Finite Group DatabasebetaFinite Group Databasebeta| order | 15296 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C22⋊Q8×C239 ⋊ C2 |
| Sylow | C239 ⋊ #562 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D239 |
| commutator, socle, cosocle | C22×C239 | C24 |
| Fitting | C22⋊Q8×C239 | C2 |
| hypercenter | C22⋊Q8 | D239 |
| hypocenter | C239 | #562 |
| LCS | G ▷ C22×C239 ▷ C239 | |
| UCS | C1 ◁ C22 ◁ C22⋊Q8 | |
220 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 220 conjugacy classes of subgroups — too many to draw. Only its 91 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 7648 |
Polycyclic presentation:
2^6,239|c^2=e,b^a=be,c^a=cf,d^a=def,c^b=cef,d^b=de,g^b=g^238,g^d=g^2381685 conjugacy classes. The character table is too large to display.
| maximal quotient of | #44487692, #44487693, #44487695, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15296, 1295);