direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×Q8×D11, C22.8C24, C44.22C23, Dic22⋊9C22, D22.10C23, Dic11.5C23, C22⋊2(C2×Q8), (Q8×C22)⋊5C2, C11⋊2(C22×Q8), (C2×C4).61D22, (Q8×C11)⋊5C22, C2.9(C23×D11), (C2×Dic22)⋊13C2, (C2×C22).66C23, (C2×C44).46C22, C4.22(C22×D11), (C4×D11).13C22, C22.31(C22×D11), (C2×Dic11).44C22, (C22×D11).36C22, (C2×C4×D11).6C2, SmallGroup(352,180)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×Q8×D11
G = < a,b,c,d,e | a2=b4=d11=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >
Subgroups: 746 in 156 conjugacy classes, 97 normal (10 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, Q8, Q8, C23, C11, C22×C4, C2×Q8, C2×Q8, D11, C22, C22, C22×Q8, Dic11, C44, D22, C2×C22, Dic22, C4×D11, C2×Dic11, C2×C44, Q8×C11, C22×D11, C2×Dic22, C2×C4×D11, Q8×D11, Q8×C22, C2×Q8×D11
Quotients: C1, C2, C22, Q8, C23, C2×Q8, C24, D11, C22×Q8, D22, C22×D11, Q8×D11, C23×D11, C2×Q8×D11
►On 176 points
Generators in S176
(1 98)(2 99)(3 89)(4 90)(5 91)(6 92)(7 93)(8 94)(9 95)(10 96)(11 97)(12 100)(13 101)(14 102)(15 103)(16 104)(17 105)(18 106)(19 107)(20 108)(21 109)(22 110)(23 111)(24 112)(25 113)(26 114)(27 115)(28 116)(29 117)(30 118)(31 119)(32 120)(33 121)(34 122)(35 123)(36 124)(37 125)(38 126)(39 127)(40 128)(41 129)(42 130)(43 131)(44 132)(45 133)(46 134)(47 135)(48 136)(49 137)(50 138)(51 139)(52 140)(53 141)(54 142)(55 143)(56 144)(57 145)(58 146)(59 147)(60 148)(61 149)(62 150)(63 151)(64 152)(65 153)(66 154)(67 155)(68 156)(69 157)(70 158)(71 159)(72 160)(73 161)(74 162)(75 163)(76 164)(77 165)(78 166)(79 167)(80 168)(81 169)(82 170)(83 171)(84 172)(85 173)(86 174)(87 175)(88 176)
(1 65 21 54)(2 66 22 55)(3 56 12 45)(4 57 13 46)(5 58 14 47)(6 59 15 48)(7 60 16 49)(8 61 17 50)(9 62 18 51)(10 63 19 52)(11 64 20 53)(23 78 34 67)(24 79 35 68)(25 80 36 69)(26 81 37 70)(27 82 38 71)(28 83 39 72)(29 84 40 73)(30 85 41 74)(31 86 42 75)(32 87 43 76)(33 88 44 77)(89 144 100 133)(90 145 101 134)(91 146 102 135)(92 147 103 136)(93 148 104 137)(94 149 105 138)(95 150 106 139)(96 151 107 140)(97 152 108 141)(98 153 109 142)(99 154 110 143)(111 166 122 155)(112 167 123 156)(113 168 124 157)(114 169 125 158)(115 170 126 159)(116 171 127 160)(117 172 128 161)(118 173 129 162)(119 174 130 163)(120 175 131 164)(121 176 132 165)
(1 131 21 120)(2 132 22 121)(3 122 12 111)(4 123 13 112)(5 124 14 113)(6 125 15 114)(7 126 16 115)(8 127 17 116)(9 128 18 117)(10 129 19 118)(11 130 20 119)(23 89 34 100)(24 90 35 101)(25 91 36 102)(26 92 37 103)(27 93 38 104)(28 94 39 105)(29 95 40 106)(30 96 41 107)(31 97 42 108)(32 98 43 109)(33 99 44 110)(45 155 56 166)(46 156 57 167)(47 157 58 168)(48 158 59 169)(49 159 60 170)(50 160 61 171)(51 161 62 172)(52 162 63 173)(53 163 64 174)(54 164 65 175)(55 165 66 176)(67 144 78 133)(68 145 79 134)(69 146 80 135)(70 147 81 136)(71 148 82 137)(72 149 83 138)(73 150 84 139)(74 151 85 140)(75 152 86 141)(76 153 87 142)(77 154 88 143)
