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 ⋊ #488 |
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)(4,6,7,11)(5,9,8,10)(13,18)(14,21)(15,24)(17,25)(20,27)(22,23);(5,8)(9,10)(12,14)(13,17)(15,21)(16,23)(18,20)(19,27)(22,24)(25,26);(2,3)(4,5,7,8)(6,10,11,9)(12,13)(14,20)(15,19)(16,18)(17,23)(21,25)(22,27)(24,26)Polycyclic presentation:
2^6,3|a^2=e,c^2=e,d^2=f,b^a=bef,c^a=cef,g^a=g^2,c^b=cd,d^b=df,d^c=df36 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#4768, #78737, #78759, … |
| maximal normal subgroup of | C2×#4768, #78737, #78759, … |
| maximal quotient of | #68804, #68812, #68845, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 327);