metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C51⋊1C4, C17⋊Dic3, D17.S3, C3⋊(C17⋊C4), (C3×D17).1C2, SmallGroup(204,6)
Series: Derived ►Chief ►Lower central ►Upper central
| C51 — C51⋊C4 |
Generators and relations for C51⋊C4
G = < a,b | a51=b4=1, bab-1=a47 >
Character table of C51⋊C4
| class | 1 | 2 | 3 | 4A | 4B | 6 | 17A | 17B | 17C | 17D | 51A | 51B | 51C | 51D | 51E | 51F | 51G | 51H | |
| size | 1 | 17 | 2 | 51 | 51 | 34 | 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 | trivial |
| ρ2 | 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 | 1 | -i | i | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ4 | 1 | -1 | 1 | i | -i | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ5 | 2 | 2 | -1 | 0 | 0 | -1 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ6 | 2 | -2 | -1 | 0 | 0 | 1 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | symplectic lifted from Dic3, Schur index 2 |
| ρ7 | 4 | 0 | 4 | 0 | 0 | 0 | ζ1716+ζ1713+ζ174+ζ17 | ζ1715+ζ179+ζ178+ζ172 | ζ1714+ζ1712+ζ175+ζ173 | ζ1711+ζ1710+ζ177+ζ176 | ζ1714+ζ1712+ζ175+ζ173 | ζ1711+ζ1710+ζ177+ζ176 | ζ1711+ζ1710+ζ177+ζ176 | ζ1715+ζ179+ζ178+ζ172 | ζ1716+ζ1713+ζ174+ζ17 | ζ1715+ζ179+ζ178+ζ172 | ζ1714+ζ1712+ζ175+ζ173 | ζ1716+ζ1713+ζ174+ζ17 | orthogonal lifted from C17⋊C4 |
| ρ8 | 4 | 0 | 4 | 0 | 0 | 0 | ζ1711+ζ1710+ζ177+ζ176 | ζ1714+ζ1712+ζ175+ζ173 | ζ1716+ζ1713+ζ174+ζ17 | ζ1715+ζ179+ζ178+ζ172 | ζ1716+ζ1713+ζ174+ζ17 | ζ1715+ζ179+ζ178+ζ172 | ζ1715+ζ179+ζ178+ζ172 | ζ1714+ζ1712+ζ175+ζ173 | ζ1711+ζ1710+ζ177+ζ176 | ζ1714+ζ1712+ζ175+ζ173 | ζ1716+ζ1713+ζ174+ζ17 | ζ1711+ζ1710+ζ177+ζ176 | orthogonal lifted from C17⋊C4 |
| ρ9 | 4 | 0 | 4 | 0 | 0 | 0 | ζ1714+ζ1712+ζ175+ζ173 | ζ1711+ζ1710+ζ177+ζ176 | ζ1715+ζ179+ζ178+ζ172 | ζ1716+ζ1713+ζ174+ζ17 | ζ1715+ζ179+ζ178+ζ172 | ζ1716+ζ1713+ζ174+ζ17 | ζ1716+ζ1713+ζ174+ζ17 | ζ1711+ζ1710+ζ177+ζ176 | ζ1714+ζ1712+ζ175+ζ173 | ζ1711+ζ1710+ζ177+ζ176 | ζ1715+ζ179+ζ178+ζ172 | ζ1714+ζ1712+ζ175+ζ173 | orthogonal lifted from C17⋊C4 |
