Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| isoclinism | isoclinic class of #826 |
| automorphism | Aut(G) = C2×D3×#442 (order 768) · Inn(G) = #213 · Out(G) = C22×C4 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C3 ⋊ #359 |
56 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 2 generators on 35 points:
(1,2,3)(4,6)(5,11)(7,16)(8,21)(9,25)(10,18)(12,26)(13,29)(14,28)(15,20)(17,22)(19,33)(23,35)(24,27)(30,31)(32,34);(2,3)(4,5,7,12,10,14,9,24)(6,15,16,31,18,32,25,29)(8,20,19,30,23,34,22,13)(11,21,26,33,28,35,27,17)Polycyclic presentation:
2^6,3|a^2=c,c^2=f,d^2=ef,e^2=f,b^a=bd,d^a=de,e^a=ef,g^a=g^2,d^b=de,e^b=ef36 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#359, #69241, #70921, … |
| maximal normal subgroup of | D3×#359, #69241, #70921, … |
| maximal quotient of | #67724, #67763, #67765, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 126);