(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)(133 134 135 136 137 138 139 140 141 142 143)(144 145 146 147 148 149 150 151 152 153 154)(155 156 157 158 159 160 161 162 163 164 165)(166 167 168 169 170 171 172 173 174 175 176)
(1 97)(2 96)(3 95)(4 94)(5 93)(6 92)(7 91)(8 90)(9 89)(10 99)(11 98)(12 106)(13 105)(14 104)(15 103)(16 102)(17 101)(18 100)(19 110)(20 109)(21 108)(22 107)(23 117)(24 116)(25 115)(26 114)(27 113)(28 112)(29 111)(30 121)(31 120)(32 119)(33 118)(34 128)(35 127)(36 126)(37 125)(38 124)(39 123)(40 122)(41 132)(42 131)(43 130)(44 129)(45 139)(46 138)(47 137)(48 136)(49 135)(50 134)(51 133)(52 143)(53 142)(54 141)(55 140)(56 150)(57 149)(58 148)(59 147)(60 146)(61 145)(62 144)(63 154)(64 153)(65 152)(66 151)(67 161)(68 160)(69 159)(70 158)(71 157)(72 156)(73 155)(74 165)(75 164)(76 163)(77 162)(78 172)(79 171)(80 170)(81 169)(82 168)(83 167)(84 166)(85 176)(86 175)(87 174)(88 173)
G:=sub<Sym(176)| (1,98)(2,99)(3,89)(4,90)(5,91)(6,92)(7,93)(8,94)(9,95)(10,96)(11,97)(12,100)(13,101)(14,102)(15,103)(16,104)(17,105)(18,106)(19,107)(20,108)(21,109)(22,110)(23,111)(24,112)(25,113)(26,114)(27,115)(28,116)(29,117)(30,118)(31,119)(32,120)(33,121)(34,122)(35,123)(36,124)(37,125)(38,126)(39,127)(40,128)(41,129)(42,130)(43,131)(44,132)(45,133)(46,134)(47,135)(48,136)(49,137)(50,138)(51,139)(52,140)(53,141)(54,142)(55,143)(56,144)(57,145)(58,146)(59,147)(60,148)(61,149)(62,150)(63,151)(64,152)(65,153)(66,154)(67,155)(68,156)(69,157)(70,158)(71,159)(72,160)(73,161)(74,162)(75,163)(76,164)(77,165)(78,166)(79,167)(80,168)(81,169)(82,170)(83,171)(84,172)(85,173)(86,174)(87,175)(88,176), (1,65,21,54)(2,66,22,55)(3,56,12,45)(4,57,13,46)(5,58,14,47)(6,59,15,48)(7,60,16,49)(8,61,17,50)(9,62,18,51)(10,63,19,52)(11,64,20,53)(23,78,34,67)(24,79,35,68)(25,80,36,69)(26,81,37,70)(27,82,38,71)(28,83,39,72)(29,84,40,73)(30,85,41,74)(31,86,42,75)(32,87,43,76)(33,88,44,77)(89,144,100,133)(90,145,101,134)(91,146,102,135)(92,147,103,136)(93,148,104,137)(94,149,105,138)(95,150,106,139)(96,151,107,140)(97,152,108,141)(98,153,109,142)(99,154,110,143)(111,166,122,155)(112,167,123,156)(113,168,124,157)(114,169,125,158)(115,170,126,159)(116,171,127,160)(117,172,128,161)(118,173,129,162)(119,174,130,163)(120,175,131,164)(121,176,132,165), (1,131,21,120)(2,132,22,121)(3,122,12,111)(4,123,13,112)(5,124,14,113)(6,125,15,114)(7,126,16,115)(8,127,17,116)(9,128,18,117)(10,129,19,118)(11,130,20,119)(23,89,34,100)(24,90,35,101)(25,91,36,102)(26,92,37,103)(27,93,38,104)(28,94,39,105)(29,95,40,106)(30,96,41,107)(31,97,42,108)(32,98,43,109)(33,99,44,110)(45,155,56,166)(46,156,57,167)(47,157,58,168)(48,158,59,169)(49,159,60,170)(50,160,61,171)(51,161,62,172)(52,162,63,173)(53,163,64,174)(54,164,65,175)(55,165,66,176)(67,144,78,133)(68,145,79,134)(69,146,80,135)(70,147,81,136)(71,148,82,137)(72,149,83,138)(73,150,84,139)(74,151,85,140)(75,152,86,141)(76,153,87,