Finite Group DatabasebetaFinite Group Databasebeta| order | 216 |
| isoclinism | isoclinic class of #634⋊C3 |
| automorphism | Aut(G) = D4×((C3⋊D3)⋊C3)⋊C2 (order 864) · Inn(G) = C2×(C3⋊D3)⋊C3 · Out(G) = C23 |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2×C3 |
| unambiguous semidirect product | C4⋊((C3⋊D3)⋊C3) |
| unambiguous semidirect product | (C4⋊(C3⋊D3))⋊C3 |
| Fitting | C4×H(3,𝔽3) ⋊ C2 |
| Sylow | H(3,𝔽3) ⋊ D4 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×(C3⋊D3)⋊C3 |
| commutator, cosocle | C2×C32 | C22×C3 |
| Frattini, socle | C2×C3 | C2×C3×D3 |
| Fitting | C4×H(3,𝔽3) | C2 |
| hypercenter | C4 | (C3⋊D3)⋊C3 |
| hypocenter | C32 | C3×D4 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ C4 | |
62 conjugacy classes of subgroups.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 36 |
A minimal faithful permutation representation — 2 generators on 13 points:
(1,2,3,4)(5,6,12)(7,9,11)(8,10,13);(2,4)(6,8)(7,10)(9,11)(12,13)Polycyclic presentation:
2^2,3,2,3^2|b^2=d,b^a=bd,e^a=e^2,f^a=f^2,e^c=ef31 conjugacy classes — view the full character table.
| maximal subgroup of | #88431, #88432, #88436, … |
| maximal normal subgroup of | #88431, #88432, #88436, … |
| maximal quotient of | #3746⋊C3, #3744⋊C3, #3745⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(216, 51);