Finite Group DatabasebetaFinite Group Databasebeta| order | 352 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C11 ⋊ #126 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×D11 |
| commutator, socle, cosocle | C22×C11 | C23 |
| Fitting | C11×C22⋊C4 | C2 |
| prime radical | C23×C11 | C22 |
| hypercenter | C22⋊C4 | D11 |
| hypocenter | C11 | #126 |
| LCS | G ▷ C22×C11 ▷ C11 | |
| UCS | C1 ◁ C22 ◁ C22⋊C4 | |
60 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 176 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,4,6,8,10,11,9,7,5,3)(13,17)(16,19)(22,26)(24,27);(12,14,15,18)(13,16,17,19)(20,22,23,26)(21,24,25,27);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,16)(15,17)(18,19)(20,21,23,25)(22,24,26,27)Polycyclic presentation:
2^5,11|a^2=e,b^2=d,c^a=cde,f^a=f^10,c^b=ce64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#66932, #10623025, #10623026, … |
| maximal normal subgroup of | C2×#66932, #10623025, #10623026, … |
| maximal quotient of | #10622264, #10622265, #10622266, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(352, 74);