Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| isoclinism | isoclinic class of #10596770⋊C2 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #10596770⋊C2 |
| unambiguous semidirect product | S42⋊C2 |
| unambiguous semidirect product | ((#1020⋊C3)⋊C2)⋊C2 |
| Fitting, socle | C24 ⋊ GO+(4,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle | A4⋊S4 | C22 |
| Fitting, socle | C24 | GO+(4,𝔽2) |
| hypocenter | A42 | D4 |
| LCS | G ▷ A4⋊S4 ▷ A42 | |
| UCS | C1 | |
221 conjugacy classes of subgroups; only the 8 normal subgroups are shown.
This group has 221 conjugacy classes of subgroups — too many to draw. Only its 8 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 8 |
| transitive degree | μt(G) = 8 |
Polycyclic presentation:
2^3,3^2,2^4|b^a=bc,d^a=d^2,e^a=d^2e,f^a=fg,d^b=de,e^b=e^2,f^b=h,g^b=i,h^b=f,i^b=g,d^c=d^2,e^c=e^2,f^c=fg,h^c=hi,f^d=g,g^d=fg,f^e=fg,g^e=f,h^e=i,i^e=hi20 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 157849);