Finite Group DatabasebetaFinite Group Databasebeta| order | 216 |
| isoclinism | isoclinic class of #6295 |
| automorphism | Aut(G) = #72276300 (order 5184) · Inn(G) = #1113 · Out(G) = C2×S4 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C33 ⋊ Q8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | #1113 |
| commutator, Fitting, socle, prime radical, cosocle | C2×C33 | C22 |
| hypocenter | C33 | Q8 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C2 | |
70 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 17 points:
(1,3,4,8)(2,5,6,7)(9,10)(13,14)(15,16,17);(1,2,4,6)(3,7,8,5)(10,11)(12,13,14)(16,17)Polycyclic presentation:
2^3,3^3|a^2=c,b^2=c,b^a=bc,d^a=d^2,f^a=f^2,d^b=d^2,e^b=e^227 conjugacy classes — view the full character table.
| derived subgroup of | SL(2,ℤ/9), #88958⋊C3, #88963⋊C3, … |
| maximal subgroup of | #88937, #88938, #88942, … |
| maximal normal subgroup of | #88937, #88938, #88942, … |
| maximal quotient of | #88815, C2×#6296, #10619895, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(216, 133);