Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C22×B(4,𝔽2) ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#1020⋊C2 |
| commutator, cosocle | C2×#997 | C23 |
| Frattini | C24 | C22×S4 |
| Fitting | C22×B(4,𝔽2) | D3 |
| hypercenter | C23 | #1020⋊C2 |
| hypocenter | #997 | C2×D4 |
| LCS | G ▷ C2×#997 ▷ #997 | |
| UCS | C1 ◁ C22 ◁ C23 | |
1863 conjugacy classes of subgroups; only the 43 normal subgroups are shown.
This group has 1863 conjugacy classes of subgroups — too many to draw. Only its 43 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)(5,13,15)(6,20,18)(7,14)(8,9,16)(10,19,12)(11,17);(1,2)(3,4)(6,10)(7,13)(8,15)(11,19)(14,16)(17,20);(5,6)(7,11)(8,12)(9,10)(13,20)(14,17)(15,18)(16,19)Polycyclic presentation:
2^4,3,2^5|f^2=j,g^2=j,b^a=bj,c^b=cd,e^b=e^2,f^b=fg,f^c=fhj,f^e=g,g^b=gj,g^c=ghij,g^e=fgj,g^f=gj,h^b=i,h^e=hi,h^g=hj,i^b=h,i^e=h,i^f=ij,i^g=ij56 conjugacy classes — view the full character table.
| maximal subgroup of | #72246216, #72246533, #72252367, … |
| maximal normal subgroup of | #72246216, #72246533, #72252367, … |
| maximal quotient of | C2×#72245477 |
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, 6, F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 5, 2, F.5^2);;
gap> SetConjugate(coll, 6, 2, F.6*F.7);;
gap> SetConjugate(coll, 6, 3, F.6*F.8*F.10);;
gap> SetConjugate(coll, 6, 5, F.7);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.8*F.9*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, 2, F.9);;
gap> SetConjugate(coll, 8, 5, F.8*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);;
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);