metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.D17, C34.11D4, C22⋊Dic17, C22.7D34, (C2×C34)⋊4C4, C17⋊3(C22⋊C4), C34.16(C2×C4), (C2×Dic17)⋊2C2, C2.3(C17⋊D4), (C2×C34).7C22, (C22×C34).2C2, C2.5(C2×Dic17), SmallGroup(272,19)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.D17
G = < a,b,c,d,e | a2=b2=c2=d17=1, e2=b, ab=ba, eae-1=ac=ca, ad=da, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >
Smallest permutation representation of C23.D17
►On 136 points
Generators in S136
(69 86)(70 87)(71 88)(72 89)(73 90)(74 91)(75 92)(76 93)(77 94)(78 95)(79 96)(80 97)(81 98)(82 99)(83 100)(84 101)(85 102)(103 120)(104 121)(105 122)(106 123)(107 124)(108 125)(109 126)(110 127)(111 128)(112 129)(113 130)(114 131)(115 132)(116 133)(117 134)(118 135)(119 136)
(1 40)(2 41)(3 42)(4 43)(5 44)(6 45)(7 46)(8 47)(9 48)(10 49)(11 50)(12 51)(13 35)(14 36)(15 37)(16 38)(17 39)(18 53)(19 54)(20 55)(21 56)(22 57)(23 58)(24 59)(25 60)(26 61)(27 62)(28 63)(29 64)(30 65)(31 66)(32 67)(33 68)(34 52)(69 103)(70 104)(71 105)(72 106)(73 107)(74 108)(75 109)(76 110)(77 111)(78 112)(79 113)(80 114)(81 115)(82 116)(83 117)(84 118)(85 119)(86 120)(87 121)(88 122)(89 123)(90 124)(91 125)(92 126)(93 127)(94 128)(95 129)(96 130)(97 131)(98 132)(99 133)(100 134)(101 135)(102 136)
(1 24)(2 25)(3 26)(4 27)(5 28)(6 29)(7 30)(8 31)(9 32)(10 33)(11 34)(12 18)(13 19)(14 20)(15 21)(16 22)(17 23)(35 54)(36 55)(37 56)(38 57)(39 58)(40 59)(41 60)(42 61)(43 62)(44 63)(45 64)(46 65)(47 66)(48 67)(49 68)(50 52)(51 53)(69 86)(70 87)(71 88)(72 89)(73 90)(74 91)(75 92)(76 93)(77 94)(78 95)(79 96)(80 97)(81 98)(82 99)(83 100)(84 101)(85 102)(103 120)(104 121)(105 122)(106 123)(107 124)(108 125)(109 126)(110 127)(111 128)(112 129)(113 130)(114 131)(115 132)(116 133)(117 134)(118 135)(119 136)
(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)
(1 119 40 85)(2 118 41 84)(3 117 42 83)(4 116 43 82)(5 115 44 81)(6 114 45 80)(7 113 46 79)(8 112 47 78)(9 111 48 77)(10 110 49 76)(11 109 50 75)(12 108 51 74)(13 107 35 73)(14 106 36 72)(15 105 37 71)(16 104 38 70)(17 103 39 69)(18 125 53 91)(19 124 54 90)(20 123 55 89)(21 122 56 88)(22 121 57 87)(23 120 58 86)(24 136 59 102)(25 135 60 101)(26 134 61 100)(27 133 62 99)(28 132 63 98)(29 131 64 97)(30 130 65 96)(31 129 66 95)(32 128 67 94)(33 127 68 93)(34 126 52 92)
G:=sub<Sym(136)| (69,86)(70,87)(71,88)(72,89)(73,90)(74,91)(75,92)(76,93)(77,94)(78,95)(79,96)(80,97)(81,98)(82,99)(83,100)(84,101)(85,102)(103,120)(104,121)(105,122)(106,123)(107,124)(108,125)(109,126)(110,127)(111,128)(112,129)(113,130)(114,131)(115,132)(116,133)(117,134)(118,135)(119,136), (1,40)(2,41)(3,42)(4,43)(5,44)(6,45)(7,46)(8,47)(9,48)(10,49)(11,50)(12,51)(13,35)(14,36)(15,37)(16,38)(17,39)(18,53)(19,54)(20,55)(21,56)(22,57)(23,58)(24,59)(25,60)(26,61)(27,62)(28,63)(29,64)(30,65)(31,66)(32,67)(33,68)(34,52)(69,103)(70,104)(71,105)(72,106)(73,107)(74,108)(75,109)(76,110)(77,111)(78,112)(79,113)(80,114)(81,115)(82,116)(83,117)(84,118)(85,119)(86,120)(87,121)(88,122)(89,123)(90,124)(91,125)(92,126)(93,127)(94,128)(95,129)(96,130)(97,131)(98,132)(99,133)(100,134)(101,135)(102,136), (1,24)(2,25)(3,26)(4,27)(5,28)(6,