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) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #440 |
120 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 120 conjugacy classes of subgroups — too many to draw. Only its 51 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)(4,5,6,7)(8,10,11,12)(9,14,15,13);(9,13)(10,12)(14,15);(2,3)(8,9,11,15)(10,14,12,13)Polycyclic presentation:
2^6,3|a^2=f,c^2=d,e^2=f,b^a=be,e^a=ef,g^a=g^2,c^b=cf,e^b=ef42 conjugacy classes — view the full character table.
| maximal subgroup of | #79587, #79589, #79594, … |
| maximal normal subgroup of | #79587, #79589, #79594, … |
| maximal quotient of | #69459, #69465, #69475, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 577);