direct product, p-group, abelian, monomial
Aliases: C2×C82, SmallGroup(128,179)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
| C1 — C2×C82 |
| C1 — C2×C82 |
| C1 — C2×C82 |
Generators and relations for C2×C82
G = < a,b,c | a2=b8=c8=1, ab=ba, ac=ca, bc=cb >
Subgroups: 140, all normal (6 characteristic)
C1, C2, C4, C22, C22, C8, C2×C4, C23, C42, C42, C2×C8, C22×C4, C4×C8, C2×C42, C22×C8, C82, C2×C4×C8, C2×C82
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, C42, C2×C8, C22×C4, C4×C8, C2×C42, C22×C8, C82, C2×C4×C8, C2×C82
►Regular action on 128 points
Generators in S128
(1 95)(2 96)(3 89)(4 90)(5 91)(6 92)(7 93)(8 94)(9 126)(10 127)(11 128)(12 121)(13 122)(14 123)(15 124)(16 125)(17 103)(18 104)(19 97)(20 98)(21 99)(22 100)(23 101)(24 102)(25 83)(26 84)(27 85)(28 86)(29 87)(30 88)(31 81)(32 82)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)(49 57)(50 58)(51 59)(52 60)(53 61)(54 62)(55 63)(56 64)(65 73)(66 74)(67 75)(68 76)(69 77)(70 78)(71 79)(72 80)(105 113)(106 114)(107 115)(108 116)(109 117)(110 118)(111 119)(112 120)
(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)(113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128)
(1 66 55 121 105 37 87 103)(2 67 56 122 106 38 88 104)(3 68 49 123 107 39 81 97)(4 69 50 124 108 40 82 98)(5 70 51 125 109 33 83 99)(6 71 52 126 110 34 84 100)(7 72 53 127 111 35 85 101)(8 65 54 128 112 36 86 102)(9 118 42 26 22 92 79 60)(10 119 43 27 23 93 80 61)(11 120 44 28 24 94 73 62)(12 113 45 29 17 95 74 63)(13 114 46 30 18 96 75 64)(14 115 47 31 19 89 76 57)(15 116 48 32 20 90 77 58)(16 117 41 25 21 91 78 59)
G:=sub<Sym(128)| (1,95)(2,96)(3,89)(4,90)(5,91)(6,92)(7,93)(8,94)(9,126)(10,127)(11,128)(12,121)(13,122)(14,123)(15,124)(16,125)(17,103)(18,104)(19,97)(20,98)(21,99)(22,100)(23,101)(24,102)(25,83)(26,84)(27,85)(28,86)(29,87)(30,88)(31,81)(32,82)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48)(49,57)(50,58)(51,59)(52,60)(53,61)(54,62)(55,63)(56,64)(65,73)(66,74)(67,75)(68,76)(69,77)(70,78)(71,79)(72,80)(105,113)(106,114)(107,115)(108,116)(109,117)(110,118)(111,119)(112,120), (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)(113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128), (1,66,55,121,105,37,87,103)(2,67,56,122,106,38,88,104)(3,68,49,123,107,39,81,97)(4,69,50,124,108,40,82,98)(5,70,51,125,109,33,83,99)(6,71,52,126,110,34,84,100)(7,72,53,127,111,35,85,101)(8,65,54,128,112,36,86,102)(9,118,42,26,22,92,79,60)(10,119,43,27,23,93,80,61)(11,120,44,28,24,94,73,62)(12,113,45,29,17,95,74,63)(13,114,46,30,18,96,75,64)(14,115,47,31,19,89,76,57)(15,116,48,32,20,90,77,58)(16,117,41,25,21,91,78,59)>;