| ρ10 | 4 | 0 | 4 | 0 | 0 | 0 | ζ1715+ζ179+ζ178+ζ172 | ζ1716+ζ1713+ζ174+ζ17 | ζ1711+ζ1710+ζ177+ζ176 | ζ1714+ζ1712+ζ175+ζ173 | ζ1711+ζ1710+ζ177+ζ176 | ζ1714+ζ1712+ζ175+ζ173 | ζ1714+ζ1712+ζ175+ζ173 | ζ1716+ζ1713+ζ174+ζ17 | ζ1715+ζ179+ζ178+ζ172 | ζ1716+ζ1713+ζ174+ζ17 | ζ1711+ζ1710+ζ177+ζ176 | ζ1715+ζ179+ζ178+ζ172 | orthogonal lifted from C17⋊C4 |
| ρ11 | 4 | 0 | -2 | 0 | 0 | 0 | ζ1711+ζ1710+ζ177+ζ176 | ζ1714+ζ1712+ζ175+ζ173 | ζ1716+ζ1713+ζ174+ζ17 | ζ1715+ζ179+ζ178+ζ172 | -ζ3ζ1716+ζ3ζ1713+ζ3ζ174-ζ3ζ17-ζ1716-ζ17 | -ζ32ζ1715+ζ32ζ179+ζ32ζ178-ζ32ζ172-ζ1715-ζ172 | ζ32ζ1715-ζ32ζ179-ζ32ζ178+ζ32ζ172-ζ179-ζ178 | ζ3ζ1714-ζ3ζ1712-ζ3ζ175+ζ3ζ173-ζ1712-ζ175 | -ζ32ζ1711+ζ32ζ1710+ζ32ζ177-ζ32ζ176-ζ1711-ζ176 | -ζ3ζ1714+ζ3ζ1712+ζ3ζ175-ζ3ζ173-ζ1714-ζ173 | ζ3ζ1716-ζ3ζ1713-ζ3ζ174+ζ3ζ17-ζ1713-ζ174 | ζ32ζ1711-ζ32ζ1710-ζ32ζ177+ζ32ζ176-ζ1710-ζ177 | complex faithful |
| ρ12 | 4 | 0 | -2 | 0 | 0 | 0 | ζ1716+ζ1713+ζ174+ζ17 | ζ1715+ζ179+ζ178+ζ172 | ζ1714+ζ1712+ζ175+ζ173 | ζ1711+ζ1710+ζ177+ζ176 | ζ3ζ1714-ζ3ζ1712-ζ3ζ175+ζ3ζ173-ζ1712-ζ175 | ζ32ζ1711-ζ32ζ1710-ζ32ζ177+ζ32ζ176-ζ1710-ζ177 | -ζ32ζ1711+ζ32ζ1710+ζ32ζ177-ζ32ζ176-ζ1711-ζ176 | -ζ32ζ1715+ζ32ζ179+ζ32ζ178-ζ32ζ172-ζ1715-ζ172 | -ζ3ζ1716+ζ3ζ1713+ζ3ζ174-ζ3ζ17-ζ1716-ζ17 | ζ32ζ1715-ζ32ζ179-ζ32ζ178+ζ32ζ172-ζ179-ζ178 | -ζ3ζ1714+ζ3ζ1712+ζ3ζ175-ζ3ζ173-ζ1714-ζ173 | ζ3ζ1716-ζ3ζ1713-ζ3ζ174+ζ3ζ17-ζ1713-ζ174 | complex faithful |
| ρ13 | 4 | 0 | -2 | 0 | 0 | 0 | ζ1711+ζ1710+ζ177+ζ176 | ζ1714+ζ1712+ζ175+ζ173 | ζ1716+ζ1713+ζ174+ζ17 | ζ1715+ζ179+ζ178+ζ172 | ζ3ζ1716-ζ3ζ1713-ζ3ζ174+ζ3ζ17-ζ1713-ζ174 | ζ32ζ1715-ζ32ζ179-ζ32ζ178+ζ32ζ172-ζ179-ζ178 | -ζ32ζ1715+ζ32ζ179+ζ32ζ178-ζ32ζ172-ζ1715-ζ172 | -ζ3ζ1714+ζ3ζ1712+ζ3ζ175-ζ3ζ173-ζ1714-ζ173 | ζ32ζ1711-ζ32ζ1710-ζ32ζ177+ζ32ζ176-ζ1710-ζ177 | ζ3ζ1714-ζ3ζ1712-ζ3ζ175+ζ3ζ173-ζ1712-ζ175 | -ζ3ζ1716+ζ3ζ1713+ζ3ζ174-ζ3ζ17-ζ1716-ζ17 | -ζ32ζ1711+ζ32ζ1710+ζ32ζ177-ζ32ζ176-ζ1711-ζ176 | complex faithful |
