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Cayley diagram

#2070⋊C3 #68038

order384
isoclinismisoclinic class of #2069⋊C3
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect product#2070⋊C3

Splittings

Fitting, Sylow#2070 ⋊ C3

Subgroups

subgroupG/H
center, socleC2#396⋊C3
commutator, cosocle, hypocenter#338C2×C3
FrattiniC22×C4C2×A4
Fitting#2070C3
prime radical#338⋊C3C2
hypercenterC4#126⋊C3
LCSG ▷ #338
UCSC1 ◁ C2 ◁ C4

56 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 32
transitive degreeμt(G) = 32

A minimal faithful permutation representation — 2 generators on 32 points:

(1,2,6,10)(3,23,4,7,12,8,14,24,18,26,11,15)(5,19,22,9)(13,32,17,29,31,30,20,27,21,25,16,28);(1,16,5,11,6,31,22,12)(2,27,9,23,10,32,19,24)(3,18,13,21,14,4,20,17)(7,15,25,30,26,8,29,28)

Polycyclic presentation:

2,3,2^6|a^2=h,c^2=efh,d^2=efgh,e^2=h,b^a=bg,c^a=cf,d^a=dg,e^a=eh,c^b=cdfgh,d^b=cfg,e^b=efgh,f^b=fg,g^b=fh,d^c=deg,e^c=eh,f^c=fh,g^c=gh,e^d=eh,g^d=gh

Conjugacy classes & character table

21 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#2070⋊C3, #23622⋊C3, #30236⋊C3, …
maximal normal subgroup ofC2×#2070⋊C3, #23622⋊C3, #30236⋊C3, …
maximal quotient of#11312⋊C3, C2×#2070⋊C3, C3×#2070⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 614);