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Cayley diagram

#6869⋊C2 #92210

order486
isoclinismisoclinic class of #6869⋊C2
automorphism?
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect product#6869⋊C2

Splittings

Fitting, derived, Sylow#6869 ⋊ C2

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, Fitting, cosocle, hypocenter#6869C2
FrattiniC3×C9C3⋊D3
socleC3#4181
LCSG ▷ #6869
UCSC1

56 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 27 points:

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

Polycyclic presentation:

2,3^5|b^3=d^2f,d^3=f^2,b^a=b^2d,c^a=c^2,d^a=d^2f,e^a=ef^2,f^a=f^2,c^b=ce,e^b=ef

Conjugacy classes & character table

30 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6869⋊C2, #10624507⋊C2, #10624510⋊C2, …
maximal normal subgroup ofC2×#6869⋊C2, #10624507⋊C2, #10624510⋊C2, …
maximal quotient of#6869⋊C4, C2×#6869⋊C2, #13090437, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(486, 55);