142)(77,154,88,143), (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)(133,134,135,136,137,138,139,140,141,142,143)(144,145,146,147,148,149,150,151,152,153,154)(155,156,157,158,159,160,161,162,163,164,165)(166,167,168,169,170,171,172,173,174,175,176), (1,97)(2,96)(3,95)(4,94)(5,93)(6,92)(7,91)(8,90)(9,89)(10,99)(11,98)(12,106)(13,105)(14,104)(15,103)(16,102)(17,101)(18,100)(19,110)(20,109)(21,108)(22,107)(23,117)(24,116)(25,115)(26,114)(27,113)(28,112)(29,111)(30,121)(31,120)(32,119)(33,118)(34,128)(35,127)(36,126)(37,125)(38,124)(39,123)(40,122)(41,132)(42,131)(43,130)(44,129)(45,139)(46,138)(47,137)(48,136)(49,135)(50,134)(51,133)(52,143)(53,142)(54,141)(55,140)(56,150)(57,149)(58,148)(59,147)(60,146)(61,145)(62,144)(63,154)(64,153)(65,152)(66,151)(67,161)(68,160)(69,159)(70,158)(71,157)(72,156)(73,155)(74,165)(75,164)(76,163)(77,162)(78,172)(79,171)(80,170)(81,169)(82,168)(83,167)(84,166)(85,176)(86,175)(87,174)(88,173)>;
G:=Group( (1,98)(2,99)(3,89)(4,90)(5,91)(6,92)(7,93)(8,94)(9,95)(10,96)(11,97)(12,100)(13,101)(14,102)(15,103)(16,104)(17,105)(18,106)(19,107)(20,108)(21,109)(22,110)(23,111)(24,112)(25,113)(26,114)(27,115)(28,116)(29,117)(30,118)(31,119)(32,120)(33,121)(34,122)(35,123)(36,124)(37,125)(38,126)(39,127)(40,128)(41,129)(42,130)(43,131)(44,132)(45,133)(46,134)(47,135)(48,136)(49,137)(50,138)(51,139)(52,140)(53,141)(54,142)(55,143)(56,144)(57,145)(58,146)(59,147)(60,148)(61,149)(62,150)(63,151)(64,152)(65,153)(66,154)(67,155)(68,156)(69,157)(70,158)(71,159)(72,160)(73,161)(74,162)(75,163)(76,164)(77,165)(78,166)(79,167)(80,168)(81,169)(82,170)(83,171)(84,172)(85,173)(86,174)(87,175)(88,176), (1,65,21,54)(2,66,22,55)(3,56,12,45)(4,57,13,46)(5,58,14,47)(6,59,15,48)(7,60,16,49)(8,61,17,50)(9,62,18,51)(10,63,19,52)(11,64,20,53)(23,78,34,67)(24,79,35,68)(25,80,36,69)(26,81,37,70)(27,82,38,71)(28,83,39,72)(29,84,40,73)(30,85,41,74)(31,86,42,75)(32,87,43,76)(33,88,44,77)(89,144,100,133)(90,145,101,134)(91,146,102,135)(92,147,103,136)(93,148,104,137)(94,149,105,138)(95,150,106,139)(96,151,107,140)(97,152,108,141)(98,153,109,142)(99,154,110,143)(111,166,122,155)(112,167,123,156)(113,168,124,157)(114,169,125,158)(115,170,126,159)(116,171,127,160)(117,172,128,161)(118,173,129,162)(119,174,130,163)(120,175,131,164)(121,176,132,165), (1,131,21,120)(2,132,22,121)(3,122,12,111)(4,123,13,112)(5,124,14,113)(6,125,15,114)(7,126,16,115)(8,127,17,116)(9,128,18,117)(10,129,19,118)(11,130,20,119)(23,89,34,100)(24,90,35,101)(25,91,36,102)(26,92,37,103)(27,93,38,104)(28,94,39,105)(29,95,40,106)(30,96,41,107)(31,97,42,108)(32,98,43,109)(33,99,44,110)(45,155,56,166)(46,156,57,167)(47,157,58,168)(48,158,59,169)(49,159,60,170)(50,160,61,171)(51,161,62,172)(52,162,63,173)(53,163,64,174)(54,164,65,175)(55,165,66,176)(67,144,78,133)(68,145,79,134)(69,146,80,135)(70,147,81,136)(71,148,82,137)(72,149,83,138)(73,150,84,139)(74,151,85,140)(75,152,86,141)(76,153,87