Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) order 6144 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C3×#117 ⋊ C2 |
| Sylow | C3 ⋊ C4×#41 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D3 |
| commutator | C2×C3 | C23×C4 |
| Frattini | C22 | C23×D3 |
| Fitting | C3×#117 | C2 |
| socle, cosocle | C22×C3 | C24 |
| prime radical | C2×#235 | C2 |
| hypercenter | #117 | D3 |
| hypocenter | C3 | C4×#41 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C4 ◁ #117 | |
310 conjugacy classes of subgroups; only the 155 normal subgroups are shown.
This group has 310 conjugacy classes of subgroups — too many to draw. Only its 155 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 15 points:
(1,2,3)(4,5,6,7)(10,14)(12,15);(8,10)(9,12)(11,14)(13,15);(8,9,11,13)(10,12,14,15);(2,3)(10,14)(12,15)Polycyclic presentation:
2^6,3|b^2=f,d^2=e,c^a=cf,g^a=g^2,d^c=df60 conjugacy classes — view the full character table.
| maximal subgroup of | #68711, #68728, #68729, … |
| maximal normal subgroup of | #68711, #68728, #68729, … |
| maximal quotient of | #73382, #73384, #73385, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1081);