Finite Group DatabasebetaFinite Group Databasebeta| order | 3600 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| derived, Hall | C32×C52 ⋊ C22×C4 |
| Sylow | C52 ⋊ C2×#624 |
| Sylow | C32 ⋊ C2×#6036 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | #14268487 |
| commutator, hypocenter | C32×C52 | C22×C4 |
| Fitting, socle, cosocle | C2×C32×C52 | C23 |
| prime radical | C2×C3⋊D3×C5⋊D5 | C2 |
| LCS | G ▷ C32×C52 | |
| UCS | C1 ◁ C2 | |
688 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 688 conjugacy classes of subgroups — too many to draw. Only its 91 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) = 60 |
A minimal faithful permutation representation — 3 generators on 20 points:
(6,7)(8,9)(10,11)(12,13)(15,16,17)(18,19,20);(5,6,8,9,7)(16,17)(19,20);(1,2,3,4)(11,12)(13,14)(19,20)Polycyclic presentation:
2^4,3^2,5^2|a^2=d,e^a=e^2,e^b=e^2,f^a=e^2f,f^b=f^2,g^a=g^4,g^c=g^4,h^a=g^3h,h^c=h^4156 conjugacy classes — view the full character table.
| maximal subgroup of | #72224612, #72224623, #72224627, … |
| maximal normal subgroup of | #72224612, #72224623, #72224627, … |
| maximal quotient of | #72259958, #72260100, C2×#72224595 |
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, 5, 1, F.5^2);;
gap> SetConjugate(coll, 5, 2, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.5^2*F.6);;
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);