Finite Group Databasebeta

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#72220488

order5184
automorphism?
homologyAbelianization (H1) = C2×C8 · Schur multiplier (H2) = C2

Aliases

No aliases for this group.

Splittings

Fitting, socle, SylowC34 ⋊ Q8⋊C8

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutatorC32×#157C2×C8
Fitting, socle, hypocenterC34Q8⋊C8
prime radicalC3⋊D32C22⋊C4
cosocle#1572C22
LCSG ▷ C32×#157 ▷ C32×C3⋊D3 ▷ C34
UCSC1

227 conjugacy classes of subgroups; only the 29 normal subgroups are shown.

normal subgroups only

This group has 227 conjugacy classes of subgroups — too many to draw. Only its 29 normal subgroups are shown.

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

Full subgroup data →

Presentation

Polycyclic presentation:

2^6,3^4|a^2=d,b^2=f,c^2=f,d^2=e,b^a=bc,c^b=cf,d^b=df,g^a=i,g^b=g^2i^2,g^c=g^2i,g^d=gi^2,g^e=g^2,g^f=g^2,h^a=i^2j,h^b=ghi^2,h^c=ghi,h^d=hij^2,h^e=h^2,h^f=gh,i^a=gi^2,i^b=g^2i,i^c=gi,i^d=g^2i^2,i^e=i^2,i^f=i^2,j^a=ghj^2,j^b=gj,j^c=g^2j,j^d=g^2h^2j^2,j^e=j^2,j^f=i^2j

Conjugacy classes & character table

42 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72220490, AGL(1,𝔽9)×AΓL(1,𝔽9), #72252880
maximal normal subgroup of#72220490, AGL(1,𝔽9)×AΓL(1,𝔽9), #72252880

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, 2, 2, 3, 3, 3, 3]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.6);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.5);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 7, 1, F.9);;
gap> SetConjugate(coll, 7, 2, F.7^2*F.9^2);;
gap> SetConjugate(coll, 7, 3, F.7^2*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.9^2);;
gap> SetConjugate(coll, 7, 5, F.7^2);;
gap> SetConjugate(coll, 7, 6, F.7^2);;
gap> SetConjugate(coll, 8, 1, F.9^2*F.10);;
gap> SetConjugate(coll, 8, 2, F.7*F.8*F.9^2);;
gap> SetConjugate(coll, 8, 3, F.7*F.8*F.9);;
gap> SetConjugate(coll, 8, 4, F.8*F.9*F.10^2);;
gap> SetConjugate(coll, 8, 5, F.8^2);;
gap> SetConjugate(coll, 8, 6, F.7*F.8);;
gap> SetConjugate(coll, 9, 1, F.7*F.9^2);;
gap> SetConjugate(coll, 9, 2, F.7^2*F.9);;
gap> SetConjugate(coll, 9, 3, F.7*F.9);;
gap> SetConjugate(coll, 9, 4, F.7^2*F.9^2);;
gap> SetConjugate(coll, 9, 5, F.9^2);;
gap> SetConjugate(coll, 9, 6, F.9^2);;
gap> SetConjugate(coll, 10, 1, F.7*F.8*F.10^2);;
gap> SetConjugate(coll, 10, 2, F.7*F.10);;
gap> SetConjugate(coll, 10, 3, F.7^2*F.10);;
gap> SetConjugate(coll, 10, 4, F.7^2*F.8^2*F.10^2);;
gap> SetConjugate(coll, 10, 5, F.10^2);;
gap> SetConjugate(coll, 10, 6, F.9^2*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);