direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C3×C17⋊C4, C17⋊C12, C51⋊2C4, D17.C6, (C3×D17).2C2, SmallGroup(204,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C17 — C3×C17⋊C4 |
Generators and relations for C3×C17⋊C4
G = < a,b,c | a3=b17=c4=1, ab=ba, ac=ca, cbc-1=b4 >
Character table of C3×C17⋊C4
| class | 1 | 2 | 3A | 3B | 4A | 4B | 6A | 6B | 12A | 12B | 12C | 12D | 17A | 17B | 17C | 17D | 51A | 51B | 51C | 51D | 51E | 51F | 51G | 51H | |
| size | 1 | 17 | 1 | 1 | 17 | 17 | 17 | 17 | 17 | 17 | 17 | 17 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | linear of order 3 |
| ρ4 | 1 | 1 | ζ32 | ζ3 | -1 | -1 | ζ32 | ζ3 | ζ6 | ζ65 | ζ65 | ζ6 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | linear of order 6 |
| ρ5 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | linear of order 3 |
| ρ6 | 1 | 1 | ζ3 | ζ32 | -1 | -1 | ζ3 | ζ32 | ζ65 | ζ6 | ζ6 | ζ65 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | linear of order 6 |
| ρ7 | 1 | -1 | 1 | 1 | -i | i | -1 | -1 | -i | i | -i | i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ8 | 1 | -1 | 1 | 1 | i | -i | -1 | -1 | i | -i | i | -i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ9 | 1 | -1 | ζ3 | ζ32 | -i | i | ζ65 | ζ6 | ζ43ζ3 | ζ4ζ32 | ζ43ζ32 | ζ4ζ3 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | linear of order 12 |
| ρ10 | 1 | -1 | ζ32 | ζ3 | -i | i | ζ6 | ζ65 | ζ43ζ32 | ζ4ζ3 | ζ43ζ3 | ζ4ζ32 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | linear of order 12 |
| ρ11 | 1 | -1 | ζ3 | ζ32 | i | -i | ζ65 | ζ6 | ζ4ζ3 | ζ43ζ32 | ζ4ζ32 | ζ43ζ3 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | linear of order 12 |
| ρ12 | 1 | -1 | ζ32 | ζ3 | i | -i | ζ6 | ζ65 | ζ4ζ32 | ζ43ζ3 | ζ4ζ3 | ζ43ζ32 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | linear of order 12 |
| ρ13 | 4 | 0 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1716+ζ1713+ζ174+ζ17 | ζ1714+ζ1712+ζ175+ζ173 | ζ1715+ζ179+ζ178+ζ172 | ζ1711+ζ1710+ζ177+ζ176 | ζ1716+ζ1713+ζ174+ζ17 | ζ1715+ζ179+ζ178+ζ172 | ζ1716+ζ1713+ζ174+ζ17 | ζ1714+ζ1712+ζ175+ζ173 | ζ1711+ζ1710+ζ177+ζ176 | ζ1714+ζ1712+ζ175+ζ173 | ζ1711+ζ1710+ζ177+ζ176 | ζ1715+ζ179+ζ178+ζ172 | orthogonal lifted from C17⋊C4 |
| ρ14 | 4 | 0 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1711+ζ1710+ζ177+ζ176 | ζ1716+ζ1713+ζ174+ζ17 | ζ1714+ζ1712+ζ175+ζ173 | ζ1715+ζ179+ζ178+ζ172 | ζ1711+ζ1710+ζ177+ζ176 | ζ1714+ζ1712+ζ175+ζ173 | ζ1711+ζ1710+ζ177+ζ176 | ζ1716+ζ1713+ζ174+ζ17 | ζ1715+ζ179+ζ178+ζ172 | ζ1716+ζ1713+ζ174+ζ17 | ζ1715+ζ179+ζ178+ζ172 | ζ1714+ζ1712+ζ175+ζ173 | orthogonal lifted from C17⋊C4 |
