Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| derived | #543⋊C3 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C23 | C2×#1020 |
| commutator, hypocenter | #543⋊C3 | C2×C4 |
| Fitting | C2×#3459 | D3 |
| prime radical | C2×#3459⋊C3 | C2 |
| cosocle | C2×#543⋊C3 | C22 |
| hypercenter | C22×C4 | #1020 |
| LCS | G ▷ #543⋊C3 | |
| UCS | C1 ◁ C23 ◁ C22×C4 | |
1484 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 1484 conjugacy classes of subgroups — too many to draw. Only its 50 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:
(6,10)(7,16,20)(8,14,15)(11,12,19,17,18,13)(21,33,31,28,29,27,25,36,24,22,32,35)(23,26,30,34);(1,2,3,4)(5,13,8,11)(6,14,12,7)(9,19,16,17)(10,20,18,15)(23,29,30,32)(26,33,34,36)Polycyclic presentation:
2^3,3,2^6|a^2=c,b^2=ij,g^2=ij,h^2=ij,b^a=bj,d^a=d^2,e^a=ef,e^d=f,f^d=ef,g^a=gh,g^d=h,g^f=gi,h^a=hij,h^d=ghij,h^e=hi,h^g=hij66 conjugacy classes — view the full character table.
| maximal subgroup of | #72268131, #72268139, #72268158, … |
| maximal normal subgroup of | #72268131, #72268139, #72268158, … |
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.9*F.10);;
gap> SetPower(coll, 7, F.9*F.10);;
gap> SetPower(coll, 8, F.9*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.10);;
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, 1, F.7*F.8);;
gap> SetConjugate(coll, 7, 4, F.8);;
gap> SetConjugate(coll, 7, 6, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 4, F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 5, F.8*F.9);;
gap> SetConjugate(coll, 8, 7, F.8*F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);