Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) order 12288 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
| direct product | C2×#879 |
| Sylow | C3 ⋊ C2×#126 |
246 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 246 conjugacy classes of subgroups — too many to draw. Only its 111 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 21 points:
(1,2)(3,4,5)(6,8)(7,10)(9,13)(11,12)(14,16)(15,18)(17,20)(19,21);(8,13)(10,12)(14,16,17,20)(15,18,19,21);(4,5)(6,7,9,11)(8,12,13,10)(14,15,17,19)(16,18,20,21);(1,2)48 conjugacy classes — view the full character table.
| maximal subgroup of | #68105, C4×#879, #73450, … |
| maximal normal subgroup of | #68105, C4×#879, #73450, … |
| maximal quotient of | C2×#4651, C2×#4652, C2×#4653, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1041);