Finite Group DatabasebetaFinite Group Databasebeta| order | 1200 |
| isoclinism | isoclinic class of C5⋊(C5⋊S4) |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C2×C5 |
| unambiguous semidirect product | A4⋊#1039 |
| derived | C52×A4 ⋊ C4 |
| Sylow | C52 ⋊ SL(2,ℤ/4) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | C5⋊(C5⋊S4) |
| commutator, hypocenter | C52×A4 | C4 |
| Fitting, socle | C23×C52 | D3 |
| prime radical, cosocle | C2×C52×A4 | C2 |
| LCS | G ▷ C52×A4 | |
| UCS | C1 ◁ C2 | |
152 conjugacy classes of subgroups.
| transitive degree | μt(G) = 300 |
Polycyclic presentation:
2^2,3,5^2,2^2|a^2=b,c^a=c^2,d^a=d^4,e^a=e^4,f^a=fg,f^c=g,g^c=fg106 conjugacy classes — view the full character table.
| maximal subgroup of | #72230064, #72230228, SL(2,ℤ/4)×C5⋊D5, … |
| maximal normal subgroup of | #72230064, #72230228, SL(2,ℤ/4)×C5⋊D5, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1200, 679);