Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×D3×S4 |
| commutator, cosocle, hypocenter | C3×SL(2,𝔽3) | C24 |
| Fitting | C3×H(5,𝔽2) | C2×D3 |
| socle | C2×C3 | C23×S4 |
| hypercenter | Q8 | D3×S4 |
| LCS | G ▷ C3×SL(2,𝔽3) | |
| UCS | C1 ◁ C2 ◁ Q8 | |
1504 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 1504 conjugacy classes of subgroups — too many to draw. Only its 127 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
Polycyclic presentation:
2^4,3^2,2^3|c^2=i,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,d^c=di,g^e=h,h^e=ghi,h^g=hi66 conjugacy classes — view the full character table.
| maximal subgroup of | #72242413, D3×#87493, #72242438, … |
| maximal normal subgroup of | #72242413, D3×#87493, #72242438, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 157613);