Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Sylow | C32 ⋊ #3006 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×GO+(4,𝔽2) |
| commutator, cosocle | C2×C3⋊Dic3 | C24 |
| Fitting | C24×C32 | D4 |
| socle | C22×C32 | C22×D4 |
| hypercenter | C24 | GO+(4,𝔽2) |
| hypocenter | C32 | #3006 |
| LCS | G ▷ C2×C3⋊Dic3 ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C22 ◁ C24 | |
1464 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 1464 conjugacy classes of subgroups — too many to draw. Only its 123 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2^7,3^2|d^2=g,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,i^a=hi^2,e^b=eg,h^b=hi,i^b=i^2,h^d=h^2,i^d=i^2,h^e=h^2,i^e=i^272 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 98179);