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#72228048

order2304
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, FrattiniC2C2×S42
commutator, cosocleC2×A42C23
FittingC26D32
socleC25C2×D32
hypercenterC22S42
hypocenterA42#41
LCSG ▷ C2×A42 ▷ A42
UCSC1 ◁ C2 ◁ C22

1981 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 1981 conjugacy classes of subgroups — too many to draw. Only its 53 normal subgroups are shown.

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

Full subgroup data →

Presentation

minimal degreeμ(G) = 16
transitive degreeμt(G) = 72

A minimal faithful permutation representation — 3 generators on 16 points:

(1,4,2,3)(5,7,8,6)(9,11,12,16)(10,14,13,15);(2,3,4)(10,13)(14,15);(7,8)(9,10,12,13)(11,15,16,14)

Polycyclic presentation:

2^4,3^2,2^4|a^2=d,c^2=d,b^a=bd,c^a=cd,e^a=e^2,e^c=e^2,f^a=e^2f,f^c=f^2,g^e=h,g^f=gh,h^a=gh,h^c=gh,h^e=gh,h^f=g,i^a=gi,i^c=gi,i^e=gi,i^f=j,j^c=hij,j^e=ghj,j^f=ij

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72228310, #72259102
maximal normal subgroup of#72228310, #72259102

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