Finite Group DatabasebetaFinite Group Databasebeta| order | 216 |
| isoclinism | isoclinic class of #6229 |
| automorphism | Aut(G) = C22×AGL(2,𝔽3) (order 1728) · Inn(G) = C2×C3⋊D3 · Out(G) = GL(2,𝔽3) |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | C22×H(3,𝔽3) ⋊ C2 |
| Sylow | H(3,𝔽3) ⋊ D4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C2×C3 | C2×C3⋊D3 |
| commutator, cosocle | C2×H(3,𝔽3) | C22 |
| Fitting | C22×H(3,𝔽3) | C2 |
| hypercenter | C22×C3 | C3⋊D3 |
| hypocenter | H(3,𝔽3) | D4 |
| LCS | G ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3 | |
88 conjugacy classes of subgroups.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 36 |
A minimal faithful permutation representation — 3 generators on 13 points:
(2,5,6)(4,8,9)(11,13);(2,4)(5,9)(6,8)(10,11)(12,13);(1,2,4)(3,6,8)(5,9,7)Polycyclic presentation:
2^3,3^3|b^a=bc,d^a=d^2,e^a=e^2,e^d=ef31 conjugacy classes — view the full character table.
| maximal subgroup of | #88671, #88676, #88740, … |
| maximal normal subgroup of | #88671, #88676, #88740, … |
| maximal quotient of | #88540, #88542, #88545, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(216, 72);