Finite Group DatabasebetaFinite Group Databasebeta| order | 216 |
| isoclinism | isoclinic class of #1088 |
| automorphism | Aut(G) = C2×AΓL(1,𝔽9) (order 288) · Inn(G) = #157 · Out(G) = C23 |
| homology | Abelianization (H1) = C8 · Schur multiplier (H2) = C1 |
No aliases for this group.
| derived, Sylow | H(3,𝔽3) ⋊ C8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, hypercenter | C2×C3 | #157 |
| commutator, hypocenter | H(3,𝔽3) | C8 |
| Fitting, prime radical | C2×H(3,𝔽3) | C4 |
| cosocle | #1084 | C2 |
| LCS | G ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C2×C3 | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 2 generators on 26 points:
(1,2)(3,7,9,14)(4,11)(5,8)(6,12,13,18)(10,15,16,17)(19,20,21,23,22,24,25,26);(1,3,16)(2,15,6)(4,12,9)(5,10,18)(7,17,11)(8,13,14)Polycyclic presentation:
2^3,3^3|a^2=b,b^2=c,d^a=de^2,e^a=d^2e^2,d^b=d^2,e^b=e^2f,e^d=ef28 conjugacy classes — view the full character table.
| derived subgroup of | #13146171, #13146172, #13146173, … |
| maximal subgroup of | #88586, #88587, #88588, … |
| maximal normal subgroup of | #88586, #88587, #88588, … |
| maximal quotient of | #88410, C2×#6188, C3×#6188, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(216, 25);