Finite Group DatabasebetaFinite Group Databasebeta| order | 3072 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting, Sylow | #72268306 ⋊ C3 |
| derived | #9481449 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #35346⋊C3 |
| commutator, cosocle, hypocenter | #9481449 | C2×C3 |
| Frattini | C25 | C2×C22⋊A4 |
| Fitting | #72268306 | C3 |
| hypercenter | C23 | #85645 |
| LCS | G ▷ #9481449 | |
| UCS | C1 ◁ C22 ◁ C23 | |
1707 conjugacy classes of subgroups; only the 31 normal subgroups are shown.
This group has 1707 conjugacy classes of subgroups — too many to draw. Only its 31 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 96 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 96 points:
(2,11,8,7,3,18)(4,5,16)(9,27,29,19,13,32)(10,28,21,20,24,30)(12,31,25,26,22,23)(14,15,17)(33,34)(35,50,46,45,36,60)(37,43,55,42,38,68)(39,44)(40,53,63,49,41,75)(47,70,81,66,52,96)(48,94,73,72,77,86)(51,87,76,67,74,93)(54,65,85,61,58,92)(56,71,84,69,59,95)(57,90,79,62,83,88)(64,78,89,80,91,82);(1,14,6,16)(2,23,7,25)(3,28,11,24)(4,17,15,5)(8,32,18,29)(9,20,19,10)(12,27,26,13)(21,31,30,22)(33,55)(34,68)(35,76)(36,77)(37,38)(39,84)(40,89)(41,90)(42,43)(44,95)(45,93)(46,81)(47,48)(49,82)(50,94)(51,52)(53,83)(54,57)(56,59)(58,80)(60,96)(61,62)(63,92)(64,65)(66,72)(67,70)(69,71)(73,74)(75,85)(78,79)(86,87)(88,91);(1,8)(2,3)(4,21)(5,22)(6,18)(7,11)(9,12)(10,13)(14,32)(15,30)(16,29)(17,31)(19,26)(20,27)(23,28)(24,25)(33,46)(34,60,44,63)(35,36)(37,79)(38,74)(39,75)(40,50,45,41)(42,86,69,88)(43,91,71,87)(47,80)(48,52)(49,53)(51,54)(55,85)(56,73)(57,58)(59,78)(61,67,66,64)(62,65,72,70)(68,92,95,96)(76,83)(77,82)(81,84)(89,90,93,94)Polycyclic presentation:
2,3,2^9|c^2=i,d^2=i,e^2=j,f^2=j,c^a=ch,c^b=cd,d^a=dg,d^b=c,d^c=di,e^b=ef,e^c=ek,e^d=eik,f^b=e,f^c=fi,f^d=fk,f^e=fj,g^b=h,g^c=gjk,g^d=gk,g^e=gk,h^b=ghj,h^c=hk,h^d=hj,h^f=hk,i^a=ik50 conjugacy classes — view the full character table.
| maximal quotient of | #72268381 |
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, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 3, F.9);;
gap> SetPower(coll, 4, F.9);;
gap> SetPower(coll, 5, F.10);;
gap> SetPower(coll, 6, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.3);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.6);;
gap> SetConjugate(coll, 5, 3, F.5*F.11);;
gap> SetConjugate(coll, 5, 4, F.5*F.9*F.11);;
gap> SetConjugate(coll, 6, 2, F.5);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.11);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.8);;
gap> SetConjugate(coll, 7, 3, F.7*F.10*F.11);;
gap> SetConjugate(coll, 7, 4, F.7*F.11);;
gap> SetConjugate(coll, 7, 5, F.7*F.11);;
gap> SetConjugate(coll, 8, 2, F.7*F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.11);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 8, 6, F.8*F.11);;
gap> SetConjugate(coll, 9, 1, F.9*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);