Finite Group DatabasebetaFinite Group Databasebeta| order | 7680 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | C5×#62817 ⋊ D3 |
| derived | C5×#11710730 ⋊ C2 |
| Sylow | C5 ⋊ #72209641 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C22 | C5⋊#87588 |
| commutator, cosocle, hypocenter | C5×#11710730 | C2 |
| Fitting | C5×#62817 | D3 |
| socle | C22×C5 | #87588 |
| LCS | G ▷ C5×#11710730 | |
| UCS | C1 ◁ C22 | |
5062 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 5062 conjugacy classes of subgroups — too many to draw. Only its 49 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 3 generators on 21 points:
(2,3)(4,5)(7,12,11,8)(10,13)(15,16,19,20)(18,21);(1,2,4,5,3)(7,8,10)(11,12,13)(15,16,18)(19,20,21);(6,7)(8,10)(9,11)(12,13)(14,15)(16,18)(17,19)(20,21)Polycyclic presentation:
2,3,5,2^8|b^a=b^2,c^a=c^4,d^a=e,d^b=e,e^a=d,e^b=de,f^a=g,f^b=g,f^e=fk,g^a=f,g^b=fg,g^d=gk,h^a=i,h^b=i,h^e=hj,i^a=h,i^b=hi,i^d=ij111 conjugacy classes — view the full character table.
| maximal subgroup of | D5×#72209641 |
| maximal normal subgroup of | D5×#72209641 |
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, 5, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetConjugate(coll, 2, 1, F.2^2);;
gap> SetConjugate(coll, 3, 1, F.3^4);;
gap> SetConjugate(coll, 4, 1, F.5);;
gap> SetConjugate(coll, 4, 2, F.5);;
gap> SetConjugate(coll, 5, 1, F.4);;
gap> SetConjugate(coll, 5, 2, F.4*F.5);;
gap> SetConjugate(coll, 6, 1, F.7);;
gap> SetConjugate(coll, 6, 2, F.7);;
gap> SetConjugate(coll, 6, 5, F.6*F.11);;
gap> SetConjugate(coll, 7, 1, F.6);;
gap> SetConjugate(coll, 7, 2, F.6*F.7);;
gap> SetConjugate(coll, 7, 4, F.7*F.11);;
gap> SetConjugate(coll, 8, 1, F.9);;
gap> SetConjugate(coll, 8, 2, F.9);;
gap> SetConjugate(coll, 8, 5, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.8);;
gap> SetConjugate(coll, 9, 2, F.8*F.9);;
gap> SetConjugate(coll, 9, 4, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);