Finite Group Databasebeta

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D3×#60207 #72225828

order1536
automorphism?
homologyAbelianization (H1) = C25×C4 · Schur multiplier (H2) = C213

Aliases

direct productD3×#60207

Splittings

FittingC3×#60207 ⋊ C2
SylowC3 ⋊ C2×#60207

Subgroups

subgroupG/H
center, FrattiniC2×C4C25×D3
commutator, socleC22×C3C25×C4
FittingC3×#60207C2
prime radicalD3×#3444C2
cosocleC2×C3×C4C26
hypercenter#60207D3
hypocenterC3C2×#60207
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ #60207

14268 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 27
transitive degreeμt(G) = 192

A minimal faithful permutation representation — 6 generators on 27 points:

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

Conjugacy classes & character table

264 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofD3×#10576928, D3×#495505
maximal normal subgroup ofD3×#10576928, D3×#495505

GAP

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

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