Finite Group DatabasebetaFinite Group Databasebeta| order | 3456 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22×C3 |
No aliases for this group.
| Fitting | C2×C3×H(5,𝔽2) ⋊ C3×D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×A4×C3⋊S4 |
| commutator, cosocle, hypocenter | C3×#994 | C22×C3 |
| Fitting | C2×C3×H(5,𝔽2) | C3×D3 |
| socle | C2×C3 | C2×A4×S4 |
| hypercenter | C22 | A4×C3⋊S4 |
| LCS | G ▷ C3×#994 | |
| UCS | C1 ◁ C2 ◁ C22 | |
1022 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 1022 conjugacy classes of subgroups — too many to draw. Only its 59 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 3 generators on 51 points:
(1,15,27,18,3,37,8,39,6,22,17,19)(2,36,43,11,10,42,9,35,24,30,29,47)(4,14,23,5,16,25,20,21,26,7,38,41)(12,33,46,31,34,45,28,48,44,13,32,40)(49,50,51);(1,31)(2,22)(3,35)(4,28)(5,11)(6,46)(7,30)(8,13)(9,18)(10,38)(12,20)(14,48)(15,32)(16,29)(17,36)(19,47)(21,33)(23,40)(24,41)(25,43)(26,45)(27,44)(34,39)(37,42)(50,51);(1,6,3)(2,10,24)(4,19,14)(5,23,15)(7,26,39)(8,27,17)(9,29,43)(11,32,40)(12,33,42)(13,36,44)(16,18,25)(20,37,21)(22,41,38)(28,48,47)(30,34,45)(31,35,46)Polycyclic presentation:
2^2,3,2^2,3^2,2^3|a^2=j,d^2=j,e^2=j,h^2=j,i^2=j,b^a=bj,d^c=e,e^c=dej,e^d=ej,f^a=f^2,g^a=g^2,h^a=hj,h^f=i,i^a=hi,i^f=hij,i^h=ij99 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#983⋊A4, #72252919, #72252930 |
| maximal normal subgroup of | D3×#983⋊A4, #72252919, #72252930 |
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, 3, 2, 2, 3, 3, 2, 2, 2]);;
gap> SetPower(coll, 1, F.10);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 5, F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.10);;
gap> SetConjugate(coll, 4, 3, F.5);;
gap> SetConjugate(coll, 5, 3, F.4*F.5*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 6, F.9);;
gap> SetConjugate(coll, 9, 1, F.8*F.9);;
gap> SetConjugate(coll, 9, 6, F.8*F.9*F.10);;
gap> SetConjugate(coll, 9, 8, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);