Finite Group DatabasebetaFinite Group Databasebeta| order | 8112 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C2×C4×C132 ⋊ C2×C3 |
| Hall | C13×#6127 ⋊ C3 |
| Hall | #92788 ⋊ #41 |
| Sylow | C132 ⋊ C3×#41 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#11759817 |
| commutator, socle, cosocle | C2×C132 | C23×C3 |
| Frattini | C2 | C22×#11759817 |
| Fitting | C2×C4×C132 | C2×C3 |
| hypercenter | C2×C4 | #11759817 |
| hypocenter | C132 | C3×#41 |
| LCS | G ▷ C2×C132 ▷ C132 | |
| UCS | C1 ◁ C4 ◁ C2×C4 | |
256 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 256 conjugacy classes of subgroups — too many to draw. Only its 63 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 104 |
Polycyclic presentation:
2^3,3,2,13^2|a^2=e,c^2=e,b^a=be,c^a=ce,f^a=f^12,f^d=f^3,g^a=f^8g,g^d=g^3278 conjugacy classes — view the full character table.
| maximal subgroup of | #72259381, #72259483, #72259500, … |
| maximal normal subgroup of | #72259381, #72259483, #72259500, … |
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, 2, 3, 2, 13, 13]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 3, F.5);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^12);;
gap> SetConjugate(coll, 6, 4, F.6^3);;
gap> SetConjugate(coll, 7, 1, F.6^8*F.7);;
gap> SetConjugate(coll, 7, 4, F.7^3);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);