29)(7,30)(8,31)(9,32)(10,33)(11,34)(12,18)(13,19)(14,20)(15,21)(16,22)(17,23)(35,54)(36,55)(37,56)(38,57)(39,58)(40,59)(41,60)(42,61)(43,62)(44,63)(45,64)(46,65)(47,66)(48,67)(49,68)(50,52)(51,53)(69,86)(70,87)(71,88)(72,89)(73,90)(74,91)(75,92)(76,93)(77,94)(78,95)(79,96)(80,97)(81,98)(82,99)(83,100)(84,101)(85,102)(103,120)(104,121)(105,122)(106,123)(107,124)(108,125)(109,126)(110,127)(111,128)(112,129)(113,130)(114,131)(115,132)(116,133)(117,134)(118,135)(119,136), (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), (1,119,40,85)(2,118,41,84)(3,117,42,83)(4,116,43,82)(5,115,44,81)(6,114,45,80)(7,113,46,79)(8,112,47,78)(9,111,48,77)(10,110,49,76)(11,109,50,75)(12,108,51,74)(13,107,35,73)(14,106,36,72)(15,105,37,71)(16,104,38,70)(17,103,39,69)(18,125,53,91)(19,124,54,90)(20,123,55,89)(21,122,56,88)(22,121,57,87)(23,120,58,86)(24,136,59,102)(25,135,60,101)(26,134,61,100)(27,133,62,99)(28,132,63,98)(29,131,64,97)(30,130,65,96)(31,129,66,95)(32,128,67,94)(33,127,68,93)(34,126,52,92)>;
G:=Group( (69,86)(70,87)(71,88)(72,89)(73,90)(74,91)(75,92)(76,93)(77,94)(78,95)(79,96)(80,97)(81,98)(82,99)(83,100)(84,101)(85,102)(103,120)(104,121)(105,122)(106,123)(107,124)(108,125)(109,126)(110,127)(111,128)(112,129)(113,130)(114,131)(115,132)(116,133)(117,134)(118,135)(119,136), (1,40)(2,41)(3,42)(4,43)(5,44)(6,45)(7,46)(8,47)(9,48)(10,49)(11,50)(12,51)(13,35)(14,36)(15,37)(16,38)(17,39)(18,53)(19,54)(20,55)(21,56)(22,57)(23,58)(24,59)(25,60)(26,61)(27,62)(28,63)(29,64)(30,65)(31,66)(32,67)(33,68)(34,52)(69,103)(70,104)(71,105)(72,106)(73,107)(74,108)(75,109)(76,110)(77,111)(78,112)(79,113)(80,114)(81,115)(82,116)(83,117)(84,118)(85,119)(86,120)(87,121)(88,122)(89,123)(90,124)(91,125)(92,126)(93,127)(94,128)(95,129)(96,130)(97,131)(98,132)(99,133)(100,134)(101,135)(102,136), (1,24)(2,25)(3,26)(4,27)(5,28)(6,29)(7,30)(8,31)(9,32)(10,33)(11,34)(12,18)(13,19)(14,20)(15,21)(16,22)(17,23)(35,54)(36,55)(37,56)(38,57)(39,58)(40,59)(41,60)(42,61)(43,62)(44,63)(45,64)(46,65)(47,66)(48,67)(49,68)(50,52)(51,53)(69,86)(70,87)(71,88)(72,89)(73,90)(74,91)(75,92)(76,93)(77,94)(78,95)(79,96)(80,97)(81,98)(82,99)(83,100)(84,101)(85,102)(103,120)(104,121)(105,122)(106,123)(107,124)(108,125)(109,126)(110,127)(111,128)(112,129)(113,130)(114,131)(115,132)(116,133)(117,134)(118,135)(119,136), (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), (1,119,40,85)(2,118,41,84)(3,117,42,83)(4,116,43,82)(5,115,44,81)(6,114,45,80)(7,113,46,79)(8,112,47,78)(9,111,48,77)(10,110,49,76)(11,109,50,75)(12,108,51,74)(13,107,35,73)(14,106,36,72)(15,105,37,71)(16,104,38,70)(17,103,39,69)(18,125,53,91)(19,124,54,90)(20,123,55,89)(21,122,56,88)(22,121,57,87)(23,120,58,86)(24,136,59,102)(25,135,60,101)(26,134,61,100)(27,133,62,99)(28,132,63,98)(29,131,64,97)(30,130,65,96)(31,129,66,95)(32,128,67,94)(33,127,68,93)(34,126,52,92) );
