Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C3 |
No aliases for this group.
| Fitting | C2×C3×2−1+4 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C23×C3⋊S4 |
| commutator, cosocle, hypocenter | C3×SL(2,𝔽3) | C24 |
| Fitting | C2×C3×2−1+4 | D3 |
| socle | C2×C3 | C23×S4 |
| hypercenter | C2×D4 | C3⋊S4 |
| LCS | G ▷ C3×SL(2,𝔽3) | |
| UCS | C1 ◁ C2 ◁ C2×D4 | |
1776 conjugacy classes of subgroups; only the 181 normal subgroups are shown.
This group has 1776 conjugacy classes of subgroups — too many to draw. Only its 181 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:
2^4,3^2,2^3|g^2=i,h^2=i,d^a=di,e^a=e^2,f^a=f^2,g^a=gh,h^a=hi,c^b=ci,g^e=h,h^e=ghi,h^g=hi81 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 157749);