Finite Group DatabasebetaFinite Group Databasebeta| order | 352 |
| isoclinism | isoclinic class of Dic22 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C2×C8×C11 ⋊ C2 |
| Sylow | C11 ⋊ #131 |
| subgroup | G/H | |
|---|---|---|
| center | C8 | C2×D11 |
| commutator, socle | C2×C11 | C22×C4 |
| Frattini | C4 | C22×D11 |
| Fitting | C2×C8×C11 | C2 |
| prime radical | #4339 | C2 |
| cosocle | C4×C11 | C23 |
| hypercenter | C2×C8 | D11 |
| hypocenter | C11 | #131 |
| LCS | G ▷ C2×C11 ▷ C11 | |
| UCS | C1 ◁ C8 ◁ C2×C8 | |
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)(12,14,15,20,16,21,22,26)(13,18,19,25,17,23,24,27);(13,17)(18,23)(19,24)(25,27);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,18)(15,19)(16,17)(20,25)(21,23)(22,24)(26,27)Polycyclic presentation:
2^5,11|c^2=d,d^2=e,b^a=be,f^a=f^10100 conjugacy classes — view the full character table.
| maximal subgroup of | #10622146, #10622153, #10622313, … |
| maximal normal subgroup of | #10622146, #10622153, #10622313, … |
| maximal quotient of | C8×Dic22, #10622292, C8×D44, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(352, 96);