G=PermutationGroup([[(69,86),(70,87),(71,88),(72,89),(73,90),(74,91),(75,92),(76,93),(77,94),(78,95),(79,96),(80,97),(81,98),(82,99),(83,100),(84,101),(85,102),(103,120),(104,121),(105,122),(106,123),(107,124),(108,125),(109,126),(110,127),(111,128),(112,129),(113,130),(114,131),(115,132),(116,133),(117,134),(118,135),(119,136)], [(1,40),(2,41),(3,42),(4,43),(5,44),(6,45),(7,46),(8,47),(9,48),(10,49),(11,50),(12,51),(13,35),(14,36),(15,37),(16,38),(17,39),(18,53),(19,54),(20,55),(21,56),(22,57),(23,58),(24,59),(25,60),(26,61),(27,62),(28,63),(29,64),(30,65),(31,66),(32,67),(33,68),(34,52),(69,103),(70,104),(71,105),(72,106),(73,107),(74,108),(75,109),(76,110),(77,111),(78,112),(79,113),(80,114),(81,115),(82,116),(83,117),(84,118),(85,119),(86,120),(87,121),(88,122),(89,123),(90,124),(91,125),(92,126),(93,127),(94,128),(95,129),(96,130),(97,131),(98,132),(99,133),(100,134),(101,135),(102,136)], [(1,24),(2,25),(3,26),(4,27),(5,28),(6,29),(7,30),(8,31),(9,32),(10,33),(11,34),(12,18),(13,19),(14,20),(15,21),(16,22),(17,23),(35,54),(36,55),(37,56),(38,57),(39,58),(40,59),(41,60),(42,61),(43,62),(44,63),(45,64),(46,65),(47,66),(48,67),(49,68),(50,52),(51,53),(69,86),(70,87),(71,88),(72,89),(73,90),(74,91),(75,92),(76,93),(77,94),(78,95),(79,96),(80,97),(81,98),(82,99),(83,100),(84,101),(85,102),(103,120),(104,121),(105,122),(106,123),(107,124),(108,125),(109,126),(110,127),(111,128),(112,129),(113,130),(114,131),(115,132),(116,133),(117,134),(118,135),(119,136)], [(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)], [(1,119,40,85),(2,118,41,84),(3,117,42,83),(4,116,43,82),(5,115,44,81),(6,114,45,80),(7,113,46,79),(8,112,47,78),(9,111,48,77),(10,110,49,76),(11,109,50,75),(12,108,51,74),(13,107,35,73),(14,106,36,72),(15,105,37,71),(16,104,38,70),(17,103,39,69),(18,125,53,91),(19,124,54,90),(20,123,55,89),(21,122,56,88),(22,121,57,87),(23,120,58,86),(24,136,59,102),(25,135,60,101),(26,134,61,100),(27,133,62,99),(28,132,63,98),(29,131,64,97),(30,130,65,96),(31,129,66,95),(32,128,67,94),(33,127,68,93),(34,126,52,92)]])
74 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 17A | ··· | 17H | 34A | ··· | 34BD |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 17 | ··· | 17 | 34 | ··· | 34 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 34 | 34 | 34 | 34 | 2 | ··· | 2 | 2 | ··· | 2 |
74 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | ||
| image | C1 | C2 | C2 | C4 | D4 | D17 | Dic17 | D34 | C17⋊D4 |
| kernel | C23.D17 | C2×Dic17 | C22×C34 | C2×C34 | C34 | C23 | C22 | C22 | C2 |
| # reps | 1 | 2 | 1 | 4 | 2 | 8 | 16 | 8 | 32 |
Matrix representation of C23.D17 ►in GL3(𝔽137) generated by
| 136 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 7 | 136 |
| 136 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 136 | 0 |
| 0 | 0 | 136 |
| 1 | 0 | 0 |
| 0 | 88 | 0 |
| 0 | 83 | 123 |
| 37 | 0 | 0 |
| 0 | 23 | 13 |
| 0 | 128 | 114 |
G:=sub<GL(3,GF(137))| [136,0,0,0,1,7,0,0,136],[136,0,0,0,1,0,0,0,1],[1,0,0,0,136,0,0,0,136],[1,0,0,0,88,83,0,0,123],[37,0,0,0,23,128,0,13,114] >;
C23.D17 in GAP, Magma, Sage, TeX
C_2^3.D_{17} % in TeX
G:=Group("C2^3.D17"); // GroupNames label
G:=SmallGroup(272,19);
// by ID
G=gap.SmallGroup(272,19);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-17,20,101,6404]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^17=1,e^2=b,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations
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