Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
| dicyclic | Dic168 |
| Hall | C3×C7 ⋊ Q32 |
| Sylow | C7 ⋊ Dic24 |
| Sylow | C3 ⋊ Dic56 |
48 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 672 |
A minimal faithful permutation representation — 2 generators on 42 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18)(33,34,35)(36,37,38,39,40,41,42);(1,17,9,25)(2,18,10,26)(3,19,11,27)(4,20,12,28)(5,21,13,29)(6,22,14,30)(7,23,15,31)(8,24,16,32)(34,35)(37,42)(38,41)(39,40)Polycyclic presentation:
2^5,3,7|a^2=e,b^2=cd,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^2,g^a=g^6171 conjugacy classes — view the full character table.
| maximal subgroup of | #13050851, #13050853, #13050856, … |
| maximal normal subgroup of | #13050851, #13050853, #13050856, … |
| maximal quotient of | #13051408, #13051413, C2×Dic168, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 222);