Finite Group DatabasebetaFinite Group Databasebeta| order | 352 |
| isoclinism | isoclinic class of #4323 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C11×#41 ⋊ C2 |
| Sylow | C11 ⋊ #135 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | C22⋊D11 |
| commutator, cosocle | C4×C11 | C23 |
| Frattini | C4 | C22×D11 |
| Fitting | C11×#41 | C2 |
| socle | C2×C11 | C2×D4 |
| hypercenter | #41 | D11 |
| hypocenter | C11 | #135 |
| LCS | G ▷ C4×C11 ▷ C2×C11 ▷ C11 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ #41 | |
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)(15,23)(17,24)(19,20)(21,27)(25,26);(12,14,16,22)(13,17,20,26)(15,21,23,27)(18,24,19,25);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,17)(15,18)(16,20)(19,23)(21,24)(22,26)(25,27)Polycyclic presentation:
2^5,11|b^2=e,d^2=e,c^a=cd,d^a=de,f^a=f^10,d^c=de64 conjugacy classes — view the full character table.
| maximal subgroup of | #10622436, #10622437, #10622757, … |
| maximal normal subgroup of | #10622436, #10622437, #10622757, … |
| maximal quotient of | #10622609, #10622611, #10622615, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(352, 145);