Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C25×D3×D4 (order 1536) · Inn(G) = D3×D4 · Out(G) = C25 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | C3×#107 ⋊ C2 |
| Sylow | C3 ⋊ #485 |
96 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3)(5,9)(8,11)(12,14,21,27,16,22,15,20)(13,18,24,25,17,23,19,26);(4,6,7,10)(5,8,9,11)(13,17)(14,20)(15,21)(18,25)(22,27)(23,26);(2,3)(4,5)(6,8)(7,9)(10,11)(12,13)(14,18)(15,19)(16,17)(20,26)(21,24)(22,23)(25,27)Polycyclic presentation:
2^6,3|b^2=d,c^2=ef,e^2=f,b^a=bf,c^a=cd,g^a=g^2,c^b=ce,e^b=ef33 conjugacy classes — view the full character table.
| maximal subgroup of | #79141, C2×#4888, #79155, … |
| maximal normal subgroup of | #79141, C2×#4888, #79155, … |
| maximal quotient of | #68793, #68817, #68824, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 447);