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Cayley diagram

D3×C64 #67431

order384
isoclinismisoclinic class of D3
automorphismAut(G) = C22×D3×C16 (order 384) · Inn(G) = D3 · Out(G) = C22×C16
homologyAbelianization (H1) = C2×C64 · Schur multiplier (H2) = C2

Aliases

direct productD3×C64

Splittings

FittingC3×C64 ⋊ C2
derived, SylowC3 ⋊ C2×C64

Subgroups

subgroupG/H
center, hypercenterC64D3
commutator, hypocenterC3C2×C64
FrattiniC32C2×D3
FittingC3×C64C2
socleC2×C3C2×C32
prime radicalC2×D3C32
cosocleC3×C32C22
LCSG ▷ C3
UCSC1 ◁ C64

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 67
transitive degreeμt(G) = 192

A minimal faithful permutation representation — 2 generators on 67 points:

(2,3);(1,2,3)(4,5,6,11,7,12,16,26,8,13,17,27,20,30,36,46,9,14,18,28,21,31,37,47,23,33,39,49,42,52,56,61,10,15,19,29,22,32,38,48,24,34,40,50,43,53,57,62,25,35,41,51,44,54,58,63,45,55,59,64,60,65,66,67)

Conjugacy classes & character table

192 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×C128, #10683097, C2×D3×C64, …
maximal normal subgroup ofD3×C128, #10683097, C2×D3×C64, …
maximal quotient ofDic3×C64, (C2×D3)⋊C64, Dic3⋊C64, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 7);