metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D34⋊1C4, C2.2D68, C34.6D4, C22.6D34, (C2×C68)⋊1C2, (C2×C4)⋊1D17, C2.5(C4×D17), C17⋊2(C22⋊C4), C34.12(C2×C4), (C2×Dic17)⋊1C2, C2.2(C17⋊D4), (C2×C34).6C22, (C22×D17).1C2, SmallGroup(272,14)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D34⋊C4
G = < a,b,c | a34=b2=c4=1, bab=a-1, ac=ca, cbc-1=a17b >
Smallest permutation representation of D34⋊C4
►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 52)(2 51)(3 50)(4 49)(5 48)(6 47)(7 46)(8 45)(9 44)(10 43)(11 42)(12 41)(13 40)(14 39)(15 38)(16 37)(17 36)(18 35)(19 68)(20 67)(21 66)(22 65)(23 64)(24 63)(25 62)(26 61)(27 60)(28 59)(29 58)(30 57)(31 56)(32 55)(33 54)(34 53)(69 124)(70 123)(71 122)(72 121)(73 120)(74 119)(75 118)(76 117)(77 116)(78 115)(79 114)(80 113)(81 112)(82 111)(83 110)(84 109)(85 108)(86 107)(87 106)(88 105)(89 104)(90 103)(91 136)(92 135)(93 134)(94 133)(95 132)(96 131)(97 130)(98 129)(99 128)(100 127)(101 126)(102 125)
(1 136 53 75)(2 103 54 76)(3 104 55 77)(4 105 56 78)(5 106 57 79)(6 107 58 80)(7 108 59 81)(8 109 60 82)(9 110 61 83)(10 111 62 84)(11 112 63 85)(12 113 64 86)(13 114 65 87)(14 115 66 88)(15 116 67 89)(16 117 68 90)(17 118 35 91)(18 119 36 92)(19 120 37 93)(20 121 38 94)(21 122 39 95)(22 123 40 96)(23 124 41 97)(24 125 42 98)(25 126 43 99)(26 127 44 100)(27 128 45 101)(28 129 46 102)(29 130 47 69)(30 131 48 70)(31 132 49 71)(32 133 50 72)(33 134 51 73)(34 135 52 74)
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), (1,52)(2,51)(3,50)(4,49)(5,48)(6,47)(7,46)(8,45)(9,44)(10,43)(11,42)(12,41)(13,40)(14,39)(15,38)(16,37)(17,36)(18,35)(19,68)(20,67)(21,66)(22,65)(23,64)(24,63)(25,62)(26,61)(27,60)(28,59)(29,58)(30,57)(31,56)(32,55)(33,54)(34,53)(69,124)(70,123)(71,122)(72,121)(73,120)(74,119)(75,118)(76,117)(77,116)(78,115)(79,114)(80,113)(81,112)(82,111)(83,110)(84,109)(85,108)(86,107)(87,106)(88,105)(89,104)(90,103)(91,136)(92,135)(93,134)(94,133)(95,132)(96,131)(97,130)(98,129)(99,128)(100,127)(101,126)(102,125), (1,136,53,75)(2,103,54,76)(3,104,55,77)(4,105,56,78)(5,106,57,79)(6,107,58,80)(7,108,59,81)(8,109,60,82)(9,110,61,83)(10,111,62,84)(11,112,63,85)(12,113,64,86)(13,114,65,87)(14,115,66,88)(15,116,67,89)(16,117,68,90)(17,118,35,91)(18,119,36,92)(19,120,37,93)(20,121,38,94)(21,122,39,95)(22,123,40,96)(23,124,41,97)(24,125,42,98)(25,126,43,99)(26,127,44,100)(27,128,45,101)(28,129,46,102)(29,130,47,69)(30,131,48,70)(31,132,49,71)(32,133,50,72)(33,134,51,73)(34,135,52,74)>;
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), (1,52)(2,51)(3,50)(4,49)(5,48)(6,47)(7,46)(8,45)(9,44)(10,43)(11,42)(12,41)(13,40)(14,39)(15,38)(16,37)(17,36)(18,35)(19,68)(20,67)(21,66)(22,65)(23,64)(24,63)(25,62)(26,61)(27,60)(28,59)(29,58)(30,57)(31,56)(32,55)(33,54)(34,53)(69,124)(70,123)(71,122)(72,121)(73,120)(74,119)(75,118)(76,117)(77,116)(78,115)(79,114)(80,113)(81,112)(82,111)(83,110)(84,109)(85,108)(86,107)(87,106)(88,105)(89,104)(90,103)(91,136)(92,135)(93,134