Finite Group DatabasebetaFinite Group Databasebeta| order | 3072 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | #7633646 ⋊ D3 |
| derived | #72239530 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #72260447 |
| commutator, cosocle, hypocenter | #72239530 | C2 |
| Frattini, socle | C23 | #87588 |
| Fitting | #7633646 | D3 |
| LCS | G ▷ #72239530 | |
| UCS | C1 ◁ C2 | |
2878 conjugacy classes of subgroups; only the 22 normal subgroups are shown.
This group has 2878 conjugacy classes of subgroups — too many to draw. Only its 22 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 — 3 generators on 24 points:
(1,4,5,12)(2,14,19,15,9,18,8,6)(3,20,11,10)(7,21,16,13,17,23,24,22);(1,2,6)(3,7,13)(4,8,18)(5,9,15)(10,16,23)(11,17,22)(12,19,14)(20,24,21);(1,3)(4,10)(5,11)(6,13)(12,20)(14,21)(15,22)(18,23)Polycyclic presentation:
2,3,2^9|e^2=j,f^2=i,b^a=b^2,c^a=cd,c^b=d,d^b=cd,e^a=fj,e^b=fj,f^a=ei,f^b=ef,g^b=h,g^c=gijk,g^d=gij,h^a=gh,h^b=gh,h^c=hjk,h^d=hk,i^a=j,i^b=ij,j^a=i,j^b=i,k^a=ijk,k^b=jk54 conjugacy classes — view the full character table.
| maximal subgroup of | #72260581, #72261861, C2×#72260486 |
| maximal normal subgroup of | #72260581, #72261861, C2×#72260486 |
| maximal quotient of | #72260916, #72261991, C2×#72260486 |
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, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 5, F.10);;
gap> SetPower(coll, 6, F.9);;
gap> SetConjugate(coll, 2, 1, F.2^2);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 3, 2, F.4);;
gap> SetConjugate(coll, 4, 2, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.6*F.10);;
gap> SetConjugate(coll, 5, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 1, F.5*F.9);;
gap> SetConjugate(coll, 6, 2, F.5*F.6);;
gap> SetConjugate(coll, 7, 2, F.8);;
gap> SetConjugate(coll, 7, 3, F.7*F.9*F.10*F.11);;
gap> SetConjugate(coll, 7, 4, F.7*F.9*F.10);;
gap> SetConjugate(coll, 8, 1, F.7*F.8);;
gap> SetConjugate(coll, 8, 2, F.7*F.8);;
gap> SetConjugate(coll, 8, 3, F.8*F.10*F.11);;
gap> SetConjugate(coll, 8, 4, F.8*F.11);;
gap> SetConjugate(coll, 9, 1, F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 10, 1, F.9);;
gap> SetConjugate(coll, 10, 2, F.9);;
gap> SetConjugate(coll, 11, 1, F.9*F.10*F.11);;
gap> SetConjugate(coll, 11, 2, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);