Finite Group DatabasebetaFinite Group Databasebeta| order | 3600 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23×C3 |
No aliases for this group.
| Hall | C32×C52 ⋊ C2×D4 |
| Sylow | C52 ⋊ C2×C4⋊(C3⋊D3) |
| Sylow | C32 ⋊ #87886 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D52×C3⋊D3 |
| commutator, Fitting, socle, cosocle | C2×C32×C52 | C23 |
| hypocenter | C32×C52 | C2×D4 |
| LCS | G ▷ C2×C32×C52 ▷ C32×C52 | |
| UCS | C1 ◁ C2 | |
840 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 840 conjugacy classes of subgroups — too many to draw. Only its 114 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 180 |
A minimal faithful permutation representation — 3 generators on 20 points:
(6,7)(8,9)(10,11)(12,13)(18,19,20);(2,4)(5,6,8,9,7)(16,17)(19,20);(1,2,3,4)(11,12)(13,14)(15,16,17)Polycyclic presentation:
2^4,3^2,5^2|a^2=d,b^a=bd,e^b=e^2,f^b=f^2,g^a=g^4,g^c=g^4,h^a=g^3h,h^c=h^4174 conjugacy classes — view the full character table.
| maximal subgroup of | C3⋊D3×#87886, #72231474, #72231476, … |
| maximal normal subgroup of | C3⋊D3×#87886, #72231474, #72231476, … |
| maximal quotient of | C2×#72229316 |
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", "h");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 3, 5, 5]);;
gap> SetPower(coll, 1, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 5, 2, F.5^2);;
gap> SetConjugate(coll, 6, 2, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.7^4);;
gap> SetConjugate(coll, 7, 3, F.7^4);;
gap> SetConjugate(coll, 8, 1, F.7^3*F.8);;
gap> SetConjugate(coll, 8, 3, F.8^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);