Finite Group DatabasebetaFinite Group Databasebeta| order | 3200 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | C52×#120 ⋊ C22 |
| Sylow | C52 ⋊ #3025 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #13106387 |
| commutator, cosocle | C23×C52 | C24 |
| Frattini | C23 | C22×D52 |
| Fitting | C52×#120 | C22 |
| socle | C2×C52 | C2×#120 |
| hypercenter | #120 | D52 |
| hypocenter | C52 | #3025 |
| LCS | G ▷ C23×C52 ▷ C2×C52 ▷ C52 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ #120 | |
1862 conjugacy classes of subgroups; only the 172 normal subgroups are shown.
This group has 1862 conjugacy classes of subgroups — too many to draw. Only its 172 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 4 generators on 26 points:
(2,3)(4,5)(6,7)(8,9)(11,14,15,22)(12,17,18,21)(13,19,20,26)(16,24,23,25);(1,2,4,5,3)(16,23)(24,25);(11,13)(12,16)(14,19)(15,20)(17,24)(18,23)(21,25)(22,26);(7,8)(9,10)(11,12,15,18)(13,16,20,23)(14,21,22,17)(19,25,26,24)Polycyclic presentation:
2^7,5^2|a^2=g,d^2=g,c^a=cf,c^b=ce,d^a=dg,e^a=eg,f^b=fg,h^a=h^4,h^d=h^4,i^a=h^3i,i^d=i^4152 conjugacy classes — view the full character table.
| maximal subgroup of | #72226507 |
| maximal normal subgroup of | #72226507 |
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", "i");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 5, 5]);;
gap> SetPower(coll, 1, F.7);;
gap> SetPower(coll, 4, F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 6, 2, F.6*F.7);;
gap> SetConjugate(coll, 8, 1, F.8^4);;
gap> SetConjugate(coll, 8, 4, F.8^4);;
gap> SetConjugate(coll, 9, 1, F.8^3*F.9);;
gap> SetConjugate(coll, 9, 4, F.9^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);