Finite Group DatabasebetaFinite Group Databasebeta| order | 4056 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Hall | #13062578 ⋊ C3 |
| Hall | #92788 ⋊ D4 |
| Sylow | C132 ⋊ C3×D4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | #11759815⋊C2 |
| commutator, Fitting, socle, cosocle | C2×C132 | C22×C3 |
| hypocenter | C132 | C3×D4 |
| LCS | G ▷ C2×C132 ▷ C132 | |
| UCS | C1 ◁ C2 | |
100 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 104 |
A minimal faithful permutation representation — 2 generators on 30 points:
(2,4)(5,6,12)(7,14,17)(8,10,13)(9,16,11)(18,19,28,27,29,25)(20,26,22,21,24,30);(1,2,3,4)(6,7)(8,10)(9,13)(11,12)(14,15)(16,17)(19,20)(21,23)(22,24)(25,26)(27,28)(29,30)Polycyclic presentation:
2^2,3,2,13^2|a^2=d,b^a=bd,e^a=e^12,e^b=e^12,e^c=e^9,f^a=f^12,f^b=e^2f,f^c=f^955 conjugacy classes — view the full character table.
| maximal subgroup of | #72229267, #72259367, #72259375, … |
| maximal normal subgroup of | #72229267, #72259367, #72259375, … |
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, 2, 3, 2, 13, 13]);;
gap> SetPower(coll, 1, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^12);;
gap> SetConjugate(coll, 5, 2, F.5^12);;
gap> SetConjugate(coll, 5, 3, F.5^9);;
gap> SetConjugate(coll, 6, 1, F.6^12);;
gap> SetConjugate(coll, 6, 2, F.5^2*F.6);;
gap> SetConjugate(coll, 6, 3, F.6^9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);