Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C32 ⋊ #1656 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | #63983 |
| commutator, cosocle | C32×C42 | C23 |
| Frattini | C42 | C2×D32 |
| Fitting | C32×#127 | C22 |
| socle | C22×C32 | #120 |
| hypercenter | #127 | D32 |
| hypocenter | C32 | #1656 |
| LCS | G ▷ C32×C42 ▷ C22×C32 ▷ C32 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #127 | |
532 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 532 conjugacy classes of subgroups — too many to draw. Only its 70 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 192 |
Polycyclic presentation:
2^7,3^2|b^2=f,d^2=f,e^2=fg,b^a=bd,c^a=ce,d^a=df,e^a=efg,d^b=df,e^b=eg,h^b=h^2,i^b=i^2,d^c=dg,e^c=efg,i^c=hi^266 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 14503);