Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C2×#3529 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#5934 |
| commutator, cosocle | C2×#5465 | C22 |
| Frattini | C23 | C2×#1020 |
| Fitting | C2×#3529 | D3 |
| hypercenter | C22×C4 | #1020 |
| hypocenter | #5465 | D4 |
| LCS | G ▷ C2×#5465 ▷ #5465 | |
| UCS | C1 ◁ C22 ◁ C22×C4 | |
1427 conjugacy classes of subgroups; only the 29 normal subgroups are shown.
This group has 1427 conjugacy classes of subgroups — too many to draw. Only its 29 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 — 2 generators on 36 points:
(2,4)(5,7,10,17)(6,11,14,24)(8,16,19,29,9,36,20,30,32,15,21,28)(12,34,31,33,26,27,25,23,18,22,13,35);(1,2,3,4)(5,18,9,12)(6,19,13,8)(7,27,16,22)(10,31,21,25)(11,28,23,15)(14,32,26,20)(17,35,30,33)(24,36,34,29)Polycyclic presentation:
2^3,3,2^6|a^2=c,b^2=j,b^a=bc,d^a=d^2,e^a=ef,e^d=f,f^d=ef,g^d=h,g^e=gi,g^f=gj,h^a=gh,h^d=gh,h^e=hij,h^f=hi,i^a=ij49 conjugacy classes — view the full character table.
| maximal subgroup of | #72261456 |
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, 3, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5*F.6);;
gap> SetConjugate(coll, 5, 4, F.6);;
gap> SetConjugate(coll, 6, 4, F.5*F.6);;
gap> SetConjugate(coll, 7, 4, F.8);;
gap> SetConjugate(coll, 7, 5, F.7*F.9);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.7*F.8);;
gap> SetConjugate(coll, 8, 4, F.7*F.8);;
gap> SetConjugate(coll, 8, 5, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 6, F.8*F.9);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);