Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C4×#585 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#1020 |
| commutator, cosocle | C2×#997 | C23 |
| Frattini, socle | C22 | C22×#1020 |
| Fitting | C4×#585 | D3 |
| hypercenter | C42 | #1020 |
| hypocenter | #997 | #41 |
| LCS | G ▷ C2×#997 ▷ #997 | |
| UCS | C1 ◁ C2×C4 ◁ C42 | |
2849 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 2849 conjugacy classes of subgroups — too many to draw. Only its 54 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) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,3,4,7)(2,6,5,8)(12,18,21)(13,20,17)(15,23,24)(16,19,22);(2,5)(6,8)(9,20,13,12)(10,24,16,22)(11,21,18,17)(14,19,23,15);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,19)(13,16)(15,20)(17,24)(18,23)(21,22)Polycyclic presentation:
2^4,3,2^5|c^2=dj,f^2=j,g^2=j,b^a=bd,e^a=e^2,e^b=e^2,f^a=fgij,f^b=fgij,f^e=g,g^a=gh,g^b=gh,g^e=fgj,g^f=gj,h^e=i,h^f=hj,i^a=hi,i^b=hi,i^e=hi,i^f=ij,i^g=ij74 conjugacy classes — view the full character table.
| maximal subgroup of | #72246309, #72246847, #72246879, … |
| maximal normal subgroup of | #72246309, #72246847, #72246879, … |
| maximal quotient of | C2×#72245747 |
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, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 3, F.4*F.10);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 5, 2, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6*F.7*F.9*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.7*F.9*F.10);;
gap> SetConjugate(coll, 6, 5, F.7);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> SetConjugate(coll, 7, 5, F.6*F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 5, F.9);;
gap> SetConjugate(coll, 8, 6, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.8*F.9);;
gap> SetConjugate(coll, 9, 2, F.8*F.9);;
gap> SetConjugate(coll, 9, 5, F.8*F.9);;
gap> SetConjugate(coll, 9, 6, F.9*F.10);;
gap> SetConjugate(coll, 9, 7, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);