Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C8 | D3×S4 |
| commutator, hypocenter | C3×SL(2,𝔽3) | C2×C8 |
| Fitting | C3×#131 | C2×D3 |
| socle | C2×C3 | C8×S4 |
| prime radical | C3×#985 | C4 |
| cosocle | C3×#131⋊C3 | C22 |
| LCS | G ▷ C3×SL(2,𝔽3) | |
| UCS | C1 ◁ C8 | |
179 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 179 conjugacy classes of subgroups — too many to draw. Only its 35 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 67 |
| transitive degree | μt(G) = 192 |
Polycyclic presentation:
2^4,3^2,2^3|a^2=c,c^2=d,d^2=i,g^2=i,h^2=i,b^a=bi,f^a=f^2,e^b=e^2,g^b=gh,h^b=hi,g^e=h,h^e=ghi,h^g=hi84 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 154606);