Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | #20609 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#1020⋊C2 |
| commutator, hypocenter | #997 | C22×C4 |
| Frattini | C24 | C22×S4 |
| Fitting | #20609 | D3 |
| prime radical | C2×#87516 | C2 |
| cosocle | C2×#997 | C23 |
| hypercenter | C23 | #1020⋊C2 |
| LCS | G ▷ #997 | |
| UCS | C1 ◁ C22 ◁ C23 | |
1671 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 1671 conjugacy classes of subgroups — too many to draw. Only its 51 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) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(5,7,14)(6,13,18,10,11,19)(9,15,20)(12,16);(1,3,4,2)(5,6)(7,13)(8,16)(9,10)(11,15)(12,17)(14,19)(18,20);(1,2,4,3)(7,8)(11,12)(13,16)(15,17)Polycyclic presentation:
2^4,3,2^5|a^2=d,b^2=d,c^b=cj,e^a=e^2,f^a=g,f^b=fhij,f^e=g,g^a=f,g^b=ghj,g^e=fg,h^a=hi,h^e=i,h^f=hj,i^e=hi,i^f=ij,i^g=ij62 conjugacy classes — view the full character table.
| maximal subgroup of | #72246783, C4×#11714170, #72341350, … |
| maximal normal subgroup of | C4×#11714170, #72341350, #72341407 |
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, 1, F.4);;
gap> SetPower(coll, 2, F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.7);;
gap> SetConjugate(coll, 6, 2, F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 5, F.7);;
gap> SetConjugate(coll, 7, 1, F.6);;
gap> SetConjugate(coll, 7, 2, F.7*F.8*F.10);;
gap> SetConjugate(coll, 7, 5, F.6*F.7);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 5, F.9);;
gap> SetConjugate(coll, 8, 6, F.8*F.10);;
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);