| ρ14 | 4 | 0 | -2 | 0 | 0 | 0 | ζ1715+ζ179+ζ178+ζ172 | ζ1716+ζ1713+ζ174+ζ17 | ζ1711+ζ1710+ζ177+ζ176 | ζ1714+ζ1712+ζ175+ζ173 | ζ32ζ1711-ζ32ζ1710-ζ32ζ177+ζ32ζ176-ζ1710-ζ177 | -ζ3ζ1714+ζ3ζ1712+ζ3ζ175-ζ3ζ173-ζ1714-ζ173 | ζ3ζ1714-ζ3ζ1712-ζ3ζ175+ζ3ζ173-ζ1712-ζ175 | ζ3ζ1716-ζ3ζ1713-ζ3ζ174+ζ3ζ17-ζ1713-ζ174 | -ζ32ζ1715+ζ32ζ179+ζ32ζ178-ζ32ζ172-ζ1715-ζ172 | -ζ3ζ1716+ζ3ζ1713+ζ3ζ174-ζ3ζ17-ζ1716-ζ17 | -ζ32ζ1711+ζ32ζ1710+ζ32ζ177-ζ32ζ176-ζ1711-ζ176 | ζ32ζ1715-ζ32ζ179-ζ32ζ178+ζ32ζ172-ζ179-ζ178 | complex faithful |
| ρ15 | 4 | 0 | -2 | 0 | 0 | 0 | ζ1714+ζ1712+ζ175+ζ173 | ζ1711+ζ1710+ζ177+ζ176 | ζ1715+ζ179+ζ178+ζ172 | ζ1716+ζ1713+ζ174+ζ17 | ζ32ζ1715-ζ32ζ179-ζ32ζ178+ζ32ζ172-ζ179-ζ178 | -ζ3ζ1716+ζ3ζ1713+ζ3ζ174-ζ3ζ17-ζ1716-ζ17 | ζ3ζ1716-ζ3ζ1713-ζ3ζ174+ζ3ζ17-ζ1713-ζ174 | -ζ32ζ1711+ζ32ζ1710+ζ32ζ177-ζ32ζ176-ζ1711-ζ176 | -ζ3ζ1714+ζ3ζ1712+ζ3ζ175-ζ3ζ173-ζ1714-ζ173 | ζ32ζ1711-ζ32ζ1710-ζ32ζ177+ζ32ζ176-ζ1710-ζ177 | -ζ32ζ1715+ζ32ζ179+ζ32ζ178-ζ32ζ172-ζ1715-ζ172 | ζ3ζ1714-ζ3ζ1712-ζ3ζ175+ζ3ζ173-ζ1712-ζ175 | complex faithful |
| ρ16 | 4 | 0 | -2 | 0 | 0 | 0 | ζ1716+ζ1713+ζ174+ζ17 | ζ1715+ζ179+ζ178+ζ172 | ζ1714+ζ1712+ζ175+ζ173 | ζ1711+ζ1710+ζ177+ζ176 | -ζ3ζ1714+ζ3ζ1712+ζ3ζ175-ζ3ζ173-ζ1714-ζ173 | -ζ32ζ1711+ζ32ζ1710+ζ32ζ177-ζ32ζ176-ζ1711-ζ176 | ζ32ζ1711-ζ32ζ1710-ζ32ζ177+ζ32ζ176-ζ1710-ζ177 | ζ32ζ1715-ζ32ζ179-ζ32ζ178+ζ32ζ172-ζ179-ζ178 | ζ3ζ1716-ζ3ζ1713-ζ3ζ174+ζ3ζ17-ζ1713-ζ174 | -ζ32ζ1715+ζ32ζ179+ζ32ζ178-ζ32ζ172-ζ1715-ζ172 | ζ3ζ1714-ζ3ζ1712-ζ3ζ175+ζ3ζ173-ζ1712-ζ175 | -ζ3ζ1716+ζ3ζ1713+ζ3ζ174-ζ3ζ17-ζ1716-ζ17 | complex faithful |
| ρ17 | 4 | 0 | -2 | 0 | 0 | 0 | ζ1715+ζ179+ζ178+ζ172 | ζ1716+ζ1713+ζ174+ζ17 | ζ1711+ζ1710+ζ177+ζ176 | ζ1714+ζ1712+ζ175+ζ173 | -ζ32ζ1711+ζ32ζ1710+ζ32ζ177-ζ32ζ176-ζ1711-ζ176 | ζ3ζ1714-ζ3ζ1712-ζ3ζ175+ζ3ζ173-ζ1712-ζ175 | -ζ3ζ1714+ζ3ζ1712+ζ3ζ175-ζ3ζ173-ζ1714-ζ173 | -ζ3ζ1716+ζ3ζ1713+ζ3ζ174-ζ3ζ17-ζ1716-ζ17 | ζ32ζ1715-ζ32ζ179-ζ32ζ178+ζ32ζ172-ζ179-ζ178 | ζ3ζ1716-ζ3ζ1713-ζ3ζ174+ζ3ζ17-ζ1713-ζ174 | ζ32ζ1711-ζ32ζ1710-ζ32ζ177+ζ32ζ176-ζ1710-ζ177 | -ζ32ζ1715+ζ32ζ179+ζ32ζ178-ζ32ζ172-ζ1715-ζ172 | complex faithful |
| ρ18 | 4 | 0 | -2 | 0 | 0 | 0 | ζ1714+ζ1712+ζ175+ζ173 | ζ1711+ζ1710+ζ177+ζ176 | ζ1715+ζ179+ζ178+ζ172 | ζ1716+ζ1713+ζ174+ζ17 | -ζ32ζ1715+ζ32ζ179+ζ32ζ178-ζ32ζ172-ζ1715-ζ172 | ζ3ζ1716-ζ3ζ1713-ζ3ζ174+ζ3ζ17-ζ1713-ζ174 | -ζ3ζ1716+ζ3ζ1713+ζ3ζ174-ζ3ζ17-ζ1716-ζ17 | ζ32ζ1711-ζ32ζ1710-ζ32ζ177+ζ32ζ176-ζ1710-ζ177 | ζ3ζ1714-ζ3ζ1712-ζ3ζ175+ζ3ζ173-ζ1712-ζ175 | -ζ32ζ1711+ζ32ζ1710+ζ32ζ177-ζ32ζ176-ζ1711-ζ176 | ζ32ζ1715-ζ32ζ179-ζ32ζ178+ζ32ζ172-ζ179-ζ178 | -ζ3ζ1714+ζ3ζ1712+ζ3ζ175-ζ3ζ173-ζ1714-ζ173 | complex faithful |