| ρ15 | 4 | 0 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1714+ζ1712+ζ175+ζ173 | ζ1715+ζ179+ζ178+ζ172 | ζ1711+ζ1710+ζ177+ζ176 | ζ1716+ζ1713+ζ174+ζ17 | ζ1714+ζ1712+ζ175+ζ173 | ζ1711+ζ1710+ζ177+ζ176 | ζ1714+ζ1712+ζ175+ζ173 | ζ1715+ζ179+ζ178+ζ172 | ζ1716+ζ1713+ζ174+ζ17 | ζ1715+ζ179+ζ178+ζ172 | ζ1716+ζ1713+ζ174+ζ17 | ζ1711+ζ1710+ζ177+ζ176 | orthogonal lifted from C17⋊C4 |
| ρ16 | 4 | 0 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1715+ζ179+ζ178+ζ172 | ζ1711+ζ1710+ζ177+ζ176 | ζ1716+ζ1713+ζ174+ζ17 | ζ1714+ζ1712+ζ175+ζ173 | ζ1715+ζ179+ζ178+ζ172 | ζ1716+ζ1713+ζ174+ζ17 | ζ1715+ζ179+ζ178+ζ172 | ζ1711+ζ1710+ζ177+ζ176 | ζ1714+ζ1712+ζ175+ζ173 | ζ1711+ζ1710+ζ177+ζ176 | ζ1714+ζ1712+ζ175+ζ173 | ζ1716+ζ1713+ζ174+ζ17 | orthogonal lifted from C17⋊C4 |
| ρ17 | 4 | 0 | -2-2√-3 | -2+2√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1715+ζ179+ζ178+ζ172 | ζ1711+ζ1710+ζ177+ζ176 | ζ1716+ζ1713+ζ174+ζ17 | ζ1714+ζ1712+ζ175+ζ173 | ζ32ζ1715+ζ32ζ179+ζ32ζ178+ζ32ζ172 | ζ3ζ1716+ζ3ζ1713+ζ3ζ174+ζ3ζ17 | ζ3ζ1715+ζ3ζ179+ζ3ζ178+ζ3ζ172 | ζ3ζ1711+ζ3ζ1710+ζ3ζ177+ζ3ζ176 | ζ3ζ1714+ζ3ζ1712+ζ3ζ175+ζ3ζ173 | ζ32ζ1711+ζ32ζ1710+ζ32ζ177+ζ32ζ176 | ζ32ζ1714+ζ32ζ1712+ζ32ζ175+ζ32ζ173 | ζ32ζ1716+ζ32ζ1713+ζ32ζ174+ζ32ζ17 | complex faithful |
| ρ18 | 4 | 0 | -2+2√-3 | -2-2√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1715+ζ179+ζ178+ζ172 | ζ1711+ζ1710+ζ177+ζ176 | ζ1716+ζ1713+ζ174+ζ17 | ζ1714+ζ1712+ζ175+ζ173 | ζ3ζ1715+ζ3ζ179+ζ3ζ178+ζ3ζ172 | ζ32ζ1716+ζ32ζ1713+ζ32ζ174+ζ32ζ17 | ζ32ζ1715+ζ32ζ179+ζ32ζ178+ζ32ζ172 | ζ32ζ1711+ζ32ζ1710+ζ32ζ177+ζ32ζ176 | ζ32ζ1714+ζ32ζ1712+ζ32ζ175+ζ32ζ173 | ζ3ζ1711+ζ3ζ1710+ζ3ζ177+ζ3ζ176 | ζ3ζ1714+ζ3ζ1712+ζ3ζ175+ζ3ζ173 | ζ3ζ1716+ζ3ζ1713+ζ3ζ174+ζ3ζ17 | complex faithful |
| ρ19 | 4 | 0 | -2-2√-3 | -2+2√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1714+ζ1712+ζ175+ζ173 | ζ1715+ζ179+ζ178+ζ172 | ζ1711+ζ1710+ζ177+ζ176 | ζ1716+ζ1713+ζ174+ζ17 | ζ32ζ1714+ζ32ζ1712+ζ32ζ175+ζ32ζ173 | ζ3ζ1711+ζ3ζ1710+ζ3ζ177+ζ3ζ176 | ζ3ζ1714+ζ3ζ1712+ζ3ζ175+ζ3ζ173 | ζ3ζ1715+ζ3ζ179+ζ3ζ178+ζ3ζ172 | ζ3ζ1716+ζ3ζ1713+ζ3ζ174+ζ3ζ17 | ζ32ζ1715+ζ32ζ179+ζ32ζ178+ζ32ζ172 | ζ32ζ1716+ζ32ζ1713+ζ32ζ174+ζ32ζ17 | ζ32ζ1711+ζ32ζ1710+ζ32ζ177+ζ32ζ176 | complex faithful |
| ρ20 | 4 | 0 | -2-2√-3 | -2+2√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1716+ζ1713+ζ174+ζ17 | ζ1714+ζ1712+ζ175+ζ173 | ζ1715+ζ179+ζ178+ζ172 | ζ1711+ζ1710+ζ177+ζ176 | ζ32ζ1716+ζ32ζ1713+ζ32ζ174+ζ32ζ17 | ζ3ζ1715+ζ3ζ179+ζ3ζ178+ζ3ζ172 | ζ3ζ1716+ζ3ζ1713+ζ3ζ174+ζ3ζ17 | ζ3ζ1714+ζ3ζ1712+ζ3ζ175+ζ3ζ173 | ζ3ζ1711+ζ3ζ1710+ζ3ζ177+ζ3ζ176 | ζ32ζ1714+ζ32ζ1712+ζ32ζ175+ζ32ζ173 | ζ32ζ1711+ζ32ζ1710+ζ32ζ177+ζ32ζ176 | ζ32ζ1715+ζ32ζ179+ζ32ζ178+ζ32ζ172 | complex faithful |
