Finite Group DatabasebetaFinite Group Databasebeta| order | 216 |
| isoclinism | isoclinic class of #6258 |
| automorphism | Aut(G) = ((C3⋊S4)⋊C3)⋊C2 (order 432) · Inn(G) = C3⋊S4 · Out(G) = D3 |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | C22×C32 ⋊ D3 |
| derived | C22⋊H(3,𝔽3) ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C3 | C3⋊S4 |
| commutator, cosocle, hypocenter | C22⋊H(3,𝔽3) | C2 |
| Fitting | C22×C32 | D3 |
| socle | C22×C3 | C3⋊D3 |
| LCS | G ▷ C22⋊H(3,𝔽3) | |
| UCS | C1 ◁ C3 | |
57 conjugacy classes of subgroups.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 18 |
A minimal faithful permutation representation — 3 generators on 13 points:
(1,2,4,3)(6,8)(9,13)(10,12);(6,9,10)(8,12,13);(2,3,4)(5,6,8)(7,10,12)(9,13,11)Polycyclic presentation:
2,3^3,2^2|b^a=b^2,c^a=c^2,e^a=f,f^a=e,c^b=cd,e^b=f,f^b=ef19 conjugacy classes — view the full character table.
| maximal subgroup of | ((C3⋊S4)⋊C3)⋊C2, C2×#6258, #10619740, … |
| maximal normal subgroup of | ((C3⋊S4)⋊C3)⋊C2, C2×#6258, #10619740, … |
| maximal quotient of | #88610, #88611, #88613, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(216, 95);