Finite Group DatabasebetaFinite Group Databasebeta| order | 3888 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2×C3 |
No aliases for this group.
| Fitting, Sylow | H(5,𝔽3) ⋊ C4⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, hypercenter | C3 | #13049783 |
| commutator | #92392 | C2×C4 |
| Fitting, hypocenter | H(5,𝔽3) | C4⋊C4 |
| prime radical, cosocle | #11758087 | C22 |
| LCS | G ▷ #92392 ▷ H(5,𝔽3) | |
| UCS | C1 ◁ C3 | |
250 conjugacy classes of subgroups; only the 18 normal subgroups are shown.
This group has 250 conjugacy classes of subgroups — too many to draw. Only its 18 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 108 |
| transitive degree | μt(G) = 108 |
A minimal faithful permutation representation — 2 generators on 108 points:
(1,74,18,29,31,81,22,28,6,26,75,84)(2,69,45,90,41,89,39,35,12,43,99,65)(3,68,49,54,71,108,60,27,17,62,102,106)(4,21,5,73,23,20,85,80,24,78,30,77)(7,98,37,46,93,53,87,44,36,10,34,100)(8,86,9,50,52,38,82,107,42,96,48,88)(11,33,51,95,25,101,47,56,58,91,57,32)(13,59,14,63,66,19,103,67,61,15,64,104)(16,105,70)(40,55,94,97,92,83)(72,79,76);(1,58,103,7,85,52,3,95,84,2,106,96)(4,47,104,35,22,99,19,69,81,9,16,55)(5,8,17,32,28,12,27,38,6,25,14,36)(10,61,87,72,46,68,50,21,40,49,91,80)(11,64,93,30,42,71,56,29,41,54,86,31)(13,34,79,44,108,107,20,92,60,33,77,98)(15,89,26,82,105,83,23,57,63,65,75,45)(18,39,59,43,74,48,70,97,24,51,67,90)(37,76,100,62,88,78,94,102,101,73,53,66)Polycyclic presentation:
2^4,3^5|a^2=c,b^2=d,b^a=bc,e^a=ef,e^b=fi^2,e^c=e^2i^2,e^d=e^2i^2,f^a=ef^2i,f^b=e^2,f^c=f^2i,f^d=f^2i,f^e=fi,g^a=efgi,g^b=gh,g^c=fgi,g^d=g^2,g^e=gi,h^a=e^2fhi,h^b=gh^2,h^c=e^2f^2hi^2,h^d=h^2,h^e=hi^2,h^f=hi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72228600 |
| maximal normal subgroup of | #72228600 |
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, 3, 3, 3, 3, 3]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 5, 1, F.5*F.6);;
gap> SetConjugate(coll, 5, 2, F.6*F.9^2);;
gap> SetConjugate(coll, 5, 3, F.5^2*F.9^2);;
gap> SetConjugate(coll, 5, 4, F.5^2*F.9^2);;
gap> SetConjugate(coll, 6, 1, F.5*F.6^2*F.9);;
gap> SetConjugate(coll, 6, 2, F.5^2);;
gap> SetConjugate(coll, 6, 3, F.6^2*F.9);;
gap> SetConjugate(coll, 6, 4, F.6^2*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.9);;
gap> SetConjugate(coll, 7, 1, F.5*F.6*F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> SetConjugate(coll, 7, 3, F.6*F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7^2);;
gap> SetConjugate(coll, 7, 5, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.5^2*F.6*F.8*F.9);;
gap> SetConjugate(coll, 8, 2, F.7*F.8^2);;
gap> SetConjugate(coll, 8, 3, F.5^2*F.6^2*F.8*F.9^2);;
gap> SetConjugate(coll, 8, 4, F.8^2);;
gap> SetConjugate(coll, 8, 5, F.8*F.9^2);;
gap> SetConjugate(coll, 8, 6, F.8*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);