| ρ21 | 4 | 0 | -2+2√-3 | -2-2√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1714+ζ1712+ζ175+ζ173 | ζ1715+ζ179+ζ178+ζ172 | ζ1711+ζ1710+ζ177+ζ176 | ζ1716+ζ1713+ζ174+ζ17 | ζ3ζ1714+ζ3ζ1712+ζ3ζ175+ζ3ζ173 | ζ32ζ1711+ζ32ζ1710+ζ32ζ177+ζ32ζ176 | ζ32ζ1714+ζ32ζ1712+ζ32ζ175+ζ32ζ173 | ζ32ζ1715+ζ32ζ179+ζ32ζ178+ζ32ζ172 | ζ32ζ1716+ζ32ζ1713+ζ32ζ174+ζ32ζ17 | ζ3ζ1715+ζ3ζ179+ζ3ζ178+ζ3ζ172 | ζ3ζ1716+ζ3ζ1713+ζ3ζ174+ζ3ζ17 | ζ3ζ1711+ζ3ζ1710+ζ3ζ177+ζ3ζ176 | complex faithful |
| ρ22 | 4 | 0 | -2-2√-3 | -2+2√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1711+ζ1710+ζ177+ζ176 | ζ1716+ζ1713+ζ174+ζ17 | ζ1714+ζ1712+ζ175+ζ173 | ζ1715+ζ179+ζ178+ζ172 | ζ32ζ1711+ζ32ζ1710+ζ32ζ177+ζ32ζ176 | ζ3ζ1714+ζ3ζ1712+ζ3ζ175+ζ3ζ173 | ζ3ζ1711+ζ3ζ1710+ζ3ζ177+ζ3ζ176 | ζ3ζ1716+ζ3ζ1713+ζ3ζ174+ζ3ζ17 | ζ3ζ1715+ζ3ζ179+ζ3ζ178+ζ3ζ172 | ζ32ζ1716+ζ32ζ1713+ζ32ζ174+ζ32ζ17 | ζ32ζ1715+ζ32ζ179+ζ32ζ178+ζ32ζ172 | ζ32ζ1714+ζ32ζ1712+ζ32ζ175+ζ32ζ173 | complex faithful |
| ρ23 | 4 | 0 | -2+2√-3 | -2-2√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1716+ζ1713+ζ174+ζ17 | ζ1714+ζ1712+ζ175+ζ173 | ζ1715+ζ179+ζ178+ζ172 | ζ1711+ζ1710+ζ177+ζ176 | ζ3ζ1716+ζ3ζ1713+ζ3ζ174+ζ3ζ17 | ζ32ζ1715+ζ32ζ179+ζ32ζ178+ζ32ζ172 | ζ32ζ1716+ζ32ζ1713+ζ32ζ174+ζ32ζ17 | ζ32ζ1714+ζ32ζ1712+ζ32ζ175+ζ32ζ173 | ζ32ζ1711+ζ32ζ1710+ζ32ζ177+ζ32ζ176 | ζ3ζ1714+ζ3ζ1712+ζ3ζ175+ζ3ζ173 | ζ3ζ1711+ζ3ζ1710+ζ3ζ177+ζ3ζ176 | ζ3ζ1715+ζ3ζ179+ζ3ζ178+ζ3ζ172 | complex faithful |
| ρ24 | 4 | 0 | -2+2√-3 | -2-2√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1711+ζ1710+ζ177+ζ176 | ζ1716+ζ1713+ζ174+ζ17 | ζ1714+ζ1712+ζ175+ζ173 | ζ1715+ζ179+ζ178+ζ172 | ζ3ζ1711+ζ3ζ1710+ζ3ζ177+ζ3ζ176 | ζ32ζ1714+ζ32ζ1712+ζ32ζ175+ζ32ζ173 | ζ32ζ1711+ζ32ζ1710+ζ32ζ177+ζ32ζ176 | ζ32ζ1716+ζ32ζ1713+ζ32ζ174+ζ32ζ17 | ζ32ζ1715+ζ32ζ179+ζ32ζ178+ζ32ζ172 | ζ3ζ1716+ζ3ζ1713+ζ3ζ174+ζ3ζ17 | ζ3ζ1715+ζ3ζ179+ζ3ζ178+ζ3ζ172 | ζ3ζ1714+ζ3ζ1712+ζ3ζ175+ζ3ζ173 | complex faithful |
►On 51 points
Generators in S51
(1 35 18)(2 36 19)(3 37 20)(4 38 21)(5 39 22)(6 40 23)(7 41 24)(8 42 25)(9 43 26)(10 44 27)(11 45 28)(12 46 29)(13 47 30)(14 48 31)(15 49 32)(16 50 33)(17 51 34)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17)(18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34)(35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51)
