Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| isoclinism | isoclinic class of Dic10 |
| automorphism | Aut(G) = C22×C8×AGL(1,𝔽5) (order 640) · Inn(G) = C2×D5 · Out(G) = C22×C8 |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting | C5×C32 ⋊ C2 |
| Sylow | C5 ⋊ M64 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C16 | C2×D5 |
| commutator, socle | C2×C5 | C2×C16 |
| Fitting | C5×C32 | C2 |
| prime radical | C2×D5 | C16 |
| cosocle | C5×C16 | C22 |
| hypercenter | C32 | D5 |
| hypocenter | C5 | M64 |
| LCS | G ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C16 ◁ C32 | |
32 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 2 generators on 37 points:
(1,2,4,5,3)(6,7,8,13,9,14,16,25,10,15,17,26,19,27,28,35,11,12,18,22,20,23,29,32,21,24,30,33,31,34,36,37);(2,3)(4,5)(7,12)(13,22)(14,23)(15,24)(25,32)(26,33)(27,34)(35,37)Polycyclic presentation:
2^6,5|b^2=c,c^2=d,d^2=e,e^2=f,b^a=bf,g^a=g^4104 conjugacy classes — view the full character table.
| maximal subgroup of | #10598487, #10598488, #10598491, … |
| maximal normal subgroup of | #10598487, #10598488, #10598491, … |
| maximal quotient of | #10598080, #10598081, #10598085, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 5);