Finite Group DatabasebetaFinite Group Databasebeta| order | 19494 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C9 · Schur multiplier (H2) = C3 |
No aliases for this group.
| Fitting, socle, derived, Sylow | C192 ⋊ C2×C3×C9 |
| Hall | #72214740 ⋊ C2 |
| Hall | C19⋊D19 ⋊ C3×C9 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C192 | C2×C3×C9 |
| prime radical | (C19⋊C3)⋊(D19⋊C3) | C3 |
| cosocle | #11762245 | C2×C32 |
| LCS | G ▷ C192 | |
| UCS | C1 | |
116 conjugacy classes of subgroups.
Polycyclic presentation:
2,3^3,19^2|c^3=d,e^a=e^18,e^b=e^12f^8,e^c=e^16,e^d=e^11,f^a=f^18,f^b=e,f^c=f^16,f^d=f^1166 conjugacy classes — view the full character table.
| maximal subgroup of | #72228749, C2×#72214746, #72244912 |
| maximal normal subgroup of | #72228749, C2×#72214746, #72244912 |
| maximal quotient of | C2×#72214746 |
| holomorph of | C19⋊C3 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:
gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f");;
gap> coll := SingleCollector(F, [2, 3, 3, 3, 19, 19]);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 5, 1, F.5^18);;
gap> SetConjugate(coll, 5, 2, F.5^12*F.6^8);;
gap> SetConjugate(coll, 5, 3, F.5^16);;
gap> SetConjugate(coll, 5, 4, F.5^11);;
gap> SetConjugate(coll, 6, 1, F.6^18);;
gap> SetConjugate(coll, 6, 2, F.5);;
gap> SetConjugate(coll, 6, 3, F.6^16);;
gap> SetConjugate(coll, 6, 4, F.6^11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);