Finite Group DatabasebetaFinite Group Databasebeta| order | 2704 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C4×C132 ⋊ C22 |
| Sylow | C132 ⋊ C2×D4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D132 |
| commutator, socle, cosocle | C2×C132 | C23 |
| Fitting | C4×C132 | C22 |
| hypercenter | C4 | D132 |
| hypocenter | C132 | C2×D4 |
| LCS | G ▷ C2×C132 ▷ C132 | |
| UCS | C1 ◁ C2 ◁ C4 | |
156 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 104 |
A minimal faithful permutation representation — 3 generators on 30 points:
(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29);(2,4)(5,6,8,10,12,14,16,17,15,13,11,9,7)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30);(1,2,3,4)Polycyclic presentation:
2^4,13^2|a^2=d,b^a=bd,e^b=e^12,e^c=e^12,f^b=e^2f,f^c=f^12214 conjugacy classes — view the full character table.
| maximal subgroup of | #72229267, #72259209, D4×D132, … |
| maximal normal subgroup of | #72229267, #72259209, D4×D132, … |
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, 2, 2, 13, 13]);;
gap> SetPower(coll, 1, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 5, 2, F.5^12);;
gap> SetConjugate(coll, 5, 3, F.5^12);;
gap> SetConjugate(coll, 6, 2, F.5^2*F.6);;
gap> SetConjugate(coll, 6, 3, F.6^12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);