direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C22×C4×D11, C44⋊3C23, C22.2C24, D22.9C23, C23.34D22, Dic11⋊3C23, C11⋊1(C23×C4), C22⋊1(C22×C4), (C22×C44)⋊10C2, (C2×C44)⋊14C22, C2.1(C23×D11), (C2×C22).63C23, (C23×D11).3C2, (C2×Dic11)⋊12C22, (C22×Dic11)⋊10C2, (C22×C22).44C22, C22.29(C22×D11), (C22×D11).35C22, (C2×C22)⋊6(C2×C4), SmallGroup(352,174)
Series: Derived ►Chief ►Lower central ►Upper central
| C11 — C22×C4×D11 |
Generators and relations for C22×C4×D11
G = < a,b,c,d,e | a2=b2=c4=d11=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >
Subgroups: 1146 in 236 conjugacy classes, 145 normal (11 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, C23, C23, C11, C22×C4, C22×C4, C24, D11, C22, C22, C23×C4, Dic11, C44, D22, C2×C22, C4×D11, C2×Dic11, C2×C44, C22×D11, C22×C22, C2×C4×D11, C22×Dic11, C22×C44, C23×D11, C22×C4×D11
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C24, D11, C23×C4, D22, C4×D11, C22×D11, C2×C4×D11, C23×D11, C22×C4×D11
►On 176 points
Generators in S176
(1 109)(2 110)(3 100)(4 101)(5 102)(6 103)(7 104)(8 105)(9 106)(10 107)(11 108)(12 89)(13 90)(14 91)(15 92)(16 93)(17 94)(18 95)(19 96)(20 97)(21 98)(22 99)(23 122)(24 123)(25 124)(26 125)(27 126)(28 127)(29 128)(30 129)(31 130)(32 131)(33 132)(34 111)(35 112)(36 113)(37 114)(38 115)(39 116)(40 117)(41 118)(42 119)(43 120)(44 121)(45 144)(46 145)(47 146)(48 147)(49 148)(50 149)(51 150)(52 151)(53 152)(54 153)(55 154)(56 133)(57 134)(58 135)(59 136)(60 137)(61 138)(62 139)(63 140)(64 141)(65 142)(66 143)(67 166)(68 167)(69 168)(70 169)(71 170)(72 171)(73 172)(74 173)(75 174)(76 175)(77 176)(78 155)(79 156)(80 157)(81 158)(82 159)(83 160)(84 161)(85 162)(86 163)(87 164)(88 165)
(1 65)(2 66)(3 56)(4 57)(5 58)(6 59)(7 60)(8 61)(9 62)(10 63)(11 64)(12 45)(13 46)(14 47)(15 48)(16 49)(17 50)(18 51)(19 52)(20 53)(21 54)(22 55)(23 78)(24 79)(25 80)(26 81)(27 82)(28 83)(29 84)(30 85)(31 86)(32 87)(33 88)(34 67)(35 68)(36 69)(37 70)(38 71)(39 72)(40 73)(41 74)(42 75)(43 76)(44 77)(89 144)(90 145)(91 146)(92 147)(93 148)(94 149)(95 150)(96 151)(97 152)(98 153)(99 154)(100 133)(101 134)(102 135)(103 136)(104 137)(105 138)(106 139)(107 140)(108 141)(109 142)(110 143)(111 166)(112 167)(113 168)(114 169)(115 170)(116 171)(117 172)(118 173)(119 174)(120 175)(121 176)(122 155)(123 156)(124 157)(125 158)(126 159)(127 160)(128 161)(129 162)(130 163)(131 164)(132 165)
