Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) order 6144 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C2×C3×C22⋊C4 ⋊ C2 |
| Sylow | C3 ⋊ #518 |
338 conjugacy classes of subgroups; only the 151 normal subgroups are shown.
This group has 338 conjugacy classes of subgroups — too many to draw. Only its 151 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) = 48 |
A minimal faithful permutation representation — 4 generators on 15 points:
(1,2,3)(4,5,6,7)(10,14)(12,15);(9,13)(12,15);(8,10)(9,12)(11,14)(13,15);(2,3)(8,9)(10,12)(11,13)(14,15)Polycyclic presentation:
2^6,3|d^2=e,c^a=cf,g^a=g^2,d^b=df54 conjugacy classes — view the full character table.
| maximal subgroup of | #68960, #68961, #68962, … |
| maximal normal subgroup of | #68960, #68961, #68962, … |
| maximal quotient of | #73322, #73333, #73334, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1045);