Finite Group DatabasebetaFinite Group Databasebeta| order | 8112 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Hall | #72214177 ⋊ C3 |
| Hall | #92789 ⋊ C22⋊C4 |
| Sylow | C132 ⋊ C3×C22⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | #72220622 |
| commutator, Fitting, socle | C2×C132 | C2×C3×C4 |
| prime radical | C2×#11759818⋊C2 | C2 |
| cosocle | C2×C13⋊D13 | C22×C3 |
| hypocenter | C132 | C3×C22⋊C4 |
| LCS | G ▷ C2×C132 ▷ C132 | |
| UCS | C1 ◁ C2 | |
152 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 152 conjugacy classes of subgroups — too many to draw. Only its 30 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 156 |
Polycyclic presentation:
2^2,3,2^2,13^2|a^2=d,b^2=de,b^a=be,f^a=f^5,f^b=f^8,f^c=f^9,f^d=f^12,g^a=f^6g^8,g^b=g^8,g^c=fg^3,g^d=g^1250 conjugacy classes — view the full character table.
| maximal subgroup of | #72241924, #72242045 |
| maximal normal subgroup of | #72241924, #72242045 |
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");;
gap> coll := SingleCollector(F, [2, 2, 3, 2, 2, 13, 13]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.4*F.5);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^5);;
gap> SetConjugate(coll, 6, 2, F.6^8);;
gap> SetConjugate(coll, 6, 3, F.6^9);;
gap> SetConjugate(coll, 6, 4, F.6^12);;
gap> SetConjugate(coll, 7, 1, F.6^6*F.7^8);;
gap> SetConjugate(coll, 7, 2, F.7^8);;
gap> SetConjugate(coll, 7, 3, F.6*F.7^3);;
gap> SetConjugate(coll, 7, 4, F.7^12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);