Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Fitting | C4×H(3,𝔽4) ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×#85559 |
| commutator, cosocle | C2×#5464 | C22 |
| Frattini, socle | C23 | C2×#1020 |
| Fitting | C4×H(3,𝔽4) | D3 |
| hypercenter | C4 | #85559 |
| hypocenter | #5464 | D4 |
| LCS | G ▷ C2×#5464 ▷ #5464 | |
| UCS | C1 ◁ C2 ◁ C4 | |
992 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 992 conjugacy classes of subgroups — too many to draw. Only its 24 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) = 48 |
A minimal faithful permutation representation — 2 generators on 20 points:
(1,2,3,4)(5,20,17)(6,18,9)(7,15,14)(8,13,11)(10,19,12);(2,4)(7,10)(8,19,13,18)(9,14,12,17)(11,20,16,15)Polycyclic presentation:
2^3,3,2^6|b^2=c,g^2=j,h^2=ij,b^a=bc,d^a=d^2,e^a=fi,e^d=f,f^a=eij,f^d=ef,g^a=eghj,g^d=hj,g^e=gj,g^f=gi,h^a=efhij,h^d=gh,h^e=hi,h^f=hij,i^a=ij,i^d=j,j^d=ij33 conjugacy classes — view the full character table.
| maximal subgroup of | #72341130, #72341149, C2×#72341129 |
| maximal normal subgroup of | #72341130, #72341149, C2×#72341129 |
| maximal quotient of | C2×#72341129 |
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, 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);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.6*F.9);;
gap> SetConjugate(coll, 5, 4, F.6);;
gap> SetConjugate(coll, 6, 1, F.5*F.9*F.10);;
gap> SetConjugate(coll, 6, 4, F.5*F.6);;
gap> SetConjugate(coll, 7, 1, F.5*F.7*F.8*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.5*F.6*F.8*F.9*F.10);;
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);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 4, F.10);;
gap> SetConjugate(coll, 10, 4, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);