, hypocenterC32×SL(2,𝔽5)C2
Fitting, socle, cosocleC2×C32C2×A5
LCSG ▷ C32×SL(2,𝔽5)
UCSC1 ◁ C2

218 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 30
transitive degreeμt(G) = 216

A minimal faithful permutation representation — 3 generators on 30 points:

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

Conjugacy classes & character table

54 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#157×SL(2,𝔽5)
maximal normal subgroup of#157×SL(2,𝔽5)

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

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