Finite Group DatabasebetaFinite Group Databasebeta| order | 216 |
| isoclinism | isoclinic class of #6196 |
| automorphism | Aut(G) = #11722315 (order 864) · Inn(G) = ((C3⋊D3)⋊C3)⋊C2 · Out(G) = D4 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | H(3,𝔽3) ⋊ Q8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | ((C3⋊D3)⋊C3)⋊C2 |
| commutator, Fitting, prime radical, cosocle | C2×H(3,𝔽3) | C22 |
| Frattini, socle | C2×C3 | D32 |
| hypocenter | H(3,𝔽3) | Q8 |
| LCS | G ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C2 | |
48 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 2 generators on 17 points:
(2,8,6,4,5,9)(3,7)(10,12,13,17)(11,14,15,16);(1,2,4)(3,5,8,7,6,9)(10,11,13,15)(12,16,17,14)Polycyclic presentation:
2^3,3^3|a^2=c,b^2=c,b^a=bc,e^a=e^2,f^a=f^2,d^b=d^2,f^b=f^2,e^d=ef19 conjugacy classes — view the full character table.
| maximal subgroup of | #88587, #88589, ((C3⋊D3)⋊C3)⋊Q8, … |
| maximal normal subgroup of | #88587, #88589, ((C3⋊D3)⋊C3)⋊Q8, … |
| maximal quotient of | #88448, #88449, C2×#6196, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(216, 33);