Finite Group DatabasebetaFinite Group Databasebeta| order | 400 |
| isoclinism | isoclinic class of #6057 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2×C5 |
No aliases for this group.
| Sylow | C52 ⋊ C4⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | #6057 |
| commutator | C5⋊D5 | C2×C4 |
| Fitting, socle | C2×C52 | Q8 |
| prime radical, cosocle | C2×C5⋊D5 | C22 |
| hypocenter | C52 | C4⋊C4 |
| LCS | G ▷ C5⋊D5 ▷ C52 | |
| UCS | C1 ◁ C2 | |
50 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 20 |
A minimal faithful permutation representation — 2 generators on 14 points:
(1,3,9,4)(2,5,10,6)(11,12,13,14);(1,2)(3,7,8,6)(4,10,9,5)Polycyclic presentation:
2^4,5^2|a^2=c,b^2=cd,b^a=bc,e^a=e^2,f^a=f^3,e^b=f,f^b=e^4,e^c=e^4,f^c=f^416 conjugacy classes — view the full character table.
| maximal subgroup of | #11717306, #11717307, #11717309, … |
| maximal normal subgroup of | #11717306, #11717307, #11717309, … |
| maximal quotient of | #11716802, #11716803, #11716804, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(400, 134);