Finite Group DatabasebetaFinite Group Databasebeta| order | 1200 |
| isoclinism | isoclinic class of A4⋊#1042 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | A4⋊#1042 |
| Fitting, socle | C22×C52 ⋊ Dic3 |
| derived | C52×A4 ⋊ C4 |
| Sylow | C52 ⋊ SL(2,ℤ/4) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, hypocenter | C52×A4 | C4 |
| Fitting, socle | C22×C52 | Dic3 |
| prime radical, cosocle | C5×D5×A4 | C2 |
| LCS | G ▷ C52×A4 | |
| UCS | C1 | |
81 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 80 |
Polycyclic presentation:
2^2,3,5^2,2^2|a^2=b,c^a=c^2,d^a=d^2,e^a=e^4,f^a=fg,d^b=d^4,f^c=g,g^c=fg46 conjugacy classes — view the full character table.
| maximal subgroup of | #72222151, D5×A4⋊AGL(1,𝔽5), AGL(1,𝔽5)×C5⋊S4, … |
| maximal normal subgroup of | D5×A4⋊AGL(1,𝔽5), AGL(1,𝔽5)×C5⋊S4, S4×#1042 |
| maximal quotient of | #72222151 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1200, 972);