Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C2×D3×D4×#120 (order 3072) · Inn(G) = C2×#67 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C2×C3×C4⋊C4 ⋊ C2 |
| Sylow | C3 ⋊ #417 |
162 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 162 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) = 15 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 15 points:
(1,2,3)(4,5,6,7)(9,14)(10,11)(13,15);(8,10,12,11)(9,13,15,14);(2,3)(8,9)(10,13)(11,14)(12,15)Polycyclic presentation:
2^6,3|b^2=f,c^2=d,e^2=f,c^a=ce,e^a=ef,g^a=g^2,c^b=cf,e^c=ef42 conjugacy classes — view the full character table.
| maximal subgroup of | #68986, #68987, #69000, … |
| maximal normal subgroup of | #68986, #68987, #69000, … |
| maximal quotient of | #69398, #69470, #69474, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 525);