Finite Group DatabasebetaFinite Group Databasebeta| order | 216 |
| isoclinism | isoclinic class of D32 |
| automorphism | Aut(G) = C2×#64241 (order 576) · Inn(G) = D32 · Out(G) = C2×D4 |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2 |
| direct product | C3×#624 |
| derived | C32 ⋊ C2×C3×C4 |
| Sylow | C33 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | D32 |
| commutator, hypocenter | C32 | C2×C3×C4 |
| Frattini | C2 | C3×D32 |
| Fitting, socle | C2×C33 | C22 |
| prime radical | C2×C3×C3⋊D3 | C2 |
| cosocle | C2×C32 | C22×C3 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C2×C3 | |
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:
(1,2,4,3)(5,6,7)(9,10)(11,12,13);(1,3,4,2)(8,9,10)(12,13)54 conjugacy classes — view the full character table.
| maximal subgroup of | C3×D32⋊C4, C3×#3773, #88932, … |
| maximal normal subgroup of | C3×D32⋊C4, C3×#3773, #88932, … |
| maximal quotient of | C3×#3710, C3×#3712, C3×Dic32, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(216, 120);