Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | #62624 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C16 | #1020 |
| commutator, hypocenter | #997 | C2×C8 |
| Frattini | C8 | C2×#1020 |
| Fitting | #62624 | D3 |
| socle | C2 | C8×#1020 |
| prime radical | #87520 | C4 |
| cosocle | #85666 | C22 |
| LCS | G ▷ #997 | |
| UCS | C1 ◁ C16 | |
663 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 663 conjugacy classes of subgroups — too many to draw. Only its 32 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,3,8,4,9,13,24,7,12,16,27,19,30,37,48)(5,33,38,25,44,53,35,63,23,32,15,52,43,29,60,62,22,31,14,51,42,28,59,61,21,11,41,50,18,56,58,47,20,10,40,49,17,55,57,46,6,34,39,26,45,54,36,64);(1,20,22,5,7,23,6,21)(2,31,33,10,12,34,11,32)(3,38,40,14,16,41,15,39)(4,42,44,17,19,45,18,43)(8,49,51,25,27,52,26,50)(9,53,55,28,30,56,29,54)(13,57,59,35,37,60,36,58)(24,61,63,46,48,64,47,62)Polycyclic presentation:
2^4,3,2^5|b^2=c,c^2=d,d^2=j,f^2=j,g^2=j,e^a=e^2,f^a=fg,f^e=g,g^a=gj,g^e=fgj,g^f=gj,h^a=hi,h^e=i,h^g=hj,i^e=hi,i^f=ij104 conjugacy classes — view the full character table.
| maximal subgroup of | #72341738 |
| maximal normal subgroup of | #72341738 |
| maximal quotient of | C16×#5935 |
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.3);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6*F.7);;
gap> SetConjugate(coll, 6, 5, F.7);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 7, 5, F.6*F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 5, F.9);;
gap> SetConjugate(coll, 8, 7, F.8*F.10);;
gap> SetConjugate(coll, 9, 5, F.8*F.9);;
gap> SetConjugate(coll, 9, 6, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);