Finite Group DatabasebetaFinite Group Databasebeta| order | 352 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C11×SD16 ⋊ C2 |
| Sylow | C11 ⋊ #135 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D4×D11 |
| commutator, cosocle | C4×C11 | C23 |
| Frattini | C4 | C22×D11 |
| Fitting | C11×SD16 | C2 |
| socle | C2×C11 | C2×D4 |
| hypercenter | SD16 | D11 |
| hypocenter | C11 | #135 |
| LCS | G ▷ C4×C11 ▷ C2×C11 ▷ C11 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ SD16 | |
62 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)(12,14,16,22)(13,18,17,20)(15,24,23,21)(19,26,25,27);(13,17)(14,21)(15,23)(18,26)(20,27)(22,24);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,20)(15,19)(16,17)(18,22)(21,26)(23,25)(24,27)Polycyclic presentation:
2^5,11|c^2=e,d^2=e,b^a=be,c^a=ce,f^a=f^10,c^b=cd,d^b=de,d^c=de49 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#66969, #10623293, D11×#135, … |
| maximal normal subgroup of | C2×#66969, #10623293, D11×#135, … |
| maximal quotient of | #10622369, #10622376, #10622381, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(352, 111);