Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C2×D3×#23900 (order 3072) · Inn(G) = C22×D3 · Out(G) = C2×D42 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Sylow | C3 ⋊ #405 |
130 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,3)(8,10,12,20,13,18,11,19)(9,14,16,23,17,21,15,22);(10,18)(14,21)(19,20)(22,23);(2,3)(4,5,6,7)(8,9,11,15,13,17,12,16)(10,14,19,22,18,21,20,23)Polycyclic presentation:
2^6,3|a^2=d,c^2=e,d^2=f,e^2=f,g^a=g^2,c^b=cf60 conjugacy classes — view the full character table.
| maximal subgroup of | #68697, #68717, #68738, … |
| maximal normal subgroup of | #68697, #68717, #68738, … |
| maximal quotient of | #69804, #69806, #69810, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 681);