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Cayley diagram

#3407⋊C3 #73276

order384
automorphism?
homologyAbelianization (H1) = C2×C3×C8 · Schur multiplier (H2) = C2

Aliases

unambiguous semidirect product#3407⋊C3

Splittings

Fitting, Sylow#3407 ⋊ C3

Subgroups

subgroupG/H
center, FrattiniC8C22×A4
commutator, hypocenterQ8C2×C3×C8
Fitting#3407C3
socleC2C2×C8×A4
prime radicalC2×#41⋊C3C4
cosocle#131C22×C3
hypercenterM32A4
LCSG ▷ Q8
UCSC1 ◁ C8 ◁ M32

100 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 64
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 2 generators on 64 points:

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

Polycyclic presentation:

2^2,3,2^5|a^2=d,d^2=e,e^2=h,f^2=h,g^2=h,b^a=bh,f^c=fg,g^c=fh,g^f=gh

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#33329⋊C3, #33337⋊C3, #33340⋊C3, …
maximal normal subgroup of#33329⋊C3, #33337⋊C3, #33340⋊C3, …
maximal quotient of#33326⋊C3, #33334⋊C3, C4⋊(#504⋊C3), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 5852);