Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | Aut(G) = C4×D4×AGL(1,𝔽5) (order 640) · Inn(G) = C22⋊AGL(1,𝔽5) · Out(G) = C2×C4 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C5 ⋊ #350 |
56 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 2 generators on 37 points:
(1,2,3,5,4)(6,8,11,19,12,20,10,25)(7,13,15,28,16,29,24,17)(9,18,22,26,23,33,21,35)(14,27,30,36,31,37,34,32);(2,3,4,5)(6,7,9,14,11,15,22,30,12,16,23,31,10,24,21,34)(8,17,18,32,19,13,26,27,20,28,33,36,25,29,35,37)Polycyclic presentation:
2^6,5|a^2=c,b^2=e,c^2=e,d^2=f,e^2=f,b^a=bd,d^a=df,g^a=g^2,d^b=df,g^c=g^438 conjugacy classes — view the full character table.
| maximal subgroup of | #10603999, #10604000, #10604083, … |
| maximal normal subgroup of | #10603999, #10604000, #10604083, … |
| maximal quotient of | #10598579, #10598580, #10598581, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 236);