Finite Group DatabasebetaFinite Group Databasebeta| order | 2304 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C24×C3 |
No aliases for this group.
| Fitting | C25×C4 ⋊ C3⋊D3 |
1911 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 1911 conjugacy classes of subgroups — too many to draw. Only its 71 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:
(1,4,3,2)(5,6,8,7)(14,16);(6,7,8)(9,10,11,12)(13,14,15,16);(2,3,4)(6,7,8)Polycyclic presentation:
2^4,3^2,2^4|a^2=c,b^a=bcd,e^b=e^2,f^b=f^2,g^b=h,g^e=h,g^f=h,h^b=g,h^e=gh,h^f=gh,i^b=gij,i^e=j,i^f=ghi,j^b=ghj,j^e=ij,j^f=gj88 conjugacy classes — view the full character table.
| maximal subgroup of | #72259032, C22⋊C4×A4⋊S4, #72259055 |
| maximal normal subgroup of | #72259032, C22⋊C4×A4⋊S4, #72259055 |
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> SetConjugate(coll, 2, 1, F.2*F.3*F.4);;
gap> SetConjugate(coll, 5, 2, F.5^2);;
gap> SetConjugate(coll, 6, 2, F.6^2);;
gap> SetConjugate(coll, 7, 2, F.8);;
gap> SetConjugate(coll, 7, 5, F.8);;
gap> SetConjugate(coll, 7, 6, F.8);;
gap> SetConjugate(coll, 8, 2, F.7);;
gap> SetConjugate(coll, 8, 5, F.7*F.8);;
gap> SetConjugate(coll, 8, 6, F.7*F.8);;
gap> SetConjugate(coll, 9, 2, F.7*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, 2, F.7*F.8*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);