Finite Group Databasebeta

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#72269065

order20736
automorphism?
homologyAbelianization (H1) = C2×C82 · Schur multiplier (H2) = C2×C8

Aliases

No aliases for this group.

Splittings

SylowC34 ⋊ C8×M32

Subgroups

subgroupG/H
center, FrattiniC2C2×AGL(1,𝔽9)2
commutator, socleC2×C34C2×C82
FittingC22×C34C82
prime radicalC22×C3⋊(C3⋊(C3⋊D3))C4×C8
cosocle#72220201C23
hypercenterC22AGL(1,𝔽9)2
hypocenterC34C8×M32
LCSG ▷ C2×C34 ▷ C34
UCSC1 ◁ C2 ◁ C22

1071 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 1071 conjugacy classes of subgroups — too many to draw. Only its 182 normal subgroups are shown.

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

Full subgroup data →

Presentation

transitive degreeμt(G) = 144

Polycyclic presentation:

2^8,3^4|a^2=e,b^2=d,d^2=f,e^2=g,g^2=h,c^a=ch,i^a=i^2j^2,i^b=j^2,i^d=ij^2,i^e=i^2j,i^f=i^2,i^g=i^2,j^a=i^2,j^b=i^2j,j^d=i^2j^2,j^e=ij,j^f=j^2,j^g=j^2,k^a=j^2k,k^b=k^2l^2,k^d=l^2,k^e=ij^2k,k^f=k^2,k^g=i^2j^2k,l^a=i^2j^2l,l^b=kl^2,l^d=k,l^e=il,l^f=l^2,l^g=jl

Conjugacy classes & character table

204 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", "k", "l");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 7, F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 9, 1, F.9^2*F.10^2);;
gap> SetConjugate(coll, 9, 2, F.10^2);;
gap> SetConjugate(coll, 9, 4, F.9*F.10^2);;
gap> SetConjugate(coll, 9, 5, F.9^2*F.10);;
gap> SetConjugate(coll, 9, 6, F.9^2);;
gap> SetConjugate(coll, 9, 7, F.9^2);;
gap> SetConjugate(coll, 10, 1, F.9^2);;
gap> SetConjugate(coll, 10, 2, F.9^2*F.10);;
gap> SetConjugate(coll, 10, 4, F.9^2*F.10^2);;
gap> SetConjugate(coll, 10, 5, F.9*F.10);;
gap> SetConjugate(coll, 10, 6, F.10^2);;
gap> SetConjugate(coll, 10, 7, F.10^2);;
gap> SetConjugate(coll, 11, 1, F.10^2*F.11);;
gap> SetConjugate(coll, 11, 2, F.11^2*F.12^2);;
gap> SetConjugate(coll, 11, 4, F.12^2);;
gap> SetConjugate(coll, 11, 5, F.9*F.10^2*F.11);;
gap> SetConjugate(coll, 11, 6, F.11^2);;
gap> SetConjugate(coll, 11, 7, F.9^2*F.10^2*F.11);;
gap> SetConjugate(coll, 12, 1, F.9^2*F.10^2*F.12);;
gap> SetConjugate(coll, 12, 2, F.11*F.12^2);;
gap> SetConjugate(coll, 12, 4, F.11);;
gap> SetConjugate(coll, 12, 5, F.9*F.12);;
gap> SetConjugate(coll, 12, 6, F.12^2);;
gap> SetConjugate(coll, 12, 7, F.10*F.12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);