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#72220744

order10368
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C26×C4

Aliases

No aliases for this group.

Splittings

FittingC22×C34 ⋊ #127
SylowC34 ⋊ #2433

Subgroups

subgroupG/H
center, FrattiniC2C2×#72213247
commutator, cosocleC2×C3⋊D32C24
FittingC22×C34#127
socleC2×C34C2×#127
hypercenterC22#72213247
hypocenterC34#2433
LCSG ▷ C2×C3⋊D32 ▷ C34
UCSC1 ◁ C2 ◁ C22

7442 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 7442 conjugacy classes of subgroups — too many to draw. Only its 165 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:

2^7,3^4|a^2=f,d^2=g,b^a=be,c^a=cf,d^c=dg,h^a=hi,h^c=h^2i^2,h^d=h^2i^2,h^f=h^2,h^g=h^2,i^a=hi^2,i^d=h^2i,i^f=i^2,i^g=i^2,j^a=hj,j^c=j^2,j^d=j^2k^2,j^f=h^2ij,j^g=j^2,k^a=ik,k^c=j^2k,k^d=j^2k,k^f=hk,k^g=k^2

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72222932
maximal normal subgroup of#72222932

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