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)(4,6,7,10)(5,9,8,11)(13,18)(15,23)(17,24)(19,20)(21,27)(25,26);(5,8)(9,11)(12,14,16,22)(13,17,20,26)(15,21,23,27)(18,24,19,25);(2,3)(4,5,7,8)(6,9,10,11)(12,13)(14,17)(15,18)(16,20)(19,23)(21,24)(22,26)(25,27)Polycyclic presentation:
2^6,3|a^2=e,b^2=f,c^2=e,d^2=f,b^a=be,c^a=cd,d^a=df,g^a=g^2,d^c=df39 conjugacy classes — view the full character table.
| maximal subgroup of | #78847, #78848, #78849, … |
| maximal normal subgroup of | #78847, #78848, #78849, … |
| maximal quotient of | #69458, #69459, #69464, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 555);