Finite Group DatabasebetaFinite Group Databasebeta| order | 352 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C2×Q8×C11 ⋊ C2 |
| Sylow | C11 ⋊ 2−1+4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C23×D11 |
| commutator, socle, cosocle | C2×C11 | C24 |
| Fitting | C2×Q8×C11 | C2 |
| hypercenter | C2×Q8 | D11 |
| hypocenter | C11 | 2−1+4 |
| LCS | G ▷ C2×C11 ▷ C11 | |
| UCS | C1 ◁ C2 ◁ C2×Q8 | |
146 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 176 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,4,6,8,10,11,9,7,5,3)(13,19)(17,25)(18,27)(24,26);(12,15,16,23)(13,18,19,27)(14,20,21,22)(17,24,25,26);(12,14,16,21)(13,17,19,25)(15,22,23,20)(18,26,27,24);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,17)(15,18)(16,19)(20,24)(21,25)(22,26)(23,27)Polycyclic presentation:
2^5,11|b^2=e,c^2=e,d^a=de,f^a=f^10,c^b=ce67 conjugacy classes — view the full character table.
| maximal subgroup of | #10622365, #10622366, #10622675, … |
| maximal normal subgroup of | #10622365, #10622366, #10622675, … |
| maximal quotient of | #10623031, #10623035, #10623038, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(352, 182);