Finite Group DatabasebetaFinite Group Databasebeta| order | 2304 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C23×C3 |
No aliases for this group.
| Fitting | 2−1+6 ⋊ C3⋊D3 |
| derived | #11923678 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, hypercenter | C2 | A4⋊#1020 |
| commutator, cosocle, hypocenter | #11923678 | C2 |
| Fitting | 2−1+6 | C3⋊D3 |
| LCS | G ▷ #11923678 | |
| UCS | C1 ◁ C2 | |
813 conjugacy classes of subgroups; only the 22 normal subgroups are shown.
This group has 813 conjugacy classes of subgroups — too many to draw. Only its 22 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 — 3 generators on 32 points:
(1,24,19,27,6,26,29,13)(2,31,23,5,8,32,12,17)(3,16,21,30,11,15,9,20)(4,10,18,25,14,22,7,28);(4,5,16)(9,10,26)(12,13,28)(14,17,15)(19,20,32)(21,22,24)(23,27,25)(29,30,31);(2,3,7)(8,11,18)(9,12,19)(10,13,20)(21,23,29)(22,27,30)(24,25,31)(26,28,32)Polycyclic presentation:
2,3^2,2^7|h^2=j,i^2=j,b^a=b^2,c^a=c^2,d^a=e,d^b=e,e^a=d,e^b=de,f^b=g,f^d=fj,f^e=fj,g^a=fg,g^b=fg,g^e=gj,h^a=hj,h^c=i,i^a=hi,i^c=hij,i^h=ij34 conjugacy classes — view the full character table.
| maximal subgroup of | #72243002 |
| maximal normal subgroup of | #72243002 |
| maximal quotient of | #72242983 |
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, 3, 3, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2^2);;
gap> SetConjugate(coll, 3, 1, F.3^2);;
gap> SetConjugate(coll, 4, 1, F.5);;
gap> SetConjugate(coll, 4, 2, F.5);;
gap> SetConjugate(coll, 5, 1, F.4);;
gap> SetConjugate(coll, 5, 2, F.4*F.5);;
gap> SetConjugate(coll, 6, 2, F.7);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.6*F.7);;
gap> SetConjugate(coll, 7, 2, F.6*F.7);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.9);;
gap> SetConjugate(coll, 9, 1, F.8*F.9);;
gap> SetConjugate(coll, 9, 3, F.8*F.9*F.10);;
gap> SetConjugate(coll, 9, 8, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);