Finite Group Databasebeta

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#72259739

order11664
automorphism?
homologyAbelianization (H1) = C2×C4 · Schur multiplier (H2) = C22

Aliases

No aliases for this group.

Splittings

Fitting, SylowH(3,𝔽3)2 ⋊ C22⋊C4

Subgroups

subgroupG/H
center, hypercenterC3#72259737
commutatorH(3,𝔽3)×#272C2×C4
Frattini, socleC32D32⋊#157
Fitting, hypocenterH(3,𝔽3)2C22⋊C4
prime radical#272×((C3⋊D3)⋊C3)⋊C2C2
cosocle#2722C22
LCSG ▷ H(3,𝔽3)×#272 ▷ H(3,𝔽3)2
UCSC1 ◁ C3

1353 conjugacy classes of subgroups; only the 25 normal subgroups are shown.

normal subgroups only

This group has 1353 conjugacy classes of subgroups — too many to draw. Only its 25 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degreeμ(G) = 36
transitive degreeμt(G) = 108

A minimal faithful permutation representation — 2 generators on 36 points:

(1,2)(3,8,9,16)(4,12)(5,6)(7,10,15,11)(13,14,18,17)(19,32,28,24,22,25,29,30,23,26,36,20)(21,34,31,35,27,33);(1,13,9)(2,15,14)(3,10,5)(4,11,18)(6,17,8)(7,12,16)(19,22)(20,24)(21,31)(25,35)(26,33)(28,36)(32,34)

Polycyclic presentation:

2^4,3^6|b^2=c,b^a=bd,e^a=eg,e^b=egj^2,e^c=e^2j^2,e^d=e^2j^2,f^a=fg,f^b=fh,f^c=f^2i^2,f^d=efj^2,g^a=g^2,g^b=eg^2,g^c=g^2,g^d=g^2,g^e=gj,g^f=gj,h^a=ghj^2,h^b=fh^2i,h^c=h^2i,h^d=gh,h^e=hj,h^f=hi,i^a=ij,j^a=j^2

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 3, 3, 3, 3, 3]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.10^2);;
gap> SetConjugate(coll, 5, 3, F.5^2*F.10^2);;
gap> SetConjugate(coll, 5, 4, F.5^2*F.10^2);;
gap> SetConjugate(coll, 6, 1, F.6*F.7);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6^2*F.9^2);;
gap> SetConjugate(coll, 6, 4, F.5*F.6*F.10^2);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 7, 2, F.5*F.7^2);;
gap> SetConjugate(coll, 7, 3, F.7^2);;
gap> SetConjugate(coll, 7, 4, F.7^2);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.7*F.8*F.10^2);;
gap> SetConjugate(coll, 8, 2, F.6*F.8^2*F.9);;
gap> SetConjugate(coll, 8, 3, F.8^2*F.9);;
gap> SetConjugate(coll, 8, 4, F.7*F.8);;
gap> SetConjugate(coll, 8, 5, F.8*F.10);;
gap> SetConjugate(coll, 8, 6, F.8*F.9);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 10, 1, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);