G:=Group( (1,95)(2,96)(3,89)(4,90)(5,91)(6,92)(7,93)(8,94)(9,126)(10,127)(11,128)(12,121)(13,122)(14,123)(15,124)(16,125)(17,103)(18,104)(19,97)(20,98)(21,99)(22,100)(23,101)(24,102)(25,83)(26,84)(27,85)(28,86)(29,87)(30,88)(31,81)(32,82)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48)(49,57)(50,58)(51,59)(52,60)(53,61)(54,62)(55,63)(56,64)(65,73)(66,74)(67,75)(68,76)(69,77)(70,78)(71,79)(72,80)(105,113)(106,114)(107,115)(108,116)(109,117)(110,118)(111,119)(112,120), (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)(113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128), (1,66,55,121,105,37,87,103)(2,67,56,122,106,38,88,104)(3,68,49,123,107,39,81,97)(4,69,50,124,108,40,82,98)(5,70,51,125,109,33,83,99)(6,71,52,126,110,34,84,100)(7,72,53,127,111,35,85,101)(8,65,54,128,112,36,86,102)(9,118,42,26,22,92,79,60)(10,119,43,27,23,93,80,61)(11,120,44,28,24,94,73,62)(12,113,45,29,17,95,74,63)(13,114,46,30,18,96,75,64)(14,115,47,31,19,89,76,57)(15,116,48,32,20,90,77,58)(16,117,41,25,21,91,78,59) );
G=PermutationGroup([[(1,95),(2,96),(3,89),(4,90),(5,91),(6,92),(7,93),(8,94),(9,126),(10,127),(11,128),(12,121),(13,122),(14,123),(15,124),(16,125),(17,103),(18,104),(19,97),(20,98),(21,99),(22,100),(23,101),(24,102),(25,83),(26,84),(27,85),(28,86),(29,87),(30,88),(31,81),(32,82),(33,41),(34,42),(35,43),(36,44),(37,45),(38,46),(39,47),(40,48),(49,57),(50,58),(51,59),(52,60),(53,61),(54,62),(55,63),(56,64),(65,73),(66,74),(67,75),(68,76),(69,77),(70,78),(71,79),(72,80),(105,113),(106,114),(107,115),(108,116),(109,117),(110,118),(111,119),(112,120)], [(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),(113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128)], [(1,66,55,121,105,37,87,103),(2,67,56,122,106,38,88,104),(3,68,49,123,107,39,81,97),(4,69,50,124,108,40,82,98),(5,70,51,125,109,33,83,99),(6,71,52,126,110,34,84,100),(7,72,53,127,111,35,85,101),(8,65,54,128,112,36,86,102),(9,118,42,26,22,92,79,60),(10,119,43,27,23,93,80,61),(11,120,44,28,24,94,73,62),(12,113,45,29,17,95,74,63),(13,114,46,30,18,96,75,64),(14,115,47,31,19,89,76,57),(15,116,48,32,20,90,77,58),(16,117,41,25,21,91,78,59)]])
128 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4X | 8A | ··· | 8CR |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
128 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | + | |||
| image | C1 | C2 | C2 | C4 | C4 | C8 |
| kernel | C2×C82 | C82 | C2×C4×C8 | C4×C8 | C22×C8 | C2×C8 |
| # reps | 1 | 4 | 3 | 12 | 12 | 96 |
Matrix representation of C2×C82 ►in GL3(𝔽17) generated by
| 1 | 0 | 0 |
| 0 | 16 | 0 |
| 0 | 0 | 16 |
| 13 | 0 | 0 |
| 0 | 13 | 0 |
| 0 | 0 | 8 |
| 15 | 0 | 0 |
| 0 | 2 | 0 |
| 0 | 0 | 13 |
G:=sub<GL(3,GF(17))| [1,0,0,0,16,0,0,0,16],[13,0,0,0,13,0,0,0,8],[15,0,0,0,2,0,0,0,13] >;
C2×C82 in GAP, Magma, Sage, TeX
C_2\times C_8^2
% in TeX
G:=Group("C2xC8^2"); // GroupNames label
G:=SmallGroup(128,179);
// by ID
G=gap.SmallGroup(128,179);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,56,120,136,172]);
// Polycyclic
G:=Group<a,b,c|a^2=b^8=c^8=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations