metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C116⋊1C4, D29.Q8, D29.1D4, Dic29⋊3C4, D58.5C22, C29⋊(C4⋊C4), C4⋊(C29⋊C4), C58.4(C2×C4), (C4×D29).4C2, (C2×C29⋊C4).C2, C2.5(C2×C29⋊C4), SmallGroup(464,31)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C116⋊C4
G = < a,b | a116=b4=1, bab-1=a75 >
Smallest permutation representation of C116⋊C4
►On 116 points
Generators in S116
(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)
(2 100 58 76)(3 83 115 35)(4 66 56 110)(5 49 113 69)(6 32 54 28)(7 15 111 103)(8 114 52 62)(9 97 109 21)(10 80 50 96)(11 63 107 55)(12 46 48 14)(13 29 105 89)(16 94 44 82)(17 77 101 41)(18 60 42 116)(19 43 99 75)(20 26 40 34)(22 108 38 68)(23 91 95 27)(24 74 36 102)(25 57 93 61)(30 88)(31 71 87 47)(33 37 85 81)(39 51 79 67)(45 65 73 53)(64 90 112 86)(70 104 106 72)(78 84 98 92)
G:=sub<Sym(116)| (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), (2,100,58,76)(3,83,115,35)(4,66,56,110)(5,49,113,69)(6,32,54,28)(7,15,111,103)(8,114,52,62)(9,97,109,21)(10,80,50,96)(11,63,107,55)(12,46,48,14)(13,29,105,89)(16,94,44,82)(17,77,101,41)(18,60,42,116)(19,43,99,75)(20,26,40,34)(22,108,38,68)(23,91,95,27)(24,74,36,102)(25,57,93,61)(30,88)(31,71,87,47)(33,37,85,81)(39,51,79,67)(45,65,73,53)(64,90,112,86)(70,104,106,72)(78,84,98,92)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116), (2,100,58,76)(3,83,115,35)(4,66,56,110)(5,49,113,69)(6,32,54,28)(7,15,111,103)(8,114,52,62)(9,97,109,21)(10,80,50,96)(11,63,107,55)(12,46,48,14)(13,29,105,89)(16,94,44,82)(17,77,101,41)(18,60,42,116)(19,43,99,75)(20,26,40,34)(22,108,38,68)(23,91,95,27)(24,74,36,102)(25,57,93,61)(30,88)(31,71,87,47)(33,37,85,81)(39,51,79,67)(45,65,73,53)(64,90,112,86)(70,104,106,72)(78,84,98,92) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116)], [(2,100,58,76),(3,83,115,35),(4,66,56,110),(5,49,113,69),(6,32,54,28),(7,15,111,103),(8,114,52,62),(9,97,109,21),(10,80,50,96),(11,63,107,55),(12,46,48,14),(13,29,105,89),(16,94,44,82),(17,77,101,41),(18,60,42,116),(19,43,99,75),(20,26,40,34),(22,108,38,68),(23,91,95,27),(24,74,36,102),(25,57,93,61),(30,88),(31,71,87,47),(33,37,85,81),(39,51,79,67),(45,65,73,53),(64,90,112,86),(70,104,106,72),(78,84,98,92)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | ··· | 4F | 29A | ··· | 29G | 58A | ··· | 58G | 116A | ··· | 116N |
| order | 1 | 2 | 2 | 2 | 4 | 4 | ··· | 4 | 29 | ··· | 29 | 58 | ··· | 58 | 116 | ··· | 116 |
| size | 1 | 1 | 29 | 29 | 2 | 58 | ··· | 58 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | - | + | + | |||
| image | C1 | C2 | C2 | C4 | C4 | D4 | Q8 | C29⋊C4 | C2×C29⋊C4 | C116⋊C4 |
| kernel | C116⋊C4 | C4×D29 | C2×C29⋊C4 | Dic29 | C116 | D29 | D29 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 7 | 7 | 14 |
Matrix representation of C116⋊C4 ►in GL4(𝔽233) generated by
| 88 | 100 | 222 | 49 |
| 153 | 225 | 137 | 207 |
| 170 | 42 | 183 | 92 |
| 188 | 82 | 127 | 26 |
| 118 | 4 | 226 | 218 |
| 116 | 200 | 94 | 104 |
| 119 | 26 | 70 | 221 |
| 202 | 212 | 210 | 78 |
G:=sub<GL(4,GF(233))| [88,153,170,188,100,225,42,82,222,137,183,127,49,207,92,26],[118,116,119,202,4,200,26,212,226,94,70,210,218,104,221,78] >;
C116⋊C4 in GAP, Magma, Sage, TeX
C_{116}\rtimes C_4 % in TeX
G:=Group("C116:C4"); // GroupNames label
G:=SmallGroup(464,31);
// by ID
G=gap.SmallGroup(464,31);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-29,20,101,46,4804,2814]);
// Polycyclic
G:=Group<a,b|a^116=b^4=1,b*a*b^-1=a^75>;
// generators/relations
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