Finite Group DatabasebetaFinite Group Databasebeta| order | 2304 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22×C3 |
No aliases for this group.
1099 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 1099 conjugacy classes of subgroups — too many to draw. Only its 69 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,4)(2,3)(5,8,7)(10,13)(14,16);(2,4)(7,8)(9,10,11,14,12,13,15,16);(2,3,4)(6,7,8)Polycyclic presentation:
2^4,3^2,2^4|a^2=c,c^2=d,b^a=bd,e^a=e^2,f^a=f^2,g^e=h,g^f=h,h^a=gh,h^e=gh,h^f=gh,i^a=hij,i^e=j,i^f=ghi,j^a=gj,j^e=ij,j^f=gj88 conjugacy classes — view the full character table.
| maximal subgroup of | M16×A4⋊S4, #72253121, #72258944, … |
| maximal normal subgroup of | M16×A4⋊S4, #72253121, #72258944, … |
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, 3, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 7, 5, F.8);;
gap> SetConjugate(coll, 7, 6, F.8);;
gap> SetConjugate(coll, 8, 1, F.7*F.8);;
gap> SetConjugate(coll, 8, 5, F.7*F.8);;
gap> SetConjugate(coll, 8, 6, F.7*F.8);;
gap> SetConjugate(coll, 9, 1, F.8*F.9*F.10);;
gap> SetConjugate(coll, 9, 5, F.10);;
gap> SetConjugate(coll, 9, 6, F.7*F.8*F.9);;
gap> SetConjugate(coll, 10, 1, F.7*F.10);;
gap> SetConjugate(coll, 10, 5, F.9*F.10);;
gap> SetConjugate(coll, 10, 6, F.7*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);