)(94,133)(95,132)(96,131)(97,130)(98,129)(99,128)(100,127)(101,126)(102,125), (1,136,53,75)(2,103,54,76)(3,104,55,77)(4,105,56,78)(5,106,57,79)(6,107,58,80)(7,108,59,81)(8,109,60,82)(9,110,61,83)(10,111,62,84)(11,112,63,85)(12,113,64,86)(13,114,65,87)(14,115,66,88)(15,116,67,89)(16,117,68,90)(17,118,35,91)(18,119,36,92)(19,120,37,93)(20,121,38,94)(21,122,39,95)(22,123,40,96)(23,124,41,97)(24,125,42,98)(25,126,43,99)(26,127,44,100)(27,128,45,101)(28,129,46,102)(29,130,47,69)(30,131,48,70)(31,132,49,71)(32,133,50,72)(33,134,51,73)(34,135,52,74) );
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)], [(1,52),(2,51),(3,50),(4,49),(5,48),(6,47),(7,46),(8,45),(9,44),(10,43),(11,42),(12,41),(13,40),(14,39),(15,38),(16,37),(17,36),(18,35),(19,68),(20,67),(21,66),(22,65),(23,64),(24,63),(25,62),(26,61),(27,60),(28,59),(29,58),(30,57),(31,56),(32,55),(33,54),(34,53),(69,124),(70,123),(71,122),(72,121),(73,120),(74,119),(75,118),(76,117),(77,116),(78,115),(79,114),(80,113),(81,112),(82,111),(83,110),(84,109),(85,108),(86,107),(87,106),(88,105),(89,104),(90,103),(91,136),(92,135),(93,134),(94,133),(95,132),(96,131),(97,130),(98,129),(99,128),(100,127),(101,126),(102,125)], [(1,136,53,75),(2,103,54,76),(3,104,55,77),(4,105,56,78),(5,106,57,79),(6,107,58,80),(7,108,59,81),(8,109,60,82),(9,110,61,83),(10,111,62,84),(11,112,63,85),(12,113,64,86),(13,114,65,87),(14,115,66,88),(15,116,67,89),(16,117,68,90),(17,118,35,91),(18,119,36,92),(19,120,37,93),(20,121,38,94),(21,122,39,95),(22,123,40,96),(23,124,41,97),(24,125,42,98),(25,126,43,99),(26,127,44,100),(27,128,45,101),(28,129,46,102),(29,130,47,69),(30,131,48,70),(31,132,49,71),(32,133,50,72),(33,134,51,73),(34,135,52,74)]])
74 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 17A | ··· | 17H | 34A | ··· | 34X | 68A | ··· | 68AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 17 | ··· | 17 | 34 | ··· | 34 | 68 | ··· | 68 |
| size | 1 | 1 | 1 | 1 | 34 | 34 | 2 | 2 | 34 | 34 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
74 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C4 | D4 | D17 | D34 | C4×D17 | D68 | C17⋊D4 |
| kernel | D34⋊C4 | C2×Dic17 | C2×C68 | C22×D17 | D34 | C34 | C2×C4 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 2 | 8 | 8 | 16 | 16 | 16 |
Matrix representation of D34⋊C4 ►in GL3(𝔽137) generated by
| 1 | 0 | 0 |
| 0 | 116 | 116 |
| 0 | 21 | 34 |
| 1 | 0 | 0 |
| 0 | 116 | 116 |
| 0 | 34 | 21 |
| 100 | 0 | 0 |
| 0 | 129 | 101 |
| 0 | 36 | 8 |
G:=sub<GL(3,GF(137))| [1,0,0,0,116,21,0,116,34],[1,0,0,0,116,34,0,116,21],[100,0,0,0,129,36,0,101,8] >;
D34⋊C4 in GAP, Magma, Sage, TeX
D_{34}\rtimes C_4 % in TeX
G:=Group("D34:C4"); // GroupNames label
G:=SmallGroup(272,14);
// by ID
G=gap.SmallGroup(272,14);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-17,101,26,6404]);
// Polycyclic
G:=Group<a,b,c|a^34=b^2=c^4=1,b*a*b=a^-1,a*c=c*a,c*b*c^-1=a^17*b>;
// generators/relations
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