Finite Group DatabasebetaFinite Group Databasebeta| order | 7938 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C32 |
No aliases for this group.
| Fitting, derived | C32×C72 ⋊ C2×C32 |
| Hall | #72224636 ⋊ C2 |
| Sylow | C72 ⋊ C3×(C3⋊D3)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C32×C72 | C2×C32 |
| Frattini | C3 | #72222210 |
| socle | C3×C72 | C32×D3 |
| LCS | G ▷ C32×C72 | |
| UCS | C1 | |
296 conjugacy classes of subgroups; only the 33 normal subgroups are shown.
This group has 296 conjugacy classes of subgroups — too many to draw. Only its 33 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 63 |
A minimal faithful permutation representation — 2 generators on 23 points:
(1,4,2,7,6,5)(9,12,13,10,11,14)(16,18)(17,20)(19,21)(22,23);(2,3,7)(4,6,5)(8,9,11,14,13,12,10)(15,16,22)(17,19,21)(18,20,23)Polycyclic presentation:
2,3^4,7^2|d^a=d^2,d^b=de,e^a=e^2,f^a=f^6,f^b=g,f^c=f^2,g^a=g^6,g^b=f^5g^3,g^c=g^270 conjugacy classes — view the full character table.
| maximal subgroup of | #72224676, #72230279, C2×#72224671 |
| maximal normal subgroup of | #72224676, #72230279, C2×#72224671 |
| maximal quotient of | C2×#72224671 |
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");;
gap> coll := SingleCollector(F, [2, 3, 3, 3, 3, 7, 7]);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 4, 2, F.4*F.5);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^6);;
gap> SetConjugate(coll, 6, 2, F.7);;
gap> SetConjugate(coll, 6, 3, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.7^6);;
gap> SetConjugate(coll, 7, 2, F.6^5*F.7^3);;
gap> SetConjugate(coll, 7, 3, F.7^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);