(1 32 21 43)(2 33 22 44)(3 23 12 34)(4 24 13 35)(5 25 14 36)(6 26 15 37)(7 27 16 38)(8 28 17 39)(9 29 18 40)(10 30 19 41)(11 31 20 42)(45 67 56 78)(46 68 57 79)(47 69 58 80)(48 70 59 81)(49 71 60 82)(50 72 61 83)(51 73 62 84)(52 74 63 85)(53 75 64 86)(54 76 65 87)(55 77 66 88)(89 111 100 122)(90 112 101 123)(91 113 102 124)(92 114 103 125)(93 115 104 126)(94 116 105 127)(95 117 106 128)(96 118 107 129)(97 119 108 130)(98 120 109 131)(99 121 110 132)(133 155 144 166)(134 156 145 167)(135 157 146 168)(136 158 147 169)(137 159 148 170)(138 160 149 171)(139 161 150 172)(140 162 151 173)(141 163 152 174)(142 164 153 175)(143 165 154 176)
(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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 22)(11 21)(23 40)(24 39)(25 38)(26 37)(27 36)(28 35)(29 34)(30 44)(31 43)(32 42)(33 41)(45 62)(46 61)(47 60)(48 59)(49 58)(50 57)(51 56)(52 66)(53 65)(54 64)(55 63)(67 84)(68 83)(69 82)(70 81)(71 80)(72 79)(73 78)(74 88)(75 87)(76 86)(77 85)(89 106)(90 105)(91 104)(92 103)(93 102)(94 101)(95 100)(96 110)(97 109)(98 108)(99 107)(111 128)(112 127)(113 126)(114 125)(115 124)(116 123)(117 122)(118 132)(119 131)(120 130)(121 129)(133 150)(134 149)(135 148)(136 147)(137 146)(138 145)(139 144)(140 154)(141 153)(142 152)(143 151)(155 172)(156 171)(157 170)(158 169)(159 168)(160 167)(161 166)(162 176)(163 175)(164 174)(165 173)
G:=sub<Sym(176)| (1,109)(2,110)(3,100)(4,101)(5,102)(6,103)(7,104)(8,105)(9,106)(10,107)(11,108)(12,89)(13,90)(14,91)(15,92)(16,93)(17,94)(18,95)(19,96)(20,97)(21,98)(22,99)(23,122)(24,123)(25,124)(26,125)(27,126)(28,127)(29,128)(30,129)(31,130)(32,131)(33,132)(34,111)(35,112)(36,113)(37,114)(38,115)(39,116)(40,117)(41,118)(42,119)(43,120)(44,121)(45,144)(46,145)(47,146)(48,147)(49,148)(50,149)(51,150)(52,151)(53,152)(54,153)(55,154)(56,133)(57,134)(58,135)(59,136)(60,137)(61,138)(62,139)(63,140)(64,141)(65,142)(66,143)(67,166)(68,167)(69,168)(70,169)(71,170)(72,171)(73,172)(74,173)(75,174)(76,175)(77,176)(78,155)(79,156)(80,157)(81,158)(82,159)(83,160)(84,161)(85,162)(86,163)(87,164)(88,165), (1,65)(2,66)(3,56)(4,57)(5,58)(6,59)(7,60)(8,61)(9,62)(10,63)(11,64)(12,45)(13,46)(14,47)(15,48)(16,49)(17,50)(18,51)(19,52)(20,53)(21,54)(22,55)(23,78)(24,79)(25,80)(26,81)(27,82)(28,83)(29,84)(30,85)(31,86)(32,87)(33,88)(34,67)(35,68)(36,69)(37,70)(38,71)(39,72)(40,73)(41,74)(42,75)(43,76)(44,77)(89,144)(90,145)(91,146)(92,147)(93,148)(94,149)(95,150)(96,151)(97,152)(98,153)(99,154)(100,133)(101,134)(102,135)(103,136)(104,137)(105,138)(106,139)(107,140)(108,141)(109,142)(110,143)(111,166)(112,167)(113,168)(114,169)(115,170)(116,171)(117,172)(118,173)(119,174)(120,175)(121,176)(122,155)(123,156)(124,157)(125,158)(126,159)(127,160)(128,161)(129,162)(130,163)(131,164)(132,165