Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| isoclinism | isoclinic class of #826 |
| automorphism | Aut(G) = C22×D3×#442 (order 1536) · Inn(G) = #213 · Out(G) = C23×C4 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C3 ⋊ #358 |
58 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 2 generators on 23 points:
(1,2,3)(8,9,12,15)(10,17,19,23)(11,20,18,14)(13,21,22,16);(2,3)(4,5,6,7)(9,13)(10,16)(11,18)(14,23)(15,22)(17,20)(19,21)Polycyclic presentation:
2^6,3|a^2=c,b^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 | #69210, #69211, #69212, … |
| maximal normal subgroup of | #69210, #69211, #69212, … |
| maximal quotient of | #67722, #67765, #67776, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 123);