Finite Group DatabasebetaFinite Group Databasebeta| order | 3072 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C2×C43 |
No aliases for this group.
| Fitting, Sylow | #72239227 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | #11709197 |
| commutator, cosocle, hypocenter | #153709 | C2×C3 |
| Frattini, socle | C25 | C2×C22⋊A4 |
| Fitting | #72239227 | C3 |
| prime radical | #72222569 | C2 |
| LCS | G ▷ #153709 | |
| UCS | C1 ◁ C4 | |
6911 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 6911 conjugacy classes of subgroups — too many to draw. Only its 75 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,5)(3,8,10)(4,6,11)(7,12,9)(13,14,22)(15,20,24)(16,17,18)(19,23,21)(25,26,29,32)(27,37,36,38,33,31,34,39,28,30,35,40);(13,21,16,15)(14,19,17,20)(18,22)(23,24)(27,34)(28,36)(30,38)(31,40);(1,9,4,3)(2,7,6,8)(5,11)(10,12)(25,35,29,33)(26,39,32,37)(27,28,34,36)(30,31,38,40)Polycyclic presentation:
2,3,2^9|a^2=j,c^2=g,d^2=h,e^2=i,f^2=k,b^a=bi,c^b=dgj,d^b=cdj,d^c=dj,e^b=fi,e^d=ej,f^b=ef,f^c=fj,g^b=h,h^b=ghj,i^b=k,k^b=ik192 conjugacy classes — view the full character table.
| maximal subgroup of | #72239354, #72239370, #72239408, … |
| maximal normal subgroup of | #72239370, #72239827, #72239930 |
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, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.10);;
gap> SetPower(coll, 3, F.7);;
gap> SetPower(coll, 4, F.8);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 6, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.9);;
gap> SetConjugate(coll, 3, 2, F.4*F.7*F.10);;
gap> SetConjugate(coll, 4, 2, F.3*F.4*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 2, F.6*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.5*F.6);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.8);;
gap> SetConjugate(coll, 8, 2, F.7*F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.11);;
gap> SetConjugate(coll, 11, 2, F.9*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);