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.
| Sylow | C3 ⋊ #465 |
124 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 124 conjugacy classes of subgroups — too many to draw. Only its 45 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)(5,7)(9,12)(10,15)(13,14);(4,5)(6,7);(2,3)(5,7)(8,9,11,14)(10,12,15,13)Polycyclic presentation:
2^6,3|a^2=f,d^2=f,b^a=be,c^a=cd,d^a=df,g^a=g^2,c^b=ce,d^c=df36 conjugacy classes — view the full character table.
| maximal subgroup of | #80015, #80016, #80022, … |
| maximal normal subgroup of | #80015, #80016, #80022, … |
| maximal quotient of | #69511, #69514, #69520, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 600);