Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C25×D3×D4 (order 1536) · Inn(G) = D3×D4 · Out(G) = C25 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C4 |
No aliases for this group.
| Fitting | C3×#102 ⋊ C2 |
| Sylow | C3 ⋊ #490 |
110 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 110 conjugacy classes of subgroups — too many to draw. Only its 37 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3)(4,6,7,11)(5,9,8,10)(12,13)(14,19)(15,16)(17,18);(5,8)(9,10)(13,17)(14,18)(16,19);(2,3)(4,5,7,8)(6,10,11,9)(13,16)(17,19)Polycyclic presentation:
2^6,3|a^2=e,c^2=e,d^2=f,b^a=be,c^a=cef,g^a=g^2,c^b=cd,d^b=df,d^c=df30 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#4764, #78737, #78762, … |
| maximal normal subgroup of | C2×#4764, #78737, #78762, … |
| maximal quotient of | #68805, #68811, #68843, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 323);