►On 51 points
Generators in S51
(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 39 17 48)(3 26 33 44)(4 13 49 40)(5 51 14 36)(6 38 30 32)(7 25 46 28)(8 12 11 24)(9 50 27 20)(10 37 43 16)(15 23 21 47)(18 35)(19 22 34 31)(29 45 41 42)
G:=sub<Sym(51)| (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), (2,39,17,48)(3,26,33,44)(4,13,49,40)(5,51,14,36)(6,38,30,32)(7,25,46,28)(8,12,11,24)(9,50,27,20)(10,37,43,16)(15,23,21,47)(18,35)(19,22,34,31)(29,45,41,42)>;
G:=Group( (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), (2,39,17,48)(3,26,33,44)(4,13,49,40)(5,51,14,36)(6,38,30,32)(7,25,46,28)(8,12,11,24)(9,50,27,20)(10,37,43,16)(15,23,21,47)(18,35)(19,22,34,31)(29,45,41,42) );
G=PermutationGroup([[(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)], [(2,39,17,48),(3,26,33,44),(4,13,49,40),(5,51,14,36),(6,38,30,32),(7,25,46,28),(8,12,11,24),(9,50,27,20),(10,37,43,16),(15,23,21,47),(18,35),(19,22,34,31),(29,45,41,42)]])
C51⋊C4 is a maximal subgroup of
S3×C17⋊C4
C51⋊C4 is a maximal quotient of C51⋊3C8
Matrix representation of C51⋊C4 ►in GL4(𝔽409) generated by
| 88 | 293 | 232 | 354 |
| 55 | 46 | 286 | 190 |
| 219 | 386 | 33 | 208 |
| 201 | 125 | 234 | 348 |
| 1 | 0 | 0 | 0 |
| 1 | 303 | 304 | 320 |
| 16 | 195 | 170 | 106 |
| 319 | 328 | 213 | 344 |
G:=sub<GL(4,GF(409))| [88,55,219,201,293,46,386,125,232,286,33,234,354,190,208,348],[1,1,16,319,0,303,195,328,0,304,170,213,0,320,106,344] >;
C51⋊C4 in GAP, Magma, Sage, TeX
C_{51}\rtimes C_4 % in TeX
G:=Group("C51:C4"); // GroupNames label
G:=SmallGroup(204,6);
// by ID
G=gap.SmallGroup(204,6);
# by ID
G:=PCGroup([4,-2,-2,-3,-17,8,98,771,1543]);
// Polycyclic
G:=Group<a,b|a^51=b^4=1,b*a*b^-1=a^47>;
// generators/relations
Export
Subgroup lattice of C51⋊C4 in TeX
Character table of C51⋊C4 in TeX