Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C3×C4 |
No aliases for this group.
| Fitting | C32×C4⋊Q8 ⋊ C4 |
| Sylow | C32 ⋊ #2369 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×#157 |
| commutator, socle | C22×C32 | C23×C4 |
| Fitting | C32×C4⋊Q8 | C4 |
| prime radical, cosocle | C22×C3⋊D3 | C24 |
| hypercenter | C4⋊Q8 | #157 |
| hypocenter | C32 | #2369 |
| LCS | G ▷ C22×C32 ▷ C32 | |
| UCS | C1 ◁ C22 ◁ C4⋊Q8 | |
786 conjugacy classes of subgroups; only the 188 normal subgroups are shown.
This group has 786 conjugacy classes of subgroups — too many to draw. Only its 188 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 192 |
Polycyclic presentation:
2^7,3^2|a^2=g,b^2=fg,c^2=f,d^2=e,b^a=be,c^a=cf,h^a=h^2i^2,i^a=h^2i,d^b=de,h^b=hi,i^b=hi^2,h^g=h^2,i^g=i^278 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 41130);