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) = C23 |
No aliases for this group.
| Sylow | C3 ⋊ #452 |
152 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 152 conjugacy classes of subgroups — too many to draw. Only its 47 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,3)(5,7)(8,11,13,12)(9,15,16,20)(10,18,19,21)(14,22,23,17);(8,10)(9,14)(11,18)(12,21)(13,19)(15,22)(16,23)(17,20);(2,3)(4,5)(6,7)(8,9,13,16)(10,17,19,22)(11,15,12,20)(14,21,23,18)Polycyclic presentation:
2^6,3|a^2=f,c^2=f,e^2=f,b^a=be,c^a=cd,e^a=ef,g^a=g^2,e^b=ef39 conjugacy classes — view the full character table.
| maximal subgroup of | #78845, #78847, #78849, … |
| maximal normal subgroup of | #78845, #78847, #78849, … |
| maximal quotient of | #71576, #71585, #71606, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 798);