Finite Group DatabasebetaFinite Group Databasebeta| order | 4374 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C33 · Schur multiplier (H2) = C34 |
No aliases for this group.
| Fitting, Sylow | #14555438 ⋊ C2 |
3964 conjugacy classes of subgroups; only the 225 normal subgroups are shown.
This group has 3964 conjugacy classes of subgroups — too many to draw. Only its 225 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 81 |
A minimal faithful permutation representation — 3 generators on 81 points:
(1,3,12)(2,8,26)(4,38,34,15,14,60)(5,13,35)(6,20,43)(7,23,47)(9,53,52,28,27,72)(10,11,31)(16,63,59,41,39,76)(17,62,61,40,37,77)(18,36,33)(19,30,55)(21,66,65,45,44,78)(22,25,50)(24,69,68,49,48,79)(29,58,57,54,32,75)(42,74,73,64,56,81)(46,71,70,67,51,80);(1,6,22,18,10,7,5,2,19)(3,8,23,36,20,30,13,11,25)(4,29,42,16,9,46,17,21,24)(12,43,50,33,31,47,35,26,55)(14,44,51,39,32,48,37,27,56)(15,28,49,41,45,64,40,54,67)(34,57,73,59,52,70,61,65,68)(38,58,74,63,53,71,62,66,69)(60,72,79,76,78,81,77,75,80);(1,4,15)(2,21,54)(3,14,38)(5,17,40)(6,29,28)(7,46,64)(8,44,58)(9,45,10)(11,27,66)(12,34,60)(13,37,62)(16,41,18)(19,24,67)(20,32,53)(22,42,49)(23,51,74)(25,56,69)(26,65,75)(30,48,71)(31,52,78)(33,59,76)(35,61,77)(36,39,63)(43,57,72)(47,70,81)(50,73,79)(55,68,80)Polycyclic presentation:
2,3^7|c^3=h^2,c^b=ch^2,d^c=de,e^d=eh^2,f^a=f^2,g^a=g^2,g^f=gh202 conjugacy classes — view the full character table.
| maximal subgroup of | #72243741, #72243831, #72243878, … |
| maximal normal subgroup of | #72243741, #72243831, #72243878, … |
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, 3, 3, 3, 3, 3, 3, 3]);;
gap> SetPower(coll, 3, F.8^2);;
gap> SetConjugate(coll, 3, 2, F.3*F.8^2);;
gap> SetConjugate(coll, 4, 3, F.4*F.5);;
gap> SetConjugate(coll, 5, 4, F.5*F.8^2);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 7, 6, F.7*F.8);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);