Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C25×D3×D4 (order 1536) · Inn(G) = C2×#67 · Out(G) = C25 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C3×C4×D4 ⋊ C2 |
| Sylow | C3 ⋊ #463 |
124 conjugacy classes of subgroups; only the 43 normal subgroups are shown.
This group has 124 conjugacy classes of subgroups — too many to draw. Only its 43 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 — 3 generators on 15 points:
(1,2,3)(9,13)(10,15)(12,14);(4,5,6,7)(9,12)(13,14);(2,3)(5,7)(8,9)(10,13)(11,12)(14,15)Polycyclic presentation:
2^6,3|b^2=ef,d^2=f,b^a=be,c^a=cd,d^a=df,g^a=g^2,d^c=df39 conjugacy classes — view the full character table.
| maximal subgroup of | #79569, #79572, #79580, … |
| maximal normal subgroup of | #79569, #79572, #79580, … |
| maximal quotient of | #69463, #69465, #69468, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 575);