Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | #62852 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×GL(2,ℤ/4) |
| commutator, cosocle | C2×#41⋊C3 | C24 |
| Frattini | C2×C4 | C23×S4 |
| Fitting | #62852 | D3 |
| hypercenter | #536 | S4 |
| hypocenter | SL(2,𝔽3) | C2×#121 |
| LCS | G ▷ C2×#41⋊C3 ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C22 ◁ #117 ◁ #536 | |
3482 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 52 points:
(5,37,42)(6,30,52,13,47,45)(7,23,39)(8,9,41)(10,36,40)(11,21,34)(12,43,50,26,32,49)(14,29,51,28,15,46)(16,27,48,25,44,31)(17,18,22)(19,33,38)(20,24,35);(7,17)(8,20)(9,22)(10,19)(12,25)(14,28)(15,31)(16,26)(18,34)(21,38)(23,41)(24,40)(27,45)(29,48)(30,49)(32,51)(33,35)(36,42)(37,39)(43,46)(44,52)(47,50);(2,4)(5,7,11,19)(6,12,13,26)(8,17,20,10)(9,18,24,36)(14,25,28,16)(15,27,29,44)(21,33,37,23)(22,35,40,41)(30,43,47,32)(31,46,48,51)(34,38,42,39)(45,49,52,50);(1,2,3,4)(5,6)(7,12)(8,14)(9,15)(10,16)(11,13)(17,25)(18,27)(19,26)(20,28)(21,30)(22,31)(23,32)(24,29)(33,43)(34,45)(35,46)(36,44)(37,47)(38,49)(39,50)(40,48)(41,51)(42,52)Polycyclic presentation:
2^6,3,2^3|a^2=e,b^2=j,f^2=j,h^2=j,i^2=j,b^a=be,c^b=cf,d^a=dj,f^b=fj,f^c=fj,g^c=g^2,h^c=ij,h^g=i,i^c=hj,i^g=hij,i^h=ij82 conjugacy classes — view the full character table.
| maximal subgroup of | #72258787 |
| maximal normal subgroup of | #72258787 |
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, 2, 2, 2, 3, 2, 2, 2]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 2, F.10);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, F.7^2);;
gap> SetConjugate(coll, 8, 3, F.9*F.10);;
gap> SetConjugate(coll, 8, 7, F.9);;
gap> SetConjugate(coll, 9, 3, F.8*F.10);;
gap> SetConjugate(coll, 9, 7, F.8*F.9*F.10);;
gap> SetConjugate(coll, 9, 8, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);