Finite Group DatabasebetaFinite Group Databasebeta| order | 352 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C11×C22⋊C4 ⋊ C2 |
| Sylow | C11 ⋊ #123 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×D11 |
| commutator, socle, cosocle | C22×C11 | C23 |
| Fitting | C11×C22⋊C4 | C2 |
| prime radical | C2×C22⋊D11 | C2 |
| hypercenter | C22⋊C4 | D11 |
| hypocenter | C11 | #123 |
| LCS | G ▷ C22×C11 ▷ C11 | |
| UCS | C1 ◁ C22 ◁ C22⋊C4 | |
78 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 176 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,4,6,8,10,11,9,7,5,3)(13,15)(17,21)(20,23);(12,13)(14,15)(16,18,19,22)(17,20,21,23);(2,3)(4,5)(6,7)(8,9)(10,11)(16,17)(18,20)(19,21)(22,23)Polycyclic presentation:
2^5,11|b^2=d,c^a=cd,f^a=f^10,c^b=ce64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#66936, #10623021, #10623025, … |
| maximal normal subgroup of | C2×#66936, #10623021, #10623025, … |
| maximal quotient of | #10622265, #10622267, #10622270, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(352, 78);