Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×#85468 |
| Fitting | C3×#576 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | D4×S4 |
| commutator, cosocle | #41⋊C3 | C23×C3 |
| Frattini | C4 | C22×C3×S4 |
| Fitting | C3×#576 | D3 |
| hypercenter | C3×Q16 | S4 |
| hypocenter | SL(2,𝔽3) | C2×C3×D4 |
| LCS | G ▷ #41⋊C3 ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C2×C3 ◁ C3×C4 ◁ C3×Q16 | |
531 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 531 conjugacy classes of subgroups — too many to draw. Only its 64 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
93 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#11923452, C3×#11715001, D3×#85468, … |
| maximal normal subgroup of | C3×#11715001, D3×#85468, #72242437 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 156763);