,142)(77,154,88,143), (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)(133,134,135,136,137,138,139,140,141,142,143)(144,145,146,147,148,149,150,151,152,153,154)(155,156,157,158,159,160,161,162,163,164,165)(166,167,168,169,170,171,172,173,174,175,176), (1,97)(2,96)(3,95)(4,94)(5,93)(6,92)(7,91)(8,90)(9,89)(10,99)(11,98)(12,106)(13,105)(14,104)(15,103)(16,102)(17,101)(18,100)(19,110)(20,109)(21,108)(22,107)(23,117)(24,116)(25,115)(26,114)(27,113)(28,112)(29,111)(30,121)(31,120)(32,119)(33,118)(34,128)(35,127)(36,126)(37,125)(38,124)(39,123)(40,122)(41,132)(42,131)(43,130)(44,129)(45,139)(46,138)(47,137)(48,136)(49,135)(50,134)(51,133)(52,143)(53,142)(54,141)(55,140)(56,150)(57,149)(58,148)(59,147)(60,146)(61,145)(62,144)(63,154)(64,153)(65,152)(66,151)(67,161)(68,160)(69,159)(70,158)(71,157)(72,156)(73,155)(74,165)(75,164)(76,163)(77,162)(78,172)(79,171)(80,170)(81,169)(82,168)(83,167)(84,166)(85,176)(86,175)(87,174)(88,173) );
G=PermutationGroup([[(1,98),(2,99),(3,89),(4,90),(5,91),(6,92),(7,93),(8,94),(9,95),(10,96),(11,97),(12,100),(13,101),(14,102),(15,103),(16,104),(17,105),(18,106),(19,107),(20,108),(21,109),(22,110),(23,111),(24,112),(25,113),(26,114),(27,115),(28,116),(29,117),(30,118),(31,119),(32,120),(33,121),(34,122),(35,123),(36,124),(37,125),(38,126),(39,127),(40,128),(41,129),(42,130),(43,131),(44,132),(45,133),(46,134),(47,135),(48,136),(49,137),(50,138),(51,139),(52,140),(53,141),(54,142),(55,143),(56,144),(57,145),(58,146),(59,147),(60,148),(61,149),(62,150),(63,151),(64,152),(65,153),(66,154),(67,155),(68,156),(69,157),(70,158),(71,159),(72,160),(73,161),(74,162),(75,163),(76,164),(77,165),(78,166),(79,167),(80,168),(81,169),(82,170),(83,171),(84,172),(85,173),(86,174),(87,175),(88,176)], [(1,65,21,54),(2,66,22,55),(3,56,12,45),(4,57,13,46),(5,58,14,47),(6,59,15,48),(7,60,16,49),(8,61,17,50),(9,62,18,51),(10,63,19,52),(11,64,20,53),(23,78,34,67),(24,79,35,68),(25,80,36,69),(26,81,37,70),(27,82,38,71),(28,83,39,72),(29,84,40,73),(30,85,41,74),(31,86,42,75),(32,87,43,76),(33,88,44,77),(89,144,100,133),(90,145,101,134),(91,146,102,135),(92,147,103,136),(93,148,104,137),(94,149,105,138),(95,150,106,139),(96,151,107,140),(97,152,108,141),(98,153,109,142),(99,154,110,143),(111,166,122,155),(112,167,123,156),(113,168,124,157),(114,169,125,158),(115,170,126,159),(116,171,127,160),(117,172,128,161),(118,173,129,162),(119,174,130,163),(120,175,131,164),(121,176,132,165)], [(1,131,21,120),(2,132,22,121),(3,122,12,111),(4,123,13,112),(5,124,14,113),(6,125,15,114),(7,126,16,115),(8,127,17,116),(9,128,18,117),(10,129,19,118),(11,130,20,119),(23,89,34,100),(24,90,35,101),(25,91,36,102),(26,92,37,103),(27,93,38,104),(28,94,39,105),(29,95,40,106),(30,96,41,107),(31,97,42,108),(32,98,43,109),(33,99,44,110),(45,155,56,166),(46,156,57,167),(47,157,58,168),(48,158,59,169),(49,159,60,170),(50,160,61,171),(51,161,62,172),(52,162,63,173),(53,163,64,174