Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | (#1020⋊C3)⋊C2 |
| commutator, cosocle | C2×A42 | C22 |
| Fitting, socle | C25 | D32 |
| hypocenter | A42 | D4 |
| LCS | G ▷ C2×A42 ▷ A42 | |
| UCS | C1 ◁ C2 | |
252 conjugacy classes of subgroups; only the 16 normal subgroups are shown.
This group has 252 conjugacy classes of subgroups — too many to draw. Only its 16 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 24 |
Polycyclic presentation:
2^3,3^2,2^4|a^2=c,b^a=bc,d^a=d^2,e^a=e^2,f^a=fg,h^a=hi,d^b=de^2,e^b=e^2,f^b=hi,g^b=i,h^b=fg,i^b=g,f^d=g,g^d=fg,f^e=g,g^e=fg,h^e=i,i^e=hi26 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 157481);