Finite Group DatabasebetaFinite Group Databasebeta| order | 3072 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | #8108057 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | AGL(2,ℤ/4) |
| commutator, cosocle | #58990⋊C3 | C22 |
| Frattini | C24 | C2×#1020 |
| Fitting | #8108057 | D3 |
| socle | C23 | #87524 |
| hypocenter | #2679⋊C3 | D4 |
| LCS | G ▷ #58990⋊C3 ▷ #2679⋊C3 | |
| UCS | C1 ◁ C2 | |
4621 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 4621 conjugacy classes of subgroups — too many to draw. Only its 28 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) = 24 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,7,5,8)(2,4)(3,18,22,10,13,24,12,19)(6,20,16,21,15,23,17,11)(9,14);(1,6,11,2,3,10)(4,16,23,7,12,18)(5,15,21,9,13,19)(8,22,24,14,17,20)Polycyclic presentation:
2^3,3,2^7|b^2=c,g^2=j,h^2=ijk,b^a=bc,d^b=d^2,e^b=f,e^d=f,f^b=e,f^d=ef,g^a=gik,g^b=h,g^c=gj,g^d=hj,g^e=gj,g^f=gi,h^a=hj,h^b=gj,h^c=hijk,h^d=gh,h^e=hi,h^f=hijk,i^d=jk,j^b=ijk,j^d=ijk47 conjugacy classes — view the full character table.
| maximal subgroup of | #72256075, #72341182 |
| maximal normal subgroup of | #72341182 |
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", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 7, F.10);;
gap> SetPower(coll, 8, F.9*F.10*F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 2, F.4^2);;
gap> SetConjugate(coll, 5, 2, F.6);;
gap> SetConjugate(coll, 5, 4, F.6);;
gap> SetConjugate(coll, 6, 2, F.5);;
gap> SetConjugate(coll, 6, 4, F.5*F.6);;
gap> SetConjugate(coll, 7, 1, F.7*F.9*F.11);;
gap> SetConjugate(coll, 7, 2, F.8);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.8*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.7*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.9*F.10*F.11);;
gap> SetConjugate(coll, 8, 4, F.7*F.8);;
gap> SetConjugate(coll, 8, 5, F.8*F.9);;
gap> SetConjugate(coll, 8, 6, F.8*F.9*F.10*F.11);;
gap> SetConjugate(coll, 9, 4, F.10*F.11);;
gap> SetConjugate(coll, 10, 2, F.9*F.10*F.11);;
gap> SetConjugate(coll, 10, 4, F.9*F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);