direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C4×C29⋊C4, C29⋊C42, C116⋊2C4, Dic29⋊2C4, D58.4C22, D29.(C2×C4), C58.3(C2×C4), (C4×D29).6C2, C2.2(C2×C29⋊C4), (C2×C29⋊C4).2C2, SmallGroup(464,30)
Series: Derived ►Chief ►Lower central ►Upper central
| C29 — C4×C29⋊C4 |
Generators and relations for C4×C29⋊C4
G = < a,b,c | a4=b29=c4=1, ab=ba, ac=ca, cbc-1=b17 >
Smallest permutation representation of C4×C29⋊C4
►On 116 points
Generators in S116
(1 88 30 59)(2 89 31 60)(3 90 32 61)(4 91 33 62)(5 92 34 63)(6 93 35 64)(7 94 36 65)(8 95 37 66)(9 96 38 67)(10 97 39 68)(11 98 40 69)(12 99 41 70)(13 100 42 71)(14 101 43 72)(15 102 44 73)(16 103 45 74)(17 104 46 75)(18 105 47 76)(19 106 48 77)(20 107 49 78)(21 108 50 79)(22 109 51 80)(23 110 52 81)(24 111 53 82)(25 112 54 83)(26 113 55 84)(27 114 56 85)(28 115 57 86)(29 116 58 87)
(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 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87)(88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116)
(1 88 30 59)(2 100 58 76)(3 112 57 64)(4 95 56 81)(5 107 55 69)(6 90 54 86)(7 102 53 74)(8 114 52 62)(9 97 51 79)(10 109 50 67)(11 92 49 84)(12 104 48 72)(13 116 47 60)(14 99 46 77)(15 111 45 65)(16 94 44 82)(17 106 43 70)(18 89 42 87)(19 101 41 75)(20 113 40 63)(21 96 39 80)(22 108 38 68)(23 91 37 85)(24 103 36 73)(25 115 35 61)(26 98 34 78)(27 110 33 66)(28 93 32 83)(29 105 31 71)
G:=sub<Sym(116)| (1,88,30,59)(2,89,31,60)(3,90,32,61)(4,91,33,62)(5,92,34,63)(6,93,35,64)(7,94,36,65)(8,95,37,66)(9,96,38,67)(10,97,39,68)(11,98,40,69)(12,99,41,70)(13,100,42,71)(14,101,43,72)(15,102,44,73)(16,103,45,74)(17,104,46,75)(18,105,47,76)(19,106,48,77)(20,107,49,78)(21,108,50,79)(22,109,51,80)(23,110,52,81)(24,111,53,82)(25,112,54,83)(26,113,55,84)(27,114,56,85)(28,115,57,86)(29,116,58,87), (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)(59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87)(88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116), (1,88,30,59)(2,100,58,76)(3,112,57,64)(4,95,56,81)(5,107,55,69)(6,90,54,86)(7,102,53,74)(8,114,52,62)(9,97,51,79)(10,109,50,67)(11,92,49,84)(12,104,48,72)(13,116,47,60)(14,99,46,77)(15,111,45,65)(16,94,44,82)(17,106,43,70)(18,89,42,87)(19,101,41,75)(20,113,40,63)(21,96,39,80)(22,108,38,68)(23,91,37,85)(24,103,36,73)(25,115,35,61)(26,98,34,78)(27,110,33,66)(28,93,32,83)(29,105,31,71)>;
G:=Group( (1,88,30,59)(2,89,31,60)(3,90,32,61)(4,91,33,62)(5,92,34,63)(6,93,35,64)(7,94,36,65)(8,95,37,66)(9,96,38,67)(10,97,39,68)(11,98,40,69)(12,99,41,70)(13,100,42,71)(14,101,43,72)(15,102,44,73)(16,103,45,74)(17,104,46,75)(18,105,47,76)(19,106,48,77)(20,107,49,78)(21,108,50,79)(22,109,51,80)(23,110,52,81)(24,111,53,82)(25,112,54,83)(26,113,55,84)(27,114,56,85)(28,115,57,86)(29,116,58,87), (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)(59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87)(88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116), (1,88,30,59)(2,100,58,76)(3,112,57,64)(4,95,56,81)(5,107,55,69)(6,90,54,86)(7,102,53,74)(8,114,52,62)(9,97,51,79)(10,109,50,67)(11,92,49,84)(12,104,48,72)(13,116,47,60)(14,99,46,77)(15,111,45,65)(16,94,44,82)(17,106,43,70)(18,89,42,87)(19,101,41,75)(20,113,40,63)(21,96,39,80)(22,108,38,68)(23,91,37,85)(24,103,36,73)(25,115,35,61)(26,98,34,78)(27,110,33,66)(28,93,32,83)(29,105,31,71) );
