Finite Group DatabasebetaFinite Group Databasebeta| order | 352 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C11×D8 ⋊ C2 |
| Sylow | C11 ⋊ #135 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D4×D11 |
| commutator, cosocle | C4×C11 | C23 |
| Frattini | C4 | C22×D11 |
| Fitting | C11×D8 | C2 |
| socle | C2×C11 | C2×D4 |
| hypercenter | D8 | D11 |
| hypocenter | C11 | #135 |
| LCS | G ▷ C4×C11 ▷ C2×C11 ▷ C11 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ D8 | |
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)(13,18)(14,21)(15,24)(17,25)(20,27)(22,23);(12,14)(13,17)(15,21)(16,23)(18,20)(19,27)(22,24)(25,26);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,20)(15,19)(16,18)(17,23)(21,25)(22,27)(24,26)Polycyclic presentation:
2^5,11|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 | #10622524, #10622525, #10622528, … |
| maximal normal subgroup of | #10622524, #10622525, #10622528, … |
| maximal quotient of | #10622371, #10622379, #10622380, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(352, 107);