direct product, abelian, monomial, 2-elementary
Aliases: C22×C34, SmallGroup(136,15)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C22×C34 |
| C1 — C22×C34 |
| C1 — C22×C34 |
Generators and relations for C22×C34
G = < a,b,c | a2=b2=c34=1, ab=ba, ac=ca, bc=cb >
Smallest permutation representation of C22×C34
►Regular action on 136 points
Generators in S136
(1 87)(2 88)(3 89)(4 90)(5 91)(6 92)(7 93)(8 94)(9 95)(10 96)(11 97)(12 98)(13 99)(14 100)(15 101)(16 102)(17 69)(18 70)(19 71)(20 72)(21 73)(22 74)(23 75)(24 76)(25 77)(26 78)(27 79)(28 80)(29 81)(30 82)(31 83)(32 84)(33 85)(34 86)(35 110)(36 111)(37 112)(38 113)(39 114)(40 115)(41 116)(42 117)(43 118)(44 119)(45 120)(46 121)(47 122)(48 123)(49 124)(50 125)(51 126)(52 127)(53 128)(54 129)(55 130)(56 131)(57 132)(58 133)(59 134)(60 135)(61 136)(62 103)(63 104)(64 105)(65 106)(66 107)(67 108)(68 109)
(1 55)(2 56)(3 57)(4 58)(5 59)(6 60)(7 61)(8 62)(9 63)(10 64)(11 65)(12 66)(13 67)(14 68)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)(21 41)(22 42)(23 43)(24 44)(25 45)(26 46)(27 47)(28 48)(29 49)(30 50)(31 51)(32 52)(33 53)(34 54)(69 112)(70 113)(71 114)(72 115)(73 116)(74 117)(75 118)(76 119)(77 120)(78 121)(79 122)(80 123)(81 124)(82 125)(83 126)(84 127)(85 128)(86 129)(87 130)(88 131)(89 132)(90 133)(91 134)(92 135)(93 136)(94 103)(95 104)(96 105)(97 106)(98 107)(99 108)(100 109)(101 110)(102 111)
(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)
G:=sub<Sym(136)| (1,87)(2,88)(3,89)(4,90)(5,91)(6,92)(7,93)(8,94)(9,95)(10,96)(11,97)(12,98)(13,99)(14,100)(15,101)(16,102)(17,69)(18,70)(19,71)(20,72)(21,73)(22,74)(23,75)(24,76)(25,77)(26,78)(27,79)(28,80)(29,81)(30,82)(31,83)(32,84)(33,85)(34,86)(35,110)(36,111)(37,112)(38,113)(39,114)(40,115)(41,116)(42,117)(43,118)(44,119)(45,120)(46,121)(47,122)(48,123)(49,124)(50,125)(51,126)(52,127)(53,128)(54,129)(55,130)(56,131)(57,132)(58,133)(59,134)(60,135)(61,136)(62,103)(63,104)(64,105)(65,106)(66,107)(67,108)(68,109), (1,55)(2,56)(3,57)(4,58)(5,59)(6,60)(7,61)(8,62)(9,63)(10,64)(11,65)(12,66)(13,67)(14,68)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(21,41)(22,42)(23,43)(24,44)(25,45)(26,46)(27,47)(28,48)(29,49)(30,50)(31,51)(32,52)(33,53)(34,54)(69,112)(70,113)(71,114)(72,115)(73,116)(74,117)(75,118)(76,119)(77,120)(78,121)(79,122)(80,123)(81,124)(82,125)(83,126)(84,127)(85,128)(86,129)(87,130)(88,131)(89,132)(90,133)(91,134)(92,135)(93,136)(94,103)(95,104)(96,105)(97,106)(98,107)(99,108)(100,109)(101,110)(102,111), (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)>;
