Finite Group DatabasebetaFinite Group Databasebeta| order | 408 |
| isoclinism | isoclinic class of #87973 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Hall | C3×C17 ⋊ Q8 |
| Sylow | C17 ⋊ Dic6 |
| Sylow | C3 ⋊ Dic34 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D3×D17 |
| commutator, Fitting, socle, prime radical, cosocle | C2×C3×C17 | C22 |
| hypocenter | C3×C17 | Q8 |
| LCS | G ▷ C2×C3×C17 ▷ C3×C17 | |
| UCS | C1 ◁ C2 | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 408 |
A minimal faithful permutation representation — 2 generators on 28 points:
(1,2,4,6,8,10,12,14,16,17,15,13,11,9,7,5,3)(19,20)(21,23,24,28)(22,25,26,27);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19,20)(21,22,24,26)(23,27,28,25)Polycyclic presentation:
2^3,3,17|a^2=c,b^2=c,b^a=bc,e^a=e^16,d^b=d^257 conjugacy classes — view the full character table.
| maximal subgroup of | Dic6×D17, D3×Dic34, D51⋊Q8, … |
| maximal normal subgroup of | Dic6×D17, D3×Dic34, D51⋊Q8, … |
| maximal quotient of | #11717759, #11717760, Dic3⋊Dic17, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(408, 13);