Finite Group DatabasebetaFinite Group Databasebeta| order | 10112 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Sylow | C79 ⋊ #2567 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D79 |
| commutator, socle, prime radical, cosocle | C23×C79 | C24 |
| Fitting | C2×C4⋊Q8×C79 | C2 |
| hypercenter | C2×C4⋊Q8 | D79 |
| hypocenter | C79 | #2567 |
| LCS | G ▷ C23×C79 ▷ C79 | |
| UCS | C1 ◁ C23 ◁ C2×C4⋊Q8 | |
500 conjugacy classes of subgroups; only the 189 normal subgroups are shown.
This group has 500 conjugacy classes of subgroups — too many to draw. Only its 189 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^7,79|a^2=eg,b^2=eg,c^2=f,d^2=e,b^a=be,c^a=cf,d^a=de,h^a=h^78,c^b=cg,h^c=h^781130 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #34185626, #34185666, #34185960, … |
| maximal normal subgroup of | #34185626, #34185666, #34185960, … |
| maximal quotient of | #34210546, #34210685, #34211143, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(10112, 9719);