Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| isoclinism | isoclinic class of #10596770 |
| automorphism | ? |
| homology | Abelianization (H1) = C8 · Schur multiplier (H2) = C2×C3 |
No aliases for this group.
| derived | A42 ⋊ C8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | #10596770 |
| commutator, hypocenter | A42 | C8 |
| Fitting, socle | C25 | #157 |
| prime radical | C2×A42 | C4 |
| cosocle | A4⋊SL(2,ℤ/4) | C2 |
| LCS | G ▷ A42 | |
| UCS | C1 ◁ C2 | |
121 conjugacy classes of subgroups; only the 8 normal subgroups are shown.
This group has 121 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.
| transitive degree | μt(G) = 24 |
Polycyclic presentation:
2^3,3^2,2^4|a^2=b,b^2=c,d^a=de,e^a=de^2,f^a=hi,g^a=i,h^a=fg,i^a=f,d^b=d^2,e^b=e^2,f^b=g,g^b=f,i^b=hi,f^d=g,g^d=fg,h^d=i,i^d=hi,h^e=i,i^e=hi26 conjugacy classes — view the full character table.
| maximal subgroup of | #72209667, #72209668, #72209669, … |
| maximal normal subgroup of | C2×#11923442 |
| maximal quotient of | C2×#11923442, #72227483, #72227890 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 157453);