(2 14 17 5)(3 10 16 9)(4 6 15 13)(7 11 12 8)(19 31 34 22)(20 27 33 26)(21 23 32 30)(24 28 29 25)(36 48 51 39)(37 44 50 43)(38 40 49 47)(41 45 46 42)
G:=sub<Sym(51)| (1,35,18)(2,36,19)(3,37,20)(4,38,21)(5,39,22)(6,40,23)(7,41,24)(8,42,25)(9,43,26)(10,44,27)(11,45,28)(12,46,29)(13,47,30)(14,48,31)(15,49,32)(16,50,33)(17,51,34), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34)(35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51), (2,14,17,5)(3,10,16,9)(4,6,15,13)(7,11,12,8)(19,31,34,22)(20,27,33,26)(21,23,32,30)(24,28,29,25)(36,48,51,39)(37,44,50,43)(38,40,49,47)(41,45,46,42)>;
G:=Group( (1,35,18)(2,36,19)(3,37,20)(4,38,21)(5,39,22)(6,40,23)(7,41,24)(8,42,25)(9,43,26)(10,44,27)(11,45,28)(12,46,29)(13,47,30)(14,48,31)(15,49,32)(16,50,33)(17,51,34), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34)(35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51), (2,14,17,5)(3,10,16,9)(4,6,15,13)(7,11,12,8)(19,31,34,22)(20,27,33,26)(21,23,32,30)(24,28,29,25)(36,48,51,39)(37,44,50,43)(38,40,49,47)(41,45,46,42) );
G=PermutationGroup([[(1,35,18),(2,36,19),(3,37,20),(4,38,21),(5,39,22),(6,40,23),(7,41,24),(8,42,25),(9,43,26),(10,44,27),(11,45,28),(12,46,29),(13,47,30),(14,48,31),(15,49,32),(16,50,33),(17,51,34)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17),(18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34),(35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51)], [(2,14,17,5),(3,10,16,9),(4,6,15,13),(7,11,12,8),(19,31,34,22),(20,27,33,26),(21,23,32,30),(24,28,29,25),(36,48,51,39),(37,44,50,43),(38,40,49,47),(41,45,46,42)]])
C3×C17⋊C4 is a maximal subgroup of
C51⋊C8
Matrix representation of C3×C17⋊C4 ►in GL4(𝔽409) generated by
| 53 | 0 | 0 | 0 |
| 0 | 53 | 0 | 0 |
| 0 | 0 | 53 | 0 |
| 0 | 0 | 0 | 53 |
| 0 | 0 | 0 | 408 |
| 1 | 0 | 0 | 392 |
| 0 | 1 | 0 | 48 |
| 0 | 0 | 1 | 392 |
| 1 | 1 | 49 | 17 |
| 0 | 377 | 14 | 288 |
| 0 | 378 | 103 | 394 |
| 0 | 18 | 32 | 337 |
G:=sub<GL(4,GF(409))| [53,0,0,0,0,53,0,0,0,0,53,0,0,0,0,53],[0,1,0,0,0,0,1,0,0,0,0,1,408,392,48,392],[1,0,0,0,1,377,378,18,49,14,103,32,17,288,394,337] >;
C3×C17⋊C4 in GAP, Magma, Sage, TeX
C_3\times C_{17}\rtimes C_4 % in TeX
G:=Group("C3xC17:C4"); // GroupNames label
G:=SmallGroup(204,5);
// by ID
G=gap.SmallGroup(204,5);
# by ID
G:=PCGroup([4,-2,-3,-2,-17,24,2499,523]);
// Polycyclic
G:=Group<a,b,c|a^3=b^17=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^4>;
// generators/relations
Export
Subgroup lattice of C3×C17⋊C4 in TeX
Character table of C3×C17⋊C4 in TeX