Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Sylow | C32 ⋊ #2369 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C22×D32 |
| commutator | C22×C32 | C23×C4 |
| Fitting | C22×C32×Q8 | C22 |
| socle, prime radical, cosocle | C23×C32 | C24 |
| hypercenter | C22×Q8 | D32 |
| hypocenter | C32 | #2369 |
| LCS | G ▷ C22×C32 ▷ C32 | |
| UCS | C1 ◁ C23 ◁ C22×Q8 | |
1137 conjugacy classes of subgroups; only the 284 normal subgroups are shown.
This group has 1137 conjugacy classes of subgroups — too many to draw. Only its 284 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 384 |
Polycyclic presentation:
2^7,3^2|a^2=g,b^2=fg,c^2=f,d^2=e,b^a=be,c^a=cf,i^a=hi^2,d^b=de,i^b=hi^2,h^c=h^2,i^c=i^2,h^d=h^2,i^d=i^2126 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 41153);