Finite Group DatabasebetaFinite Group Databasebeta| order | 3072 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | #8062881 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×#11715753 |
| commutator, cosocle | C2×#5464 | C23 |
| Frattini, socle | C23 | C22×#1020 |
| Fitting | #8062881 | D3 |
| hypercenter | C4 | #11715753 |
| hypocenter | #5464 | C2×D4 |
| LCS | G ▷ C2×#5464 ▷ #5464 | |
| UCS | C1 ◁ C2 ◁ C4 | |
5999 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 5999 conjugacy classes of subgroups — too many to draw. Only its 52 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2,3,4)(5,20,7)(6,15,12)(9,10,18)(11,13,17)(14,16,19);(5,13,7,6)(8,10,20,17)(9,12,19,18)(11,14,16,15);(2,4)(6,12)(7,16)(8,15)(9,17)(13,20)(14,18)Polycyclic presentation:
2^4,3,2^6|c^2=d,h^2=k,i^2=jk,c^a=cd,e^b=e^2,f^a=fj,f^b=fg,f^e=g,g^a=gk,g^e=fg,h^a=hj,h^b=fik,h^e=ik,h^f=hk,h^g=hj,i^a=ik,i^b=fghk,i^e=hi,i^f=ij,i^g=ijk,j^b=jk,j^e=k,k^e=jk60 conjugacy classes — view the full character table.
| maximal subgroup of | #72341172, C2×#72341130 |
| maximal normal subgroup of | #72341172, C2×#72341130 |
| maximal quotient of | C2×#72341130 |
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, 3, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 8, F.11);;
gap> SetPower(coll, 9, F.10*F.11);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 2, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.7);;
gap> SetConjugate(coll, 6, 5, F.7);;
gap> SetConjugate(coll, 7, 1, F.7*F.11);;
gap> SetConjugate(coll, 7, 5, F.6*F.7);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.6*F.9*F.11);;
gap> SetConjugate(coll, 8, 5, F.9*F.11);;
gap> SetConjugate(coll, 8, 6, F.8*F.11);;
gap> SetConjugate(coll, 8, 7, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.11);;
gap> SetConjugate(coll, 9, 2, F.6*F.7*F.8*F.11);;
gap> SetConjugate(coll, 9, 5, F.8*F.9);;
gap> SetConjugate(coll, 9, 6, F.9*F.10);;
gap> SetConjugate(coll, 9, 7, F.9*F.10*F.11);;
gap> SetConjugate(coll, 10, 2, F.10*F.11);;
gap> SetConjugate(coll, 10, 5, F.11);;
gap> SetConjugate(coll, 11, 5, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);