Finite Group DatabasebetaFinite Group Databasebeta| order | 216 |
| isoclinism | isoclinic class of (C22⋊C27)⋊C2 |
| automorphism | Aut(G) = #11757417⋊C2 (order 1944) · Inn(G) = G · Out(G) = C9 |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | (C22⋊C27)⋊C2 |
| derived | C22⋊C27 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C22⋊C27 | C2 |
| Frattini | C9 | S4 |
| Fitting | C22×C9 | D3 |
| socle | C22×C3 | D9 |
| LCS | G ▷ C22⋊C27 | |
| UCS | C1 | |
25 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 54 |
A minimal faithful permutation representation — 2 generators on 31 points:
(1,2,4,3)(6,9)(7,13)(8,16)(10,11)(12,22)(14,15)(17,24)(18,23)(19,29)(20,28)(21,27)(25,26)(30,31);(2,3,4)(5,24,18,25,29,27,7,6,28,8,31,22,14,11,10,15,12,30,16,20,9,13,21,19,26,23,17)Polycyclic presentation:
2,3^3,2^2|b^3=c^2d,c^3=d^2,b^a=b^2c,c^a=c^2d,d^a=d^2,e^a=f,f^a=e,e^b=f,f^b=ef21 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(216, 21);