Finite Group DatabasebetaFinite Group Databasebeta| order | 352 |
| isoclinism | isoclinic class of Dic22 |
| automorphism | Aut(G) = C22×C4×AGL(1,𝔽11) (order 1760) · Inn(G) = C2×D11 · Out(G) = C2×C4×C5 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting | C11×C16 ⋊ C2 |
| Sylow | C11 ⋊ M32 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C8 | C2×D11 |
| commutator, socle | C2×C11 | C2×C8 |
| Fitting | C11×C16 | C2 |
| prime radical | C2×D11 | C8 |
| cosocle | C8×C11 | C22 |
| hypercenter | C16 | D11 |
| hypocenter | C11 | M32 |
| LCS | G ▷ C2×C11 ▷ C11 | |
| UCS | C1 ◁ C8 ◁ C16 | |
26 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 176 |
A minimal faithful permutation representation — 2 generators on 27 points:
(1,2,4,6,8,10,11,9,7,5,3)(12,13,14,18,15,19,20,25,16,17,21,23,22,24,26,27);(2,3)(4,5)(6,7)(8,9)(10,11)(13,17)(18,23)(19,24)(25,27)Polycyclic presentation:
2^5,11|b^2=c,c^2=d,d^2=e,b^a=be,f^a=f^10100 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#66862, #10622514, D11×M32, … |
| maximal normal subgroup of | C2×#66862, #10622514, D11×M32, … |
| maximal quotient of | Dic11⋊C16, #10622140, (C2×D11)⋊C16, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(352, 4);