metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D68⋊4C2, Q8⋊2D17, C4.7D34, C34.8C23, C68.7C22, D34.3C22, Dic17.9C22, (C4×D17)⋊3C2, C17⋊3(C4○D4), (Q8×C17)⋊3C2, C2.9(C22×D17), SmallGroup(272,43)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D68⋊C2
G = < a,b,c | a68=b2=c2=1, bab=a-1, cac=a33, cbc=a66b >
Smallest permutation representation of D68⋊C2
►On 136 points
Generators in S136
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136)
(1 68)(2 67)(3 66)(4 65)(5 64)(6 63)(7 62)(8 61)(9 60)(10 59)(11 58)(12 57)(13 56)(14 55)(15 54)(16 53)(17 52)(18 51)(19 50)(20 49)(21 48)(22 47)(23 46)(24 45)(25 44)(26 43)(27 42)(28 41)(29 40)(30 39)(31 38)(32 37)(33 36)(34 35)(69 94)(70 93)(71 92)(72 91)(73 90)(74 89)(75 88)(76 87)(77 86)(78 85)(79 84)(80 83)(81 82)(95 136)(96 135)(97 134)(98 133)(99 132)(100 131)(101 130)(102 129)(103 128)(104 127)(105 126)(106 125)(107 124)(108 123)(109 122)(110 121)(111 120)(112 119)(113 118)(114 117)(115 116)
(1 99)(2 132)(3 97)(4 130)(5 95)(6 128)(7 93)(8 126)(9 91)(10 124)(11 89)(12 122)(13 87)(14 120)(15 85)(16 118)(17 83)(18 116)(19 81)(20 114)(21 79)(22 112)(23 77)(24 110)(25 75)(26 108)(27 73)(28 106)(29 71)(30 104)(31 69)(32 102)(33 135)(34 100)(35 133)(36 98)(37 131)(38 96)(39 129)(40 94)(41 127)(42 92)(43 125)(44 90)(45 123)(46 88)(47 121)(48 86)(49 119)(50 84)(51 117)(52 82)(53 115)(54 80)(55 113)(56 78)(57 111)(58 76)(59 109)(60 74)(61 107)(62 72)(63 105)(64 70)(65 103)(66 136)(67 101)(68 134)
G:=sub<Sym(136)| (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)(69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136), (1,68)(2,67)(3,66)(4,65)(5,64)(6,63)(7,62)(8,61)(9,60)(10,59)(11,58)(12,57)(13,56)(14,55)(15,54)(16,53)(17,52)(18,51)(19,50)(20,49)(21,48)(22,47)(23,46)(24,45)(25,44)(26,43)(27,42)(28,41)(29,40)(30,39)(31,38)(32,37)(33,36)(34,35)(69,94)(70,93)(71,92)(72,91)(73,90)(74,89)(75,88)(76,87)(77,86)(78,85)(79,84)(80,83)(81,82)(95,136)(96,135)(97,134)(98,133)(99,132)(100,131)(101,130)(102,129)(103,128)(104,127)(105,126)(106,125)(107,124)(108,123)(109,122)(110,121)(111,120)(112,119)(113,118)(114,117)(115,116), (1,99)(2,132)(3,97)(4,130)(5,95)(6,128)(7,93)(8,126)(9,91)(10,124)(11,89)(12,122)(13,87)(14,120)(15,85)(16,118)(17,83)(18,116)(19,81)(20,114)(21,79)(22,112)(23,77)(24,110)(25,75)(26,108)(27,73)(28,106)(29,71)(30,104)(31,69)(32,102)(33,135)(34,100)(35,133)(36,98)(37,131)(38,96)(39,129)(40,94)(41,127)(42,92)(43,125)(44,90)(45,123)(46,88)(47,121)(48,86)(49,119)(50,84)(51,117)(52,82)(53,115)(54,80)(55,113)(56,78)(57,111)(58,76)(59,109)(60,74)(61,107)(62,72)(63,105)(64,70)(65,103)(66,136)(67,101)(68,134)>;
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)(69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136), (1,68)(2,67)(3,66)(4,65)(5,64)(6,63)(7,62)(8,61)(9,60)(10,59)(11,58)(12,57)(13,56)(14,55)(15,54)(16,53)(17,52)(18,51)(19,50)(20,49)(21,48)(22,47)(23,46)(24,45)(25,44)(26,43)(27,42)(28,41)(29,40)(30,39)(31,38)(32,37)(33,36)(34,35)(69,94)(70,93)(71,92)(72,91)(73,90)(74,89)(75,88)(76,87)(77,86)(78,85)(79,84)(80,83)(81,82)(95,136)(96,135)(97,134)(98,133)(99,132)(100,131)(101,130)(102,129)(103,128)(104,127)(105,126)(106,125)(107,124