Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Fitting | #32898 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#1020⋊C2 |
| commutator, cosocle | C2×#997 | C23 |
| Frattini | C24 | C22×S4 |
| Fitting | #32898 | D3 |
| hypercenter | C23 | #1020⋊C2 |
| hypocenter | #997 | C2×D4 |
| LCS | G ▷ C2×#997 ▷ #997 | |
| UCS | C1 ◁ C22 ◁ C23 | |
2124 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 2124 conjugacy classes of subgroups — too many to draw. Only its 39 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 20 points:
(2,4)(5,7,14)(6,11,18)(9,15,20)(10,13,19);(6,10)(7,8)(11,16)(12,13)(15,17)(18,19);(1,2)(3,4)(5,6)(7,13)(8,16)(9,10)(11,15)(12,17)(14,19)(18,20)Polycyclic presentation:
2^4,3,2^5|b^a=bj,c^a=cd,e^b=e^2,f^a=fhj,f^b=g,f^e=g,g^a=gij,g^b=f,g^e=fg,h^b=i,h^e=i,h^g=hj,i^b=h,i^e=hi,i^f=ij47 conjugacy classes — view the full character table.
| maximal subgroup of | #72227803 |
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> SetConjugate(coll, 2, 1, F.2*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 2, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6*F.8*F.10);;
gap> SetConjugate(coll, 6, 2, F.7);;
gap> SetConjugate(coll, 6, 5, F.7);;
gap> SetConjugate(coll, 7, 1, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 2, F.6);;
gap> SetConjugate(coll, 7, 5, F.6*F.7);;
gap> SetConjugate(coll, 8, 2, F.9);;
gap> SetConjugate(coll, 8, 5, F.9);;
gap> SetConjugate(coll, 8, 7, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.8);;
gap> SetConjugate(coll, 9, 5, F.8*F.9);;
gap> SetConjugate(coll, 9, 6, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);