Finite Group DatabasebetaFinite Group Databasebeta| order | 2496 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C13×#120 ⋊ C2×C3 |
| Hall | #11718693 ⋊ C3 |
| Hall | C13⋊C3 ⋊ B(4,𝔽2) |
| Sylow | C13 ⋊ C3×B(4,𝔽2) |
396 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 396 conjugacy classes of subgroups — too many to draw. Only its 66 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 104 |
A minimal faithful permutation representation — 3 generators on 21 points:
(2,6,7,3,4,8)(5,12,11,9,10,13)(14,16)(15,18)(17,20)(19,21);(1,2,5,4,8,11,10,12,13,7,6,9,3)(18,21);(14,15,17,19)(16,18,20,21)Polycyclic presentation:
2^3,3,2^3,13|a^2=g,b^a=bf,c^b=ce,e^a=eg,f^c=fg,h^c=h^12,h^d=h^376 conjugacy classes — view the full character table.
| maximal subgroup of | B(4,𝔽2)×D13⋊C3 |
| maximal normal subgroup of | B(4,𝔽2)×D13⋊C3 |
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");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 2, 2, 2, 13]);;
gap> SetPower(coll, 1, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 6, 3, F.6*F.7);;
gap> SetConjugate(coll, 8, 3, F.8^12);;
gap> SetConjugate(coll, 8, 4, F.8^3);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);