G=PermutationGroup([[(1,88,30,59),(2,89,31,60),(3,90,32,61),(4,91,33,62),(5,92,34,63),(6,93,35,64),(7,94,36,65),(8,95,37,66),(9,96,38,67),(10,97,39,68),(11,98,40,69),(12,99,41,70),(13,100,42,71),(14,101,43,72),(15,102,44,73),(16,103,45,74),(17,104,46,75),(18,105,47,76),(19,106,48,77),(20,107,49,78),(21,108,50,79),(22,109,51,80),(23,110,52,81),(24,111,53,82),(25,112,54,83),(26,113,55,84),(27,114,56,85),(28,115,57,86),(29,116,58,87)], [(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),(59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87),(88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116)], [(1,88,30,59),(2,100,58,76),(3,112,57,64),(4,95,56,81),(5,107,55,69),(6,90,54,86),(7,102,53,74),(8,114,52,62),(9,97,51,79),(10,109,50,67),(11,92,49,84),(12,104,48,72),(13,116,47,60),(14,99,46,77),(15,111,45,65),(16,94,44,82),(17,106,43,70),(18,89,42,87),(19,101,41,75),(20,113,40,63),(21,96,39,80),(22,108,38,68),(23,91,37,85),(24,103,36,73),(25,115,35,61),(26,98,34,78),(27,110,33,66),(28,93,32,83),(29,105,31,71)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | ··· | 4L | 29A | ··· | 29G | 58A | ··· | 58G | 116A | ··· | 116N |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 29 | ··· | 29 | 58 | ··· | 58 | 116 | ··· | 116 |
| size | 1 | 1 | 29 | 29 | 1 | 1 | 29 | ··· | 29 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 |
| type | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C4 | C4 | C4 | C29⋊C4 | C2×C29⋊C4 | C4×C29⋊C4 |
| kernel | C4×C29⋊C4 | C4×D29 | C2×C29⋊C4 | Dic29 | C116 | C29⋊C4 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 8 | 7 | 7 | 14 |
Matrix representation of C4×C29⋊C4 ►in GL5(𝔽233)
| 89 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 127 | 120 | 85 | 232 |
| 0 | 37 | 12 | 125 | 163 |
| 0 | 214 | 150 | 57 | 24 |
| 0 | 153 | 51 | 8 | 52 |
| 232 | 0 | 0 | 0 | 0 |
| 0 | 216 | 168 | 112 | 14 |
| 0 | 13 | 9 | 214 | 47 |
| 0 | 172 | 187 | 200 | 182 |
| 0 | 167 | 186 | 5 | 41 |
G:=sub<GL(5,GF(233))| [89,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,127,37,214,153,0,120,12,150,51,0,85,125,57,8,0,232,163,24,52],[232,0,0,0,0,0,216,13,172,167,0,168,9,187,186,0,112,214,200,5,0,14,47,182,41] >;
C4×C29⋊C4 in GAP, Magma, Sage, TeX
C_4\times C_{29}\rtimes C_4 % in TeX
G:=Group("C4xC29:C4"); // GroupNames label
G:=SmallGroup(464,30);
// by ID
G=gap.SmallGroup(464,30);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-29,20,46,4804,2814]);
// Polycyclic
G:=Group<a,b,c|a^4=b^29=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^17>;
// generators/relations
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