Finite Group DatabasebetaFinite Group Databasebeta| order | 29400 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting, socle, derived, Hall | C52×C72 ⋊ C23×C3 |
| Hall | #72225761 ⋊ C3 |
| Hall | C52×#3861 ⋊ C23 |
| Hall | #3861 ⋊ C2×D52 |
| Sylow | C72 ⋊ C2×C3×D52 |
| Sylow | C52 ⋊ C2×#64423⋊C2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C52×C72 | C23×C3 |
| LCS | G ▷ C52×C72 | |
| UCS | C1 | |
832 conjugacy classes of subgroups; only the 81 normal subgroups are shown.
This group has 832 conjugacy classes of subgroups — too many to draw. Only its 81 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 70 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,7)(4,5,6)(8,9,14)(11,12,13)(16,17)(18,19)(20,21)(22,23);(2,3)(4,5)(6,7)(9,10)(11,12)(13,14)(15,16,18,19,17);(9,10)(11,12)(13,14)(21,22)(23,24)Polycyclic presentation:
2^3,3,5^2,7^2|e^a=e^4,e^c=e^4,f^a=e^3f,f^c=f^4,g^a=g^6,g^b=g^6,g^d=g^2,h^a=g^2h,h^b=h^6,h^d=h^2170 conjugacy classes — view the full character table.
| maximal subgroup of | #72225770 |
| maximal normal subgroup of | #72225770 |
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, 3, 5, 5, 7, 7]);;
gap> SetConjugate(coll, 5, 1, F.5^4);;
gap> SetConjugate(coll, 5, 3, F.5^4);;
gap> SetConjugate(coll, 6, 1, F.5^3*F.6);;
gap> SetConjugate(coll, 6, 3, F.6^4);;
gap> SetConjugate(coll, 7, 1, F.7^6);;
gap> SetConjugate(coll, 7, 2, F.7^6);;
gap> SetConjugate(coll, 7, 4, F.7^2);;
gap> SetConjugate(coll, 8, 1, F.7^2*F.8);;
gap> SetConjugate(coll, 8, 2, F.8^6);;
gap> SetConjugate(coll, 8, 4, F.8^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);