metacyclic, supersoluble, monomial
Aliases: C63⋊1C6, D63⋊1C3, C32.D21, 3- 1+2⋊D7, C9⋊(C3×D7), C7⋊5(C9⋊C6), (C3×C21).1S3, C3.3(C3×D21), C21.13(C3×S3), (C7×3- 1+2)⋊1C2, SmallGroup(378,39)
Series: Derived ►Chief ►Lower central ►Upper central
| C63 — D63⋊C3 |
Generators and relations for D63⋊C3
G = < a,b,c | a63=b2=c3=1, bab=a-1, cac-1=a22, bc=cb >
Smallest permutation representation of D63⋊C3
►On 63 points
Generators in S63
(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 52 53 54 55 56 57 58 59 60 61 62 63)
(2 63)(3 62)(4 61)(5 60)(6 59)(7 58)(8 57)(9 56)(10 55)(11 54)(12 53)(13 52)(14 51)(15 50)(16 49)(17 48)(18 47)(19 46)(20 45)(21 44)(22 43)(23 42)(24 41)(25 40)(26 39)(27 38)(28 37)(29 36)(30 35)(31 34)(32 33)
(2 44 23)(3 24 45)(5 47 26)(6 27 48)(8 50 29)(9 30 51)(11 53 32)(12 33 54)(14 56 35)(15 36 57)(17 59 38)(18 39 60)(20 62 41)(21 42 63)
G:=sub<Sym(63)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63), (2,63)(3,62)(4,61)(5,60)(6,59)(7,58)(8,57)(9,56)(10,55)(11,54)(12,53)(13,52)(14,51)(15,50)(16,49)(17,48)(18,47)(19,46)(20,45)(21,44)(22,43)(23,42)(24,41)(25,40)(26,39)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33), (2,44,23)(3,24,45)(5,47,26)(6,27,48)(8,50,29)(9,30,51)(11,53,32)(12,33,54)(14,56,35)(15,36,57)(17,59,38)(18,39,60)(20,62,41)(21,42,63)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63), (2,63)(3,62)(4,61)(5,60)(6,59)(7,58)(8,57)(9,56)(10,55)(11,54)(12,53)(13,52)(14,51)(15,50)(16,49)(17,48)(18,47)(19,46)(20,45)(21,44)(22,43)(23,42)(24,41)(25,40)(26,39)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33), (2,44,23)(3,24,45)(5,47,26)(6,27,48)(8,50,29)(9,30,51)(11,53,32)(12,33,54)(14,56,35)(15,36,57)(17,59,38)(18,39,60)(20,62,41)(21,42,63) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63)], [(2,63),(3,62),(4,61),(5,60),(6,59),(7,58),(8,57),(9,56),(10,55),(11,54),(12,53),(13,52),(14,51),(15,50),(16,49),(17,48),(18,47),(19,46),(20,45),(21,44),(22,43),(23,42),(24,41),(25,40),(26,39),(27,38),(28,37),(29,36),(30,35),(31,34),(32,33)], [(2,44,23),(3,24,45),(5,47,26),(6,27,48),(8,50,29),(9,30,51),(11,53,32),(12,33,54),(14,56,35),(15,36,57),(17,59,38),(18,39,60),(20,62,41),(21,42,63)]])
43 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 6A | 6B | 7A | 7B | 7C | 9A | 9B | 9C | 21A | ··· | 21F | 21G | ··· | 21L | 63A | ··· | 63R |
| order | 1 | 2 | 3 | 3 | 3 | 6 | 6 | 7 | 7 | 7 | 9 | 9 | 9 | 21 | ··· | 21 | 21 | ··· | 21 | 63 | ··· | 63 |
| size | 1 | 63 | 2 | 3 | 3 | 63 | 63 | 2 | 2 | 2 | 6 | 6 | 6 | 2 | ··· | 2 | 6 | ··· | 6 | 6 | ··· | 6 |
43 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | 6 |
| type | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C3 | C6 | S3 | D7 | C3×S3 | C3×D7 | D21 | C3×D21 | C9⋊C6 | D63⋊C3 |
| kernel | D63⋊C3 | C7×3- 1+2 | D63 | C63 | C3×C21 | 3- 1+2 | C21 | C9 | C32 | C3 | C7 | C1 |
| # reps | 1 | 1 | 2 | 2 | 1 | 3 | 2 | 6 | 6 | 12 | 1 | 6 |
Matrix representation of D63⋊C3 ►in GL6(𝔽127)
| 0 | 0 | 54 | 90 | 0 | 0 |
| 0 | 0 | 37 | 17 | 0 | 0 |
| 37 | 37 | 37 | 37 | 20 | 91 |
| 110 | 110 | 110 | 110 | 56 | 20 |
| 0 | 17 | 90 | 0 | 17 | 90 |
| 90 | 0 | 90 | 0 | 17 | 90 |
| 110 | 90 | 0 | 0 | 0 | 0 |
| 73 | 17 | 0 | 0 | 0 | 0 |
| 17 | 17 | 17 | 17 | 107 | 71 |
| 54 | 54 | 54 | 54 | 71 | 91 |
| 0 | 110 | 73 | 0 | 110 | 73 |
| 0 | 110 | 0 | 110 | 110 | 73 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 126 | 0 | 0 |
| 0 | 0 | 1 | 126 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 1 |
| 126 | 126 | 126 | 0 | 126 | 126 |
G:=sub<GL(6,GF(127))| [0,0,37,110,0,90,0,0,37,110,17,0,54,37,37,110,90,90,90,17,37,110,0,0,0,0,20,56,17,17,0,0,91,20,90,90],[110,73,17,54,0,0,90,17,17,54,110,110,0,0,17,54,73,0,0,0,17,54,0,110,0,0,107,71,110,110,0,0,71,91,73,73],[1,0,0,0,0,126,0,1,0,0,0,126,0,0,0,1,0,126,0,0,126,126,1,0,0,0,0,0,0,126,0,0,0,0,1,126] >;
D63⋊C3 in GAP, Magma, Sage, TeX
D_{63}\rtimes C_3 % in TeX
G:=Group("D63:C3"); // GroupNames label
G:=SmallGroup(378,39);
// by ID
G=gap.SmallGroup(378,39);
# by ID
G:=PCGroup([5,-2,-3,-3,-7,-3,2072,997,642,2163,6304]);
// Polycyclic
G:=Group<a,b,c|a^63=b^2=c^3=1,b*a*b=a^-1,c*a*c^-1=a^22,b*c=c*b>;
// generators/relations
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