Finite Group DatabasebetaFinite Group Databasebeta| order | 2304 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C24×C3 |
No aliases for this group.
| Fitting, Sylow | #63103 ⋊ C32 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | C22×A42 |
| commutator, cosocle, hypocenter | H(5,𝔽2) | C23×C32 |
| Frattini, socle | C2 | C23×A42 |
| Fitting | #63103 | C32 |
| hypercenter | #41 | A42 |
| LCS | G ▷ H(5,𝔽2) | |
| UCS | C1 ◁ C4 ◁ #41 | |
4560 conjugacy classes of subgroups; only the 177 normal subgroups are shown.
This group has 4560 conjugacy classes of subgroups — too many to draw. Only its 177 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 96 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 96 points:
(1,50,58,53,61,96)(2,70,75,44,65,95)(3,19,82,5,85,57)(4,84,6,49,62,54)(7,47,63,15,27,81)(8,67,78,33,42,90)(9,36,91,11,80,74)(10,79,12,69,77,72)(13,39,94,30,73,89)(14,18,16,20,86,24)(17,22,87,25,55,88)(21,83,26,51,60,56)(23,46,28,48,59,52)(29,66,31,68,76,71)(32,35,34,37,93,40)(38,92,41,43,64,45);(2,8)(5,6,26)(9,13)(10,29)(11,31,41,30,12,40)(15,16,54)(19,20,27)(21,61,22)(24,25,28)(32,43)(33,45,72,44,34,71)(35,64)(36,38,42,39,37,65)(47,48,55)(49,85,50)(52,53,56)(57,58,59)(62,88,63)(66,79)(67,69,73,70,68,80)(74,78,76,89,75,77)(81,82,83)(86,96,87)(90,94,92,95,91,93);(1,2,7,8)(3,13,17,9)(4,10,23,29)(5,11,25,30)(6,12,28,31)(14,43,51,32)(15,44,53,33)(16,45,56,34)(18,35,60,64)(19,36,22,39)(20,37,21,38)(24,40,26,41)(27,42,61,65)(46,79,84,66)(47,70,50,67)(48,69,49,68)(52,72,54,71)(55,80,85,73)(57,74,88,89)(58,75,63,78)(59,76,62,77)(81,95,96,90)(82,94,87,91)(83,93,86,92)Polycyclic presentation:
2^3,3^2,2^5|a^2=j,c^2=j,f^2=j,g^2=j,b^a=bj,c^a=cj,f^d=fg,f^e=fg,g^d=f,g^e=f,g^f=gj,h^d=i,h^e=fhj,h^f=hj,h^g=hj,i^d=hi,i^e=fgij,i^f=ij146 conjugacy classes — view the full character table.
| maximal subgroup of | #72258914 |
| maximal normal subgroup of | #72258914 |
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", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 3, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.10);;
gap> SetPower(coll, 3, F.10);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.7);;
gap> SetConjugate(coll, 6, 5, F.6*F.7);;
gap> SetConjugate(coll, 7, 4, F.6);;
gap> SetConjugate(coll, 7, 5, F.6);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 4, F.9);;
gap> SetConjugate(coll, 8, 5, F.6*F.8*F.10);;
gap> SetConjugate(coll, 8, 6, F.8*F.10);;
gap> SetConjugate(coll, 8, 7, F.8*F.10);;
gap> SetConjugate(coll, 9, 4, F.8*F.9);;
gap> SetConjugate(coll, 9, 5, F.6*F.7*F.9*F.10);;
gap> SetConjugate(coll, 9, 6, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);