Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C23×D3×D4 (order 384) · Inn(G) = D3×D4 · Out(G) = C23 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C3×#104 ⋊ C2 |
| Sylow | C3 ⋊ #421 |
154 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 154 conjugacy classes of subgroups — too many to draw. Only its 51 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3)(4,7,8,6,9,14,17,15)(5,11,12,18,13,16,19,10);(4,6)(5,10)(7,17)(8,14)(9,15)(11,12)(13,18)(16,19);(2,3)(4,5)(6,10)(7,16)(8,12)(9,13)(11,14)(15,18)(17,19)Polycyclic presentation:
2^6,3|c^2=d,d^2=f,e^2=f,c^a=cf,g^a=g^2,c^b=ce,e^b=ef,e^c=ef36 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#4821, #78983, D3×#421, … |
| maximal normal subgroup of | C2×#4821, #78983, D3×#421, … |
| maximal quotient of | #68166, #68167, #68168, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 380);