Finite Group DatabasebetaFinite Group Databasebeta| order | 400 |
| isoclinism | isoclinic class of D5×AGL(1,𝔽5) |
| automorphism | Aut(G) = C22×AGL(1,𝔽5)2 (order 1600) · Inn(G) = D5×AGL(1,𝔽5) · Out(G) = C23 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| derived, Sylow | C52 ⋊ C2×C8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D5×AGL(1,𝔽5) |
| commutator, hypocenter | C52 | C2×C8 |
| Fitting, socle | C2×C52 | C2×C4 |
| prime radical | C2×C5⋊D5 | C4 |
| cosocle | C5×Dic5 | C22 |
| LCS | G ▷ C52 | |
| UCS | C1 ◁ C2 | |
49 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 40 |
A minimal faithful permutation representation — 2 generators on 18 points:
(1,3,5,4)(2,6,8,7)(10,11)(12,13)(14,15,16,18,17);(1,2,4,7,5,8,3,6)(9,10,12,13,11)(15,16,17,18)Polycyclic presentation:
2^4,5^2|a^2=c,b^2=d,c^2=d,f^a=f^2,e^b=e^4,f^c=f^440 conjugacy classes — view the full character table.
| maximal subgroup of | #11717300, #11717302, D5×D5⋊C8, … |
| maximal normal subgroup of | #11717300, #11717302, D5×D5⋊C8, … |
| maximal quotient of | #11716747, #11716749, Dic5×#170, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(400, 121);