Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Fitting | #20610 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#1020⋊C2 |
| commutator, hypocenter | #997 | C22×C4 |
| Frattini | C24 | C22×S4 |
| Fitting | #20610 | D3 |
| prime radical | C2×#87520 | C2 |
| cosocle | C2×#997 | C23 |
| hypercenter | C2×C4 | #1020⋊C2 |
| LCS | G ▷ #997 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
1493 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 1493 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) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,17,24,7,20,34,10,31,21,18,9,32)(6,26,28,12,23,36,11,30,27,15,14,35)(8,33,22,16)(13,29,19,25);(6,11)(8,20)(9,22)(12,15)(13,14)(16,31)(17,33)(19,23)(21,24)(25,26)(29,30)(32,34);(1,2,3,4)(5,6)(7,15)(8,19)(9,23)(10,11)(12,18)(13,22)(14,20)(16,25)(17,26)(21,28)(24,27)(29,33)(30,31)(32,35)(34,36)Polycyclic presentation:
2^4,3,2^5|a^2=d,c^2=j,f^2=j,g^2=j,b^a=bj,c^a=cj,e^b=e^2,f^a=fhj,f^b=gj,f^e=fg,g^a=ghij,g^b=fj,g^e=f,g^f=gj,h^b=hi,h^e=i,h^g=hj,i^e=hi,i^f=ij,i^g=ij62 conjugacy classes — view the full character table.
| maximal subgroup of | #72341494, #72341800, #72341834, … |
| maximal normal subgroup of | #72341494, #72341800, #72341834, … |
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, 3, 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, 3, 1, F.3*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.7*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.7);;
gap> SetConjugate(coll, 7, 1, F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 7, 2, F.6*F.10);;
gap> SetConjugate(coll, 7, 5, F.6);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 5, F.9);;
gap> SetConjugate(coll, 8, 7, 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);