Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C25×D3×D4 (order 1536) · Inn(G) = C2×#67 · Out(G) = C25 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| 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)(6,10)(8,11)(12,15,17,23)(13,19,20,27)(14,21,16,25)(18,26,24,22);(4,6)(5,8)(7,10)(9,11)(12,14,17,16)(13,18,20,24)(15,21,23,25)(19,26,27,22);(2,3)(4,5,7,9)(6,8,10,11)(12,13)(14,18)(15,22)(16,24)(17,20)(19,21)(23,26)(25,27)Polycyclic presentation:
2^6,3|a^2=e,b^2=f,c^2=f,d^2=f,c^a=cd,d^a=df,g^a=g^2,c^b=ce,d^c=df33 conjugacy classes — view the full character table.
| maximal subgroup of | #80136, #80137, #80154, … |
| maximal normal subgroup of | #80136, #80137, #80154, … |
| maximal quotient of | #69503, #69506, #69518, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 604);