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Cayley diagram

#733⋊C2 #4186

order162
isoclinismisoclinic class of #733⋊C2
automorphismAut(G) = (#6880⋊C2)⋊C3 (order 1458) · Inn(G) = G · Out(G) = C32
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect product#733⋊C2

Splittings

Fitting, derived, Sylow#733 ⋊ C2

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, Fitting, cosocle, hypocenter#733C2
FrattiniC32C3⋊D3
socleC3#272
LCSG ▷ #733
UCSC1

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

Polycyclic presentation:

2,3^4|b^3=e^2,c^3=e^2,b^a=b^2e,c^a=c^2e,d^a=de^2,e^a=e^2,c^b=cd,d^b=de

Conjugacy classes & character table

13 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#733⋊C2, #6880⋊C2, #6881⋊C2, …
maximal normal subgroup ofC2×#733⋊C2, #6880⋊C2, #6881⋊C2, …
maximal quotient of#733⋊C4, C2×#733⋊C2, #6858⋊C2, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(162, 22);