Finite Group DatabasebetaFinite Group Databasebeta| order | 2304 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22×C3 |
No aliases for this group.
| Fitting | C24×Q8 ⋊ C3⋊D3 |
1234 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 1234 conjugacy classes of subgroups — too many to draw. Only its 51 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) = 96 |
A minimal faithful permutation representation — 3 generators on 16 points:
(2,4)(5,8)(6,7)(9,11,10,12,16,13)(14,15);(1,2,3,4)(7,8)(10,13)(11,14)(15,16);(6,7,8)Polycyclic presentation:
2^3,3^2,2^5|a^2=c,h^2=j,i^2=j,b^a=bc,d^a=d^2,e^a=e^2,f^d=g,g^a=fg,g^d=fg,h^a=hj,h^e=i,i^a=hi,i^e=hij,i^h=ij74 conjugacy classes — view the full character table.
| maximal subgroup of | GL(2,𝔽3)×GL(2,ℤ/4) |
| maximal normal subgroup of | GL(2,𝔽3)×GL(2,ℤ/4) |
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, 3, 3, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 6, 4, F.7);;
gap> SetConjugate(coll, 7, 1, F.6*F.7);;
gap> SetConjugate(coll, 7, 4, F.6*F.7);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 5, F.9);;
gap> SetConjugate(coll, 9, 1, F.8*F.9);;
gap> SetConjugate(coll, 9, 5, F.8*F.9*F.10);;
gap> SetConjugate(coll, 9, 8, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);