Finite Group Databasebeta

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#72209799

order1536
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C22×C4

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, hypercenterC2#11715754
commutator, cosocle, hypocenter#2844⋊C3C22
Frattini, socleC23C2×#1020
Fitting#37642D3
prime radical#37642⋊C3C2
LCSG ▷ #2844⋊C3
UCSC1 ◁ C2

559 conjugacy classes of subgroups; only the 23 normal subgroups are shown.

normal subgroups only

This group has 559 conjugacy classes of subgroups — too many to draw. Only its 23 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:

2^2,3,2^7|b^2=hj,d^2=i,e^2=h,f^2=hj,g^2=hj,c^a=ch,c^b=c^2,d^b=ei,d^c=ei,e^b=dh,e^c=de,f^a=fh,f^b=ghj,f^c=fg,f^d=fhi,f^e=fh,g^a=gi,g^b=fhj,g^c=f,g^d=gi,g^e=ghi,g^f=ghj,h^b=i,h^c=hi,i^b=h,i^c=h,j^b=hij,j^c=ij

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72209803
maximal normal subgroup of#72209803
maximal quotient of#72227669

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