Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| Fitting | #60398 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×#1020⋊C2 |
| commutator, cosocle, hypocenter | #997 | C24 |
| Frattini | C23 | C23×S4 |
| Fitting | #60398 | D3 |
| hypercenter | C2×C4 | #1020⋊C2 |
| LCS | G ▷ #997 | |
| UCS | C1 ◁ C2 ◁ C2×C4 | |
3278 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 3278 conjugacy classes of subgroups — too many to draw. Only its 105 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,25,17,11,14,27,6,12,15,3,4,28)(2,29,24,18,19,31,7,21,23,8,9,32)(5,13,16,26)(10,22,20,30);(2,7)(4,5)(8,18)(9,20)(10,19)(12,13)(14,16)(21,30)(22,29)(23,24)(25,26)(31,32);(1,2,6,7)(3,8,11,18)(4,9,14,19)(5,10,16,20)(12,21,25,29)(13,22,26,30)(15,23,17,24)(27,31,28,32);(3,11)(4,14)(5,16)(8,18)(9,19)(10,20)(15,17)(23,24)Polycyclic presentation:
2^4,3,2^5|b^2=j,d^2=j,f^2=j,g^2=j,c^b=cj,d^a=dj,e^c=e^2,f^a=fhij,f^c=gh,f^e=g,g^a=ghj,g^c=fhi,g^e=fgj,g^f=gj,h^c=hi,h^e=i,h^f=hj,i^e=hi,i^f=ij,i^g=ij59 conjugacy classes — view the full character table.
| maximal subgroup of | #72344080 |
| maximal normal subgroup of | #72344080 |
| maximal quotient of | #72344091, #72344121 |
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, 3, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.10);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 5, 3, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 3, F.7*F.8);;
gap> SetConjugate(coll, 6, 5, F.7);;
gap> SetConjugate(coll, 7, 1, F.7*F.8*F.10);;
gap> SetConjugate(coll, 7, 3, F.6*F.8*F.9);;
gap> SetConjugate(coll, 7, 5, F.6*F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
gap> SetConjugate(coll, 8, 5, F.9);;
gap> SetConjugate(coll, 8, 6, F.8*F.10);;
gap> SetConjugate(coll, 9, 5, F.8*F.9);;
gap> SetConjugate(coll, 9, 6, F.9*F.10);;
gap> SetConjugate(coll, 9, 7, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);