Finite Group DatabasebetaFinite Group Databasebeta| order | 216 |
| isoclinism | isoclinic class of (C3⋊D3)⋊C3 |
| automorphism | Aut(G) = C22×((C3⋊D3)⋊C3)⋊C2 (order 432) · Inn(G) = (C3⋊D3)⋊C3 · Out(G) = C23 |
| homology | Abelianization (H1) = C3×C8 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | (C3⋊(C3⋊C8))⋊C3 |
| derived | C32 ⋊ C3×C8 |
| Sylow | H(3,𝔽3) ⋊ C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | (C3⋊D3)⋊C3 |
| commutator, hypocenter | C32 | C3×C8 |
| Frattini | C3×C4 | C3×D3 |
| Fitting | C4×H(3,𝔽3) | C2 |
| socle | C2×C3 | C3×Dic3 |
| prime radical | C2×H(3,𝔽3) | C4 |
| cosocle | C32×C4 | C2×C3 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C4 | |
34 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 2 generators on 17 points:
(1,2,3,5,4,6,7,8)(10,15,17,12,13,16)(11,14);(9,10,12)(11,13,16)(14,17,15)Polycyclic presentation:
2,3,2^2,3^2|a^2=c,c^2=d,e^a=e^2,f^a=f^2,e^b=ef40 conjugacy classes — view the full character table.
| maximal subgroup of | ((C3⋊D3)⋊C3)⋊C8, #88430, #88434, … |
| maximal normal subgroup of | ((C3⋊D3)⋊C3)⋊C8, #88430, #88434, … |
| maximal quotient of | (C3⋊(C3⋊C16))⋊C3, C2×(C3⋊(C3⋊C8))⋊C3, (C3⋊(C3⋊C8))⋊C9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(216, 14);