Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | #21483 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C8 | #1020⋊C2 |
| commutator, hypocenter | #997 | C22×C4 |
| Frattini | C22×C4 | C22×S4 |
| Fitting | #21483 | D3 |
| socle | C2 | C4×#1020⋊C2 |
| prime radical | #11714169 | C2 |
| cosocle | #5950 | C23 |
| LCS | G ▷ #997 | |
| UCS | C1 ◁ C8 | |
918 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 918 conjugacy classes of subgroups — too many to draw. Only its 47 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 — 3 generators on 64 points:
(1,54,6,21)(2,55,11,36)(3,63,16,42)(4,29,20,30)(5,25,22,7)(8,28,26,23)(9,64,32,57)(10,39,35,27)(12,19,37,52)(13,38,24,53)(14,50,41,51)(15,46,43,17)(18,49,47,44)(31,60,56,48)(33,40,58,61)(34,59,45,62);(1,3,5,15,8,18,4,14)(2,9,10,31,13,34,19,40)(6,17,54,43,30,46,20,42,29,16,23,63,26,51,28,41,7,50,22,44,25,47,21,49)(11,33,38,56,27,58,52,62,39,32,36,59,24,48,55,61,12,60,35,57,37,45,53,64);(1,2,8,13)(3,9,18,34)(4,19,5,10)(6,24,26,11)(7,27,30,12)(14,40,15,31)(16,45,47,32)(17,48,51,33)(20,35,22,52)(21,36,23,53)(25,39,29,37)(28,38,54,55)(41,56,43,61)(42,57,44,62)(46,60,50,58)(49,59,63,64)Polycyclic presentation:
2^4,3,2^5|a^2=j,b^2=d,c^2=j,d^2=j,f^2=j,g^2=j,c^a=cj,e^c=e^2,f^a=fij,f^c=fi,f^e=g,g^a=ghj,g^c=fghij,g^e=fgj,g^f=gj,h^c=hi,h^e=hi,h^f=hj,i^e=h,i^g=ij68 conjugacy classes — view the full character table.
| maximal subgroup of | #72342728, #72342787, #72342793 |
| maximal normal subgroup of | #72342728, #72342787, #72342793 |
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, 1, F.10);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 3, F.10);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.10);;
gap> SetConjugate(coll, 5, 3, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 6, 5, F.7);;
gap> SetConjugate(coll, 7, 1, F.7*F.8*F.10);;
gap> SetConjugate(coll, 7, 3, F.6*F.7*F.8*F.9*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, 3, F.8*F.9);;
gap> SetConjugate(coll, 8, 5, F.8*F.9);;
gap> SetConjugate(coll, 8, 6, F.8*F.10);;
gap> SetConjugate(coll, 9, 5, F.8);;
gap> SetConjugate(coll, 9, 7, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);