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#72228326

order1536
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Splittings

Fitting#33301 ⋊ D3

Subgroups

subgroupG/H
centerC2#11713628
commutator, cosocleC4×C42⋊C3C23
FrattiniC22×C4C22×S4
Fitting#33301D3
socleC23D4×S4
hypercenterC8#857⋊C2
hypocenterC42⋊C3C2×D8
LCSG ▷ C4×C42⋊C3 ▷ C2×C42⋊C3 ▷ C42⋊C3
UCSC1 ◁ C2 ◁ C4 ◁ C8

1238 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 1238 conjugacy classes of subgroups — too many to draw. Only its 40 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) = 20
transitive degreeμt(G) = 24

A minimal faithful permutation representation — 3 generators on 20 points:

(2,5)(3,8)(6,7)(9,14,19)(10,18,11)(12,15,13,16,20,17);(1,2,3,6,4,7,8,5)(10,13,14,17)(15,18,20,19);(10,13)(14,17)(15,18)(19,20)

Polycyclic presentation:

2^5,3,2^4|b^2=d,d^2=e,g^2=i,h^2=j,b^a=bi,c^a=ci,c^b=cd,d^c=de,f^a=f^2,f^b=f^2ij,f^c=fi,f^d=fij,g^a=gi,g^b=gi,g^c=gi,g^f=hi,h^a=gh,h^b=gh,h^c=hj,h^f=gh,i^f=j,j^a=ij,j^b=ij,j^f=ij

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72228331
maximal normal subgroup of#72228331

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