G:=Group( (1,87)(2,88)(3,89)(4,90)(5,91)(6,92)(7,93)(8,94)(9,95)(10,96)(11,97)(12,98)(13,99)(14,100)(15,101)(16,102)(17,69)(18,70)(19,71)(20,72)(21,73)(22,74)(23,75)(24,76)(25,77)(26,78)(27,79)(28,80)(29,81)(30,82)(31,83)(32,84)(33,85)(34,86)(35,110)(36,111)(37,112)(38,113)(39,114)(40,115)(41,116)(42,117)(43,118)(44,119)(45,120)(46,121)(47,122)(48,123)(49,124)(50,125)(51,126)(52,127)(53,128)(54,129)(55,130)(56,131)(57,132)(58,133)(59,134)(60,135)(61,136)(62,103)(63,104)(64,105)(65,106)(66,107)(67,108)(68,109), (1,55)(2,56)(3,57)(4,58)(5,59)(6,60)(7,61)(8,62)(9,63)(10,64)(11,65)(12,66)(13,67)(14,68)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(21,41)(22,42)(23,43)(24,44)(25,45)(26,46)(27,47)(28,48)(29,49)(30,50)(31,51)(32,52)(33,53)(34,54)(69,112)(70,113)(71,114)(72,115)(73,116)(74,117)(75,118)(76,119)(77,120)(78,121)(79,122)(80,123)(81,124)(82,125)(83,126)(84,127)(85,128)(86,129)(87,130)(88,131)(89,132)(90,133)(91,134)(92,135)(93,136)(94,103)(95,104)(96,105)(97,106)(98,107)(99,108)(100,109)(101,110)(102,111), (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) );
G=PermutationGroup([[(1,87),(2,88),(3,89),(4,90),(5,91),(6,92),(7,93),(8,94),(9,95),(10,96),(11,97),(12,98),(13,99),(14,100),(15,101),(16,102),(17,69),(18,70),(19,71),(20,72),(21,73),(22,74),(23,75),(24,76),(25,77),(26,78),(27,79),(28,80),(29,81),(30,82),(31,83),(32,84),(33,85),(34,86),(35,110),(36,111),(37,112),(38,113),(39,114),(40,115),(41,116),(42,117),(43,118),(44,119),(45,120),(46,121),(47,122),(48,123),(49,124),(50,125),(51,126),(52,127),(53,128),(54,129),(55,130),(56,131),(57,132),(58,133),(59,134),(60,135),(61,136),(62,103),(63,104),(64,105),(65,106),(66,107),(67,108),(68,109)], [(1,55),(2,56),(3,57),(4,58),(5,59),(6,60),(7,61),(8,62),(9,63),(10,64),(11,65),(12,66),(13,67),(14,68),(15,35),(16,36),(17,37),(18,38),(19,39),(20,40),(21,41),(22,42),(23,43),(24,44),(25,45),(26,46),(27,47),(28,48),(29,49),(30,50),(31,51),(32,52),(33,53),(34,54),(69,112),(70,113),(71,114),(72,115),(73,116),(74,117),(75,118),(76,119),(77,120),(78,121),(79,122),(80,123),(81,124),(82,125),(83,126),(84,127),(85,128),(86,129),(87,130),(88,131),(89,132),(90,133),(91,134),(92,135),(93,136),(94,103),(95,104),(96,105),(97,106),(98,107),(99,108),(100,109),(101,110),(102,111)], [(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)]])
C22×C34 is a maximal subgroup of
C23.D17
136 conjugacy classes
| class | 1 | 2A | ··· | 2G | 17A | ··· | 17P | 34A | ··· | 34DH |
| order | 1 | 2 | ··· | 2 | 17 | ··· | 17 | 34 | ··· | 34 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
136 irreducible representations
| dim | 1 | 1 | 1 | 1 |
| type | + | + | ||
| image | C1 | C2 | C17 | C34 |
| kernel | C22×C34 | C2×C34 | C23 | C22 |
| # reps | 1 | 7 | 16 | 112 |
Matrix representation of C22×C34 ►in GL3(𝔽103) generated by
| 1 | 0 | 0 |
| 0 | 102 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 102 |
| 31 | 0 | 0 |
| 0 | 27 | 0 |
| 0 | 0 | 22 |
G:=sub<GL(3,GF(103))| [1,0,0,0,102,0,0,0,1],[1,0,0,0,1,0,0,0,102],[31,0,0,0,27,0,0,0,22] >;
C22×C34 in GAP, Magma, Sage, TeX
C_2^2\times C_{34} % in TeX
G:=Group("C2^2xC34"); // GroupNames label
G:=SmallGroup(136,15);
// by ID
G=gap.SmallGroup(136,15);
# by ID
G:=PCGroup([4,-2,-2,-2,-17]);
// Polycyclic
G:=Group<a,b,c|a^2=b^2=c^34=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations
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