),(54,164,65,175),(55,165,66,176),(67,144,78,133),(68,145,79,134),(69,146,80,135),(70,147,81,136),(71,148,82,137),(72,149,83,138),(73,150,84,139),(74,151,85,140),(75,152,86,141),(76,153,87,142),(77,154,88,143)], [(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),(133,134,135,136,137,138,139,140,141,142,143),(144,145,146,147,148,149,150,151,152,153,154),(155,156,157,158,159,160,161,162,163,164,165),(166,167,168,169,170,171,172,173,174,175,176)], [(1,97),(2,96),(3,95),(4,94),(5,93),(6,92),(7,91),(8,90),(9,89),(10,99),(11,98),(12,106),(13,105),(14,104),(15,103),(16,102),(17,101),(18,100),(19,110),(20,109),(21,108),(22,107),(23,117),(24,116),(25,115),(26,114),(27,113),(28,112),(29,111),(30,121),(31,120),(32,119),(33,118),(34,128),(35,127),(36,126),(37,125),(38,124),(39,123),(40,122),(41,132),(42,131),(43,130),(44,129),(45,139),(46,138),(47,137),(48,136),(49,135),(50,134),(51,133),(52,143),(53,142),(54,141),(55,140),(56,150),(57,149),(58,148),(59,147),(60,146),(61,145),(62,144),(63,154),(64,153),(65,152),(66,151),(67,161),(68,160),(69,159),(70,158),(71,157),(72,156),(73,155),(74,165),(75,164),(76,163),(77,162),(78,172),(79,171),(80,170),(81,169),(82,168),(83,167),(84,166),(85,176),(86,175),(87,174),(88,173)]])
70 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4F | 4G | ··· | 4L | 11A | ··· | 11E | 22A | ··· | 22O | 44A | ··· | 44AD |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 11 | ··· | 11 | 22 | ··· | 22 | 44 | ··· | 44 |
| size | 1 | 1 | 1 | 1 | 11 | 11 | 11 | 11 | 2 | ··· | 2 | 22 | ··· | 22 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
70 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | - | + | + | + | - |
| image | C1 | C2 | C2 | C2 | C2 | Q8 | D11 | D22 | D22 | Q8×D11 |
| kernel | C2×Q8×D11 | C2×Dic22 | C2×C4×D11 | Q8×D11 | Q8×C22 | D22 | C2×Q8 | C2×C4 | Q8 | C2 |
| # reps | 1 | 3 | 3 | 8 | 1 | 4 | 5 | 15 | 20 | 10 |
Matrix representation of C2×Q8×D11 ►in GL4(𝔽89) generated by
| 88 | 0 | 0 | 0 |
| 0 | 88 | 0 | 0 |
| 0 | 0 | 88 | 0 |
| 0 | 0 | 0 | 88 |
| 88 | 0 | 0 | 0 |
| 0 | 88 | 0 | 0 |
| 0 | 0 | 45 | 33 |
| 0 | 0 | 33 | 44 |
| 88 | 0 | 0 | 0 |
| 0 | 88 | 0 | 0 |
| 0 | 0 | 0 | 88 |
| 0 | 0 | 1 | 0 |
| 53 | 1 | 0 | 0 |
| 16 | 2 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 3 | 5 | 0 | 0 |
| 34 | 86 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(89))| [88,0,0,0,0,88,0,0,0,0,88,0,0,0,0,88],[88,0,0,0,0,88,0,0,0,0,45,33,0,0,33,44],[88,0,0,0,0,88,0,0,0,0,0,1,0,0,88,0],[53,16,0,0,1,2,0,0,0,0,1,0,0,0,0,1],[3,34,0,0,5,86,0,0,0,0,1,0,0,0,0,1] >;
C2×Q8×D11 in GAP, Magma, Sage, TeX
C_2\times Q_8\times D_{11} % in TeX
G:=Group("C2xQ8xD11"); // GroupNames label
G:=SmallGroup(352,180);
// by ID
G=gap.SmallGroup(352,180);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-11,86,159,69,11525]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=d^11=e^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations