Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
2046 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 2046 conjugacy classes of subgroups — too many to draw. Only its 57 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,4,3)(6,7,8)(9,11,13,20,14,21,12,19)(10,15,16,18,17,23,22,24);(2,4)(5,6,8,7)(9,10,12,16,14,17,13,22)(11,18,19,23,21,24,20,15)Polycyclic presentation:
2^5,3,2^4|a^2=c,b^2=d,c^2=e,d^2=e,b^a=bd,d^a=de,f^a=f^2,g^a=h,g^f=h,h^a=g,h^f=gh,i^a=gij,i^f=j,j^a=ghj,j^f=ij88 conjugacy classes — view the full character table.
| maximal subgroup of | #72253121 |
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, 2, 3, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 4, F.5);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.8);;
gap> SetConjugate(coll, 7, 6, F.8);;
gap> SetConjugate(coll, 8, 1, F.7);;
gap> SetConjugate(coll, 8, 6, F.7*F.8);;
gap> SetConjugate(coll, 9, 1, F.7*F.9*F.10);;
gap> SetConjugate(coll, 9, 6, F.10);;
gap> SetConjugate(coll, 10, 1, F.7*F.8*F.10);;
gap> SetConjugate(coll, 10, 6, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);