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#72209648

order1536
automorphism?
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Splittings

Fitting#31076 ⋊ D3
derived#31076⋊C3 ⋊ C2

Subgroups

subgroupG/H
center, socleC2#11714279
commutator, cosocle, hypocenter#31076⋊C3C2
FrattiniC22×C4#1020
Fitting#31076D3
hypercenterC4#85559
LCSG ▷ #31076⋊C3
UCSC1 ◁ C2 ◁ C4

527 conjugacy classes of subgroups; only the 11 normal subgroups are shown.

normal subgroups only

This group has 527 conjugacy classes of subgroups — too many to draw. Only its 11 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:

2,3,2^8|a^2=i,c^2=g,d^2=gi,e^2=hj,f^2=hi,h^2=j,b^a=b^2g,c^a=di,c^b=cd,d^a=ci,d^b=cgi,e^a=cfj,e^b=fgi,e^c=eghj,e^d=egij,f^a=degj,f^b=efghij,f^c=fg,f^d=fghj,f^e=fghij,g^a=gi,g^b=i,g^e=gj,h^a=hij,h^b=hi,h^f=hj,i^b=gi,i^e=ij,i^f=ij

Conjugacy classes & character table

27 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72209649, #72209910
maximal normal subgroup of#72209910
maximal quotient of#72209802, #72209810, #72209818, …

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