Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| isoclinism | isoclinic class of #213 |
| automorphism | Aut(G) = C2×D3×#3484 (order 1536) · Inn(G) = #67 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
| direct product | C2×#825 |
| Fitting | C2×C3×M16 ⋊ C2 |
| Sylow | C3 ⋊ C2×#104 |
170 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 170 conjugacy classes of subgroups — too many to draw. Only its 63 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 13 points:
(3,4,5)(6,8,9,12,10,11,13,7);(4,5)(6,7)(8,9)(10,12)(11,13);(1,2)48 conjugacy classes — view the full character table.
| maximal subgroup of | #68714, #68719, #68728, … |
| maximal normal subgroup of | #68714, #68719, #68728, … |
| maximal quotient of | C2×#4464, #69797, C2×#4465, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 697);