Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C212 |
| direct product | C2×S4×#120 |
| Fitting | C23×#120 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C23 | C23×S4 |
| commutator, cosocle | C22×A4 | C25 |
| Frattini | C22 | C24×S4 |
| Fitting | C23×#120 | D3 |
| socle | C25 | C23×D3 |
| hypercenter | C2×#120 | S4 |
| hypocenter | A4 | C22×#120 |
| LCS | G ▷ C22×A4 ▷ A4 | |
| UCS | C1 ◁ C23 ◁ C2×#120 | |
38823 conjugacy classes of subgroups.
| transitive degree | μt(G) = 48 |
140 conjugacy classes — view the full character table.
| maximal subgroup of | #120×GL(2,ℤ/4), #72259267, C2×#72222918, … |
| maximal normal subgroup of | #120×GL(2,ℤ/4), #72259267, C2×S4×B(4,𝔽2), … |
| maximal quotient of | #120×GL(2,ℤ/4), #72259267, C2×#72222918, … |
Past the SmallGroups library coverage — no SmallGroups id.