Finite Group DatabasebetaFinite Group Databasebeta| order | 216 |
| isoclinism | isoclinic class of #6226 |
| automorphism | Aut(G) = D4×#72210336 (order 23328) · Inn(G) = C2×C3⋊D9 · Out(G) = C22×#272 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C3 |
No aliases for this group.
| Sylow | C3×C9 ⋊ Q8 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×C3⋊D9 |
| commutator, cosocle | C2×C3×C9 | C22 |
| Frattini | C2×C3 | C2×C3⋊D3 |
| Fitting | C3×C4×C9 | C2 |
| socle, prime radical | C2×C32 | C2×D3 |
| hypercenter | C4 | C3⋊D9 |
| hypocenter | C3×C9 | Q8 |
| LCS | G ▷ C2×C3×C9 ▷ C3×C9 | |
| UCS | C1 ◁ C2 ◁ C4 | |
60 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 216 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2,3)(13,15,16,20)(14,17,18,19);(2,3)(5,7)(6,10)(8,9)(11,12)(13,14,16,18)(15,19,20,17);(4,9,11,10,5,7,6,12,8)Polycyclic presentation:
2^3,3^3|a^2=c,b^2=c,d^3=f^2,b^a=bc,d^a=d^2f,e^a=e^2,f^a=f^257 conjugacy classes — view the full character table.
| maximal subgroup of | #88415, #88418, #88424, … |
| maximal normal subgroup of | #88415, #88418, #88424, … |
| maximal quotient of | #88534, C4⋊(C3⋊Dic9), C2×#6226, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(216, 63);