Finite Group DatabasebetaFinite Group Databasebeta| order | 400 |
| isoclinism | isoclinic class of (C2×AGL(1,𝔽5))⋊D5 |
| automorphism | Aut(G) = C22×AGL(1,𝔽5)2 (order 1600) · Inn(G) = D5×AGL(1,𝔽5) · Out(G) = C2×C4 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C52 ⋊ M16 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D5×AGL(1,𝔽5) |
| commutator, Fitting, socle | C2×C52 | C2×C4 |
| prime radical | C2×C5×D5 | C4 |
| cosocle | C5×Dic5 | C22 |
| hypocenter | C52 | M16 |
| LCS | G ▷ C2×C52 ▷ C52 | |
| UCS | C1 ◁ C2 | |
43 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 2 generators on 18 points:
(2,5)(6,8)(10,11)(12,13)(14,15,16,18,17);(1,2,3,6,4,5,7,8)(9,10,12,13,11)(15,16,17,18)Polycyclic presentation:
2^4,5^2|a^2=c,c^2=d,b^a=bd,f^a=f^2,e^b=e^4,f^c=f^434 conjugacy classes — view the full character table.
| maximal subgroup of | #11717327, #11717329, #11717331, … |
| maximal normal subgroup of | #11717327, #11717329, #11717331, … |
| maximal quotient of | #11716752, #11716753, #11716760, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(400, 122);