Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | D4×H(5,𝔽2) ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C22 | C22×#1020 |
| commutator, cosocle | C2×#997 | C23 |
| Fitting | D4×H(5,𝔽2) | D3 |
| hypercenter | C2×D4 | #1020 |
| hypocenter | #997 | C2×D4 |
| LCS | G ▷ C2×#997 ▷ #997 | |
| UCS | C1 ◁ C22 ◁ C2×D4 | |
3153 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 3153 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) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,3,4,2)(5,15,14,13,7,16,9,8)(6,19,18,17,10,20,12,11);(1,2)(3,4)(6,10)(8,9,16)(11,18,20,17,12,19)(13,14,15);(2,3)(5,6)(7,10)(8,11)(9,12)(13,17)(14,18)(15,19)(16,20)Polycyclic presentation:
2^4,3,2^5|c^2=dj,f^2=j,g^2=j,b^a=bd,c^a=cdj,c^b=cdj,e^c=e^2,f^c=fg,f^e=g,g^c=gj,g^e=fgj,g^f=gj,h^c=i,h^e=hi,h^g=hj,i^c=h,i^e=h,i^f=ij,i^g=ij59 conjugacy classes — view the full character table.
| maximal subgroup of | #72246076, #72246216, #72246265, … |
| maximal normal subgroup of | #72246076, #72246216, #72246265, … |
| maximal quotient of | C2×#72245388 |
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, 3, 1, F.3*F.4*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.4*F.10);;
gap> SetConjugate(coll, 5, 3, F.5^2);;
gap> SetConjugate(coll, 6, 3, F.6*F.7);;
gap> SetConjugate(coll, 6, 5, F.7);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 5, F.6*F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 3, F.9);;
gap> SetConjugate(coll, 8, 5, F.8*F.9);;
gap> SetConjugate(coll, 8, 7, F.8*F.10);;
gap> SetConjugate(coll, 9, 3, F.8);;
gap> SetConjugate(coll, 9, 5, F.8);;
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);