Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | #1020⋊C2 |
| commutator, hypocenter | #997 | C42 |
| Frattini | C23×C4 | C2×S4 |
| Fitting | C2×#1791 | D3 |
| socle | C22 | #1020⋊C4 |
| prime radical, cosocle | C2×#5950 | C22 |
| LCS | G ▷ #997 | |
| UCS | C1 ◁ C2×C4 | |
865 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 865 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) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 36 points:
(1,2,3,4)(5,19,22,8,10,23,9,20)(6,28,21,18,11,29,14,13)(7,32,30,15,17,34,16,27)(12,35,33,31,24,36,26,25);(5,6,7,12,10,11,17,24)(8,21,34,31,22,28,27,26,23,18,16,36,20,14,32,25,9,29,15,33,19,13,30,35)Polycyclic presentation:
2^4,3,2^5|a^2=d,b^2=c,d^2=j,f^2=j,g^2=j,b^a=bj,e^b=e^2,f^a=fhj,f^b=fg,f^e=g,g^a=ghij,g^b=gj,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=ij68 conjugacy classes — view the full character table.
| maximal subgroup of | #72341881, #72341898, #72341911, … |
| maximal normal subgroup of | #72341881, #72341898, #72341911, … |
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.3);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.10);;
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.6*F.7);;
gap> SetConjugate(coll, 6, 5, F.7);;
gap> SetConjugate(coll, 7, 1, F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 7, 2, 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, 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);