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) = C4 |
No aliases for this group.
| Sylow | C3 ⋊ #491 |
90 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3)(4,6,7,11)(5,9,8,10)(13,18)(14,20)(15,23)(17,24)(21,22)(25,27);(5,8)(9,10)(12,14,16,22)(13,17,18,25)(15,20,23,21)(19,27,26,24);(2,3)(4,5,7,8)(6,10,11,9)(12,13)(14,17)(15,19)(16,18)(20,27)(21,24)(22,25)(23,26)Polycyclic presentation:
2^6,3|a^2=e,b^2=f,c^2=e,d^2=f,b^a=be,c^a=cef,g^a=g^2,c^b=cd,d^b=df,d^c=df30 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#4797, #78785, #78810, … |
| maximal normal subgroup of | C2×#4797, #78785, #78810, … |
| maximal quotient of | #68805, #68810, #68842, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 356);