Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | #62654 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C22 | C22×#1020 |
| commutator, hypocenter | #997 | C22×C4 |
| Fitting | #62654 | D3 |
| prime radical | C2×#87521 | C2 |
| cosocle | C2×#997 | C23 |
| hypercenter | C4⋊C4 | #1020 |
| LCS | G ▷ #997 | |
| UCS | C1 ◁ C22 ◁ C4⋊C4 | |
3257 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 3257 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,3,4,2)(5,6)(7,17,13,11,15,18)(8,20,14,12,19,16)(9,10);(1,2,4,3)(5,7)(6,16)(8,13)(9,14)(10,11)(12,20)(15,19)(17,18);(6,10)(7,8)(11,17)(12,16)(14,15)(18,20)Polycyclic presentation:
2^4,3,2^5|b^2=d,c^2=d,c^a=cj,c^b=cj,e^a=e^2,f^a=g,f^e=g,g^a=f,g^e=fg,h^a=hi,h^e=i,h^f=hj,i^e=hi,i^f=ij,i^g=ij86 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#11715839, #72341368, #72341385, … |
| maximal normal subgroup of | C4×#11715839, #72341368, #72341385, … |
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, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 3, 1, F.3*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.7);;
gap> SetConjugate(coll, 6, 5, F.7);;
gap> SetConjugate(coll, 7, 1, F.6);;
gap> SetConjugate(coll, 7, 5, F.6*F.7);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 5, F.9);;
gap> SetConjugate(coll, 8, 6, F.8*F.10);;
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);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);