Finite Group DatabasebetaFinite Group Databasebeta| order | 2352 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22×C7 |
| direct product | Q8×#64425 |
| Fitting | C72×Q8 ⋊ C2×C3 |
| Hall | Q8×C7⋊D7 ⋊ C3 |
| Hall | #3862 ⋊ C2×Q8 |
| Sylow | C72 ⋊ C2×C3×Q8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×#64425 |
| commutator, socle, cosocle | C2×C72 | C23×C3 |
| Fitting | C72×Q8 | C2×C3 |
| prime radical | C2×#64425 | C22 |
| hypercenter | Q8 | #64425 |
| hypocenter | C72 | C2×C3×Q8 |
| LCS | G ▷ C2×C72 ▷ C72 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
190 conjugacy classes of subgroups.
| transitive degree | μt(G) = 168 |
70 conjugacy classes — view the full character table.
| maximal subgroup of | #41×#64425, SD16×#64425, Q8×ΓL(1,𝔽8)⋊AGL(1,𝔽7) |
| maximal normal subgroup of | #41×#64425, SD16×#64425, Q8×ΓL(1,𝔽8)⋊AGL(1,𝔽7) |
Past the SmallGroups library coverage — no SmallGroups id.