)(108,123)(109,122)(110,121)(111,120)(112,119)(113,118)(114,117)(115,116), (1,99)(2,132)(3,97)(4,130)(5,95)(6,128)(7,93)(8,126)(9,91)(10,124)(11,89)(12,122)(13,87)(14,120)(15,85)(16,118)(17,83)(18,116)(19,81)(20,114)(21,79)(22,112)(23,77)(24,110)(25,75)(26,108)(27,73)(28,106)(29,71)(30,104)(31,69)(32,102)(33,135)(34,100)(35,133)(36,98)(37,131)(38,96)(39,129)(40,94)(41,127)(42,92)(43,125)(44,90)(45,123)(46,88)(47,121)(48,86)(49,119)(50,84)(51,117)(52,82)(53,115)(54,80)(55,113)(56,78)(57,111)(58,76)(59,109)(60,74)(61,107)(62,72)(63,105)(64,70)(65,103)(66,136)(67,101)(68,134) );
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),(69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136)], [(1,68),(2,67),(3,66),(4,65),(5,64),(6,63),(7,62),(8,61),(9,60),(10,59),(11,58),(12,57),(13,56),(14,55),(15,54),(16,53),(17,52),(18,51),(19,50),(20,49),(21,48),(22,47),(23,46),(24,45),(25,44),(26,43),(27,42),(28,41),(29,40),(30,39),(31,38),(32,37),(33,36),(34,35),(69,94),(70,93),(71,92),(72,91),(73,90),(74,89),(75,88),(76,87),(77,86),(78,85),(79,84),(80,83),(81,82),(95,136),(96,135),(97,134),(98,133),(99,132),(100,131),(101,130),(102,129),(103,128),(104,127),(105,126),(106,125),(107,124),(108,123),(109,122),(110,121),(111,120),(112,119),(113,118),(114,117),(115,116)], [(1,99),(2,132),(3,97),(4,130),(5,95),(6,128),(7,93),(8,126),(9,91),(10,124),(11,89),(12,122),(13,87),(14,120),(15,85),(16,118),(17,83),(18,116),(19,81),(20,114),(21,79),(22,112),(23,77),(24,110),(25,75),(26,108),(27,73),(28,106),(29,71),(30,104),(31,69),(32,102),(33,135),(34,100),(35,133),(36,98),(37,131),(38,96),(39,129),(40,94),(41,127),(42,92),(43,125),(44,90),(45,123),(46,88),(47,121),(48,86),(49,119),(50,84),(51,117),(52,82),(53,115),(54,80),(55,113),(56,78),(57,111),(58,76),(59,109),(60,74),(61,107),(62,72),(63,105),(64,70),(65,103),(66,136),(67,101),(68,134)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 17A | ··· | 17H | 34A | ··· | 34H | 68A | ··· | 68X |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 17 | ··· | 17 | 34 | ··· | 34 | 68 | ··· | 68 |
| size | 1 | 1 | 34 | 34 | 34 | 2 | 2 | 2 | 17 | 17 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C4○D4 | D17 | D34 | D68⋊C2 |
| kernel | D68⋊C2 | C4×D17 | D68 | Q8×C17 | C17 | Q8 | C4 | C1 |
| # reps | 1 | 3 | 3 | 1 | 2 | 8 | 24 | 8 |
Matrix representation of D68⋊C2 ►in GL4(𝔽137) generated by
| 106 | 21 | 0 | 0 |
| 116 | 107 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 136 | 0 |
| 106 | 21 | 0 | 0 |
| 13 | 31 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 136 | 0 | 0 | 0 |
| 13 | 1 | 0 | 0 |
| 0 | 0 | 0 | 100 |
| 0 | 0 | 37 | 0 |
G:=sub<GL(4,GF(137))| [106,116,0,0,21,107,0,0,0,0,0,136,0,0,1,0],[106,13,0,0,21,31,0,0,0,0,0,1,0,0,1,0],[136,13,0,0,0,1,0,0,0,0,0,37,0,0,100,0] >;
D68⋊C2 in GAP, Magma, Sage, TeX
D_{68}\rtimes C_2 % in TeX
G:=Group("D68:C2"); // GroupNames label
G:=SmallGroup(272,43);
// by ID
G=gap.SmallGroup(272,43);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-17,46,182,97,42,6404]);
// Polycyclic
G:=Group<a,b,c|a^68=b^2=c^2=1,b*a*b=a^-1,c*a*c=a^33,c*b*c=a^66*b>;
// generators/relations
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