Finite Group DatabasebetaFinite Group Databasebeta| order | 2304 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C23×C3 |
No aliases for this group.
| derived | #10596782 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | #11923851 |
| commutator, hypocenter | #10596782 | C4 |
| Fitting, socle | C27 | C3⋊D3 |
| prime radical, cosocle | C2×#10596782 | C2 |
| LCS | G ▷ #10596782 | |
| UCS | C1 ◁ C2 | |
2778 conjugacy classes of subgroups; only the 33 normal subgroups are shown.
This group has 2778 conjugacy classes of subgroups — too many to draw. Only its 33 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 36 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2,3,4)(5,6,7,8)(9,11,12,10)(13,14,15,16);(10,11,12)(14,15,16);(6,8,7)(10,12,11)(14,15,16)Polycyclic presentation:
2^2,3^2,2^6|a^2=b,c^a=c^2,d^a=d^2,e^a=gh,e^c=egh,e^d=f,f^a=h,f^c=fg,f^d=ef,g^a=ef,g^c=f,g^d=h,h^a=f,h^c=ef,h^d=gh,i^a=ij,i^c=j,j^c=ij48 conjugacy classes — view the full character table.
| maximal subgroup of | #72247788, #72248184, #72248196, … |
| maximal normal subgroup of | #72248184, #72248196, C4×#11923851 |
| maximal quotient of | #72247788 |
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, 3, 3, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.2);;
gap> SetConjugate(coll, 3, 1, F.3^2);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.7*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.8);;
gap> SetConjugate(coll, 5, 4, F.6);;
gap> SetConjugate(coll, 6, 1, F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.7);;
gap> SetConjugate(coll, 6, 4, F.5*F.6);;
gap> SetConjugate(coll, 7, 1, F.5*F.6);;
gap> SetConjugate(coll, 7, 3, F.6);;
gap> SetConjugate(coll, 7, 4, F.8);;
gap> SetConjugate(coll, 8, 1, F.6);;
gap> SetConjugate(coll, 8, 3, F.5*F.6);;
gap> SetConjugate(coll, 8, 4, F.7*F.8);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.10);;
gap> SetConjugate(coll, 10, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);