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C3×D3×#13183634 #72230550

order31752
automorphism?
homologyAbelianization (H1) = C23×C32 · Schur multiplier (H2) = C23×C3

Aliases

direct productC3×D3×#13183634

Splittings

Fitting, socleC33×C72 ⋊ C23×C3
derivedC32×C72 ⋊ C23×C32
HallC33×#3861 ⋊ C23
SylowC72 ⋊ C2×C32×D32

Subgroups

subgroupG/H
center, hypercenterC3D3×#13183634
commutator, cosocle, hypocenterC32×C72C23×C32
FrattiniC1G
Fitting, socleC33×C72C23×C3
LCSG ▷ C32×C72
UCSC1 ◁ C3

3828 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 23
transitive degreeμt(G) = 84

A minimal faithful permutation representation — 3 generators on 23 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup ofD3×#72213636
maximal normal subgroup ofD3×#72213636

GAP

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

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