Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C23×#127 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #11715690 |
| commutator, cosocle | C2×C22⋊(C42⋊C3) | C22 |
| Frattini | C23 | C2×#1020 |
| Fitting | C23×#127 | D3 |
| socle | C25 | C2×S4 |
| hypocenter | C22⋊(C42⋊C3) | D4 |
| LCS | G ▷ C2×C22⋊(C42⋊C3) ▷ C22⋊(C42⋊C3) | |
| UCS | C1 ◁ C2 | |
3358 conjugacy classes of subgroups; only the 27 normal subgroups are shown.
This group has 3358 conjugacy classes of subgroups — too many to draw. Only its 27 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) = 24 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,2,4,3)(8,16,14,15,11,12,10,13);(2,3,4)(5,8,13)(6,14,12,9,10,16)(7,11,15)Polycyclic presentation:
2^3,3,2^6|b^2=c,e^2=i,f^2=j,b^a=bc,d^b=d^2,e^a=ei,e^b=fij,e^d=fi,f^a=fj,f^b=eij,f^d=ef,g^b=h,g^d=h,h^b=g,h^d=gh,i^b=j,i^d=j,j^b=i,j^d=ij48 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72231237, #72260915, #72261808, … |
| maximal normal subgroup of | C2×#72231237, #72262228, #72262234, … |
| maximal quotient of | #72231239, C2×#72231237, #72234243, … |
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, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 6, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 2, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 2, F.6*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.6*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 2, F.5*F.9*F.10);;
gap> SetConjugate(coll, 6, 4, F.5*F.6);;
gap> SetConjugate(coll, 7, 2, F.8);;
gap> SetConjugate(coll, 7, 4, F.8);;
gap> SetConjugate(coll, 8, 2, F.7);;
gap> SetConjugate(coll, 8, 4, F.7*F.8);;
gap> SetConjugate(coll, 9, 2, F.10);;
gap> SetConjugate(coll, 9, 4, F.10);;
gap> SetConjugate(coll, 10, 2, F.9);;
gap> SetConjugate(coll, 10, 4, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);