Finite Group DatabasebetaFinite Group Databasebeta| order | 4374 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C38 |
No aliases for this group.
| Fitting, derived, Sylow | #14555252 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, hypercenter | C32 | C3⋊(C3⋊(C3⋊(C3⋊D3))) |
| commutator, Fitting, cosocle, hypocenter | #14555252 | C2 |
| LCS | G ▷ #14555252 | |
| UCS | C1 ◁ C32 | |
10867 conjugacy classes of subgroups; only the 2674 normal subgroups are shown.
This group has 10867 conjugacy classes of subgroups — too many to draw. Only its 2674 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 54 |
| transitive degree | μt(G) = 81 |
A minimal faithful permutation representation — 6 generators on 54 points:
(2,5)(3,9)(6,14)(7,18)(8,13)(10,20)(12,15)(16,24)(17,23)(19,21)(22,26)(25,27)(29,32)(30,35)(33,42)(34,41)(36,49)(37,48)(39,44)(40,45)(43,51)(46,50)(47,53)(52,54);(29,34,45)(32,40,41)(33,36,52)(39,50,43)(42,54,49)(44,51,46);(3,10,21)(6,17,16)(9,19,20)(12,22,27)(14,24,23)(15,25,26)(30,37,53)(33,36,52)(35,47,48)(39,43,50)(42,54,49)(44,46,51);(2,7,8)(5,13,18)(6,16,17)(12,22,27)(14,23,24)(15,25,26)(28,30,35)(29,33,44)(31,37,48)(32,39,42)(34,36,51)(38,53,47)(40,50,54)(41,43,49)(45,52,46);(1,3,9)(2,6,15)(4,10,20)(5,12,14)(7,16,25)(8,17,26)(11,21,19)(13,22,23)(18,27,24)(28,29,32)(30,36,50)(31,34,41)(33,43,53)(35,46,49)(37,52,39)(38,45,40)(42,47,51)(44,54,48);(1,2,5)(3,6,12)(4,8,13)(7,18,11)(9,15,14)(10,17,22)(16,27,21)(19,25,24)(20,26,23)Polycyclic presentation:
2,3^7|b^a=b^2,c^a=c^2,d^a=d^2,d^b=dg,d^c=dh,e^a=e^2,e^c=eg^2,e^d=eh^2,f^a=f^2,f^c=fh^2147 conjugacy classes — view the full character table.
| maximal subgroup of | #72277588 |
| maximal normal subgroup of | #72277588 |
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> SetConjugate(coll, 2, 1, F.2^2);;
gap> SetConjugate(coll, 3, 1, F.3^2);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 5, 3, F.5*F.7^2);;
gap> SetConjugate(coll, 5, 4, F.5*F.8^2);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 6, 3, F.6*F.8^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);