Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | #36066 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #11713204 |
| commutator, cosocle | #376⋊C3 | C23 |
| Frattini | C22×C4 | C22×S4 |
| Fitting | #36066 | D3 |
| socle | C23 | C2×GL(2,ℤ/4) |
| hypercenter | Q8 | #857⋊C2 |
| hypocenter | #95⋊C3 | C2×D4 |
| LCS | G ▷ #376⋊C3 ▷ #95⋊C3 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ Q8 | |
1102 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 1102 conjugacy classes of subgroups — too many to draw. Only its 42 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) = 48 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,6,10,4,2,12)(3,11,5,8,7,9)(13,30,29,14,31,34,19,27,36,18,23,24)(15,25,28,22,16,33,21,20,35,17,26,32);(3,8)(7,11)(10,12);(2,5)(6,9)(7,12)(10,11)(14,17)(15,21)(16,24)(18,22)(20,28)(23,32)(25,35)(26,34)(27,29)(30,36)(31,33)Polycyclic presentation:
2^4,3,2^5|c^2=j,d^2=j,f^2=h,g^2=i,c^a=cd,d^a=dj,d^c=dj,e^a=e^2,e^c=eij,e^d=ehj,f^a=ghij,f^b=fhj,f^e=fg,g^a=fhij,g^b=gij,g^e=fij,g^f=gj,h^a=i,h^e=hij,i^a=h,i^e=h45 conjugacy classes — view the full character table.
| maximal quotient of | #72258501 |
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, 3, F.10);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 5, 3, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.8*F.10);;
gap> SetConjugate(coll, 6, 1, F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.8*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.7);;
gap> SetConjugate(coll, 7, 1, F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 5, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.9);;
gap> SetConjugate(coll, 8, 5, F.8*F.9*F.10);;
gap> SetConjugate(coll, 9, 1, F.8);;
gap> SetConjugate(coll, 9, 5, F.8);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);