Finite Group DatabasebetaFinite Group Databasebeta| order | 216 |
| isoclinism | isoclinic class of #1113 |
| automorphism | Aut(G) = D3×#64241 (order 1728) · Inn(G) = #1113 · Out(G) = C2×D4 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| derived, Sylow | C33 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | #1113 |
| commutator, hypocenter | C33 | C2×C4 |
| Fitting, socle, cosocle | C2×C33 | C22 |
| prime radical | C2×C3×C3⋊D3 | C2 |
| LCS | G ▷ C33 | |
| UCS | C1 ◁ C2 | |
90 conjugacy classes of subgroups.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 13 points:
(1,2,3,4)(5,6)(9,10)(11,12,13);(6,7)(8,9,10)(12,13)Polycyclic presentation:
2^3,3^3|b^2=c,d^a=d^2,f^a=f^2,d^b=d^2,e^b=e^230 conjugacy classes — view the full character table.
| maximal subgroup of | D32⋊Dic3, #88934, #88939, … |
| maximal normal subgroup of | D32⋊Dic3, #88934, #88939, … |
| maximal quotient of | #88808, #88809, #88813, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(216, 131);