Finite Group DatabasebetaFinite Group Databasebeta| order | 216 |
| isoclinism | isoclinic class of #6290 |
| automorphism | Aut(G) = C22×D3×AGL(2,𝔽3) (order 10368) · Inn(G) = D3×C3⋊D3 · Out(G) = C2×GL(2,𝔽3) |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2×C3 |
No aliases for this group.
| Sylow | C33 ⋊ D4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D3×C3⋊D3 |
| commutator, Fitting, socle, cosocle | C2×C33 | C22 |
| hypocenter | C33 | D4 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C2 | |
136 conjugacy classes of subgroups.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 36 |
A minimal faithful permutation representation — 3 generators on 13 points:
(1,2,3,4)(6,7)(9,10)(11,12,13);(2,4)(8,9,10)(12,13);(5,6,7)Polycyclic presentation:
2^3,3^3|b^2=c,b^a=bc,f^a=f^2,d^b=d^2,e^b=e^233 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#626, #88952, #88959, … |
| maximal normal subgroup of | D3×#626, #88952, #88959, … |
| maximal quotient of | #88790, #88794, #88795, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(216, 128);