Finite Group DatabasebetaFinite Group Databasebeta| order | 2312 |
| automorphism | ? |
| homology | Abelianization (H1) = C8 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting, socle, derived, Sylow | C172 ⋊ C8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C172 | C8 |
| prime radical | C17⋊D17 | C4 |
| cosocle | #11923883 | C2 |
| LCS | G ▷ C172 | |
| UCS | C1 | |
40 conjugacy classes of subgroups.
Polycyclic presentation:
2^3,17^2|a^2=b,b^2=c,d^a=d^9,d^b=d^13,d^c=d^16,e^a=d^10e^8,e^b=e^13,e^c=e^1644 conjugacy classes — view the full character table.
| maximal subgroup of | #72213374, C2×#72213369, #72218903, … |
| maximal normal subgroup of | #72213374, C2×#72213369, #72218903, … |
| maximal quotient of | C2×#72213369, C3×#72213369, #72229175 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:
gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e");;
gap> coll := SingleCollector(F, [2, 2, 2, 17, 17]);;
gap> SetPower(coll, 1, F.2);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 4, 1, F.4^9);;
gap> SetConjugate(coll, 4, 2, F.4^13);;
gap> SetConjugate(coll, 4, 3, F.4^16);;
gap> SetConjugate(coll, 5, 1, F.4^10*F.5^8);;
gap> SetConjugate(coll, 5, 2, F.5^13);;
gap> SetConjugate(coll, 5, 3, F.5^16);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);