), (1,32,21,43)(2,33,22,44)(3,23,12,34)(4,24,13,35)(5,25,14,36)(6,26,15,37)(7,27,16,38)(8,28,17,39)(9,29,18,40)(10,30,19,41)(11,31,20,42)(45,67,56,78)(46,68,57,79)(47,69,58,80)(48,70,59,81)(49,71,60,82)(50,72,61,83)(51,73,62,84)(52,74,63,85)(53,75,64,86)(54,76,65,87)(55,77,66,88)(89,111,100,122)(90,112,101,123)(91,113,102,124)(92,114,103,125)(93,115,104,126)(94,116,105,127)(95,117,106,128)(96,118,107,129)(97,119,108,130)(98,120,109,131)(99,121,110,132)(133,155,144,166)(134,156,145,167)(135,157,146,168)(136,158,147,169)(137,159,148,170)(138,160,149,171)(139,161,150,172)(140,162,151,173)(141,163,152,174)(142,164,153,175)(143,165,154,176), (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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,22)(11,21)(23,40)(24,39)(25,38)(26,37)(27,36)(28,35)(29,34)(30,44)(31,43)(32,42)(33,41)(45,62)(46,61)(47,60)(48,59)(49,58)(50,57)(51,56)(52,66)(53,65)(54,64)(55,63)(67,84)(68,83)(69,82)(70,81)(71,80)(72,79)(73,78)(74,88)(75,87)(76,86)(77,85)(89,106)(90,105)(91,104)(92,103)(93,102)(94,101)(95,100)(96,110)(97,109)(98,108)(99,107)(111,128)(112,127)(113,126)(114,125)(115,124)(116,123)(117,122)(118,132)(119,131)(120,130)(121,129)(133,150)(134,149)(135,148)(136,147)(137,146)(138,145)(139,144)(140,154)(141,153)(142,152)(143,151)(155,172)(156,171)(157,170)(158,169)(159,168)(160,167)(161,166)(162,176)(163,175)(164,174)(165,173)>;
G:=Group( (1,109)(2,110)(3,100)(4,101)(5,102)(6,103)(7,104)(8,105)(9,106)(10,107)(11,108)(12,89)(13,90)(14,91)(15,92)(16,93)(17,94)(18,95)(19,96)(20,97)(21,98)(22,99)(23,122)(24,123)(25,124)(26,125)(27,126)(28,127)(29,128)(30,129)(31,130)(32,131)(33,132)(34,111)(35,112)(36,113)(37,114)(38,115)(39,116)(40,117)(41,118)(42,119)(43,120)(44,121)(45,144)(46,145)(47,146)(48,147)(49,148)(50,149)(51,150)(52,151)(53,152)(54,153)(55,154)(56,133)(57,134)(58,135)(59,136)(60,137)(61,138)(62,139)(63,140)(64,141)(65,142)(66,143)(67,166)(68,167)(69,168)(70,169)(71,170)(72,171)(73,172)(74,173)(75,174)(76,175)(77,176)(78,155)(79,156)(80,157)(81,158)(82,159)(83,160)(84,161)(85,162)(86,163)(87,164)(88,165), (1,65)(2,66)(3,56)(4,57)(5,58)(6,59)(7,60)(8,61)(9,62)(10,63)(11,64)(12,45)(13,46)(14,47)(15,48)(16,49)(17,50)(18,51)(19,52)(20,53)(21,54)(22,55)(23,78)(24,79)(25,80)(26,81)(27,82)(28,83)(29,84)(30,85)(31,86)(32,87)(33,88)(34,67)(35,68)(36,69)(37,70)(38,71)(39,72)(40,73)(41,74)(42,75)(43,76)(44,77)(89,144)(90,145)(91,146)(92,147)(93,148)(94,149)(95,150)(96,151)(97,152)(98,153)(99,154)(100,133)(101,134)(102,135)(103,136)(104,137)(105,138)(106,139)(107,140)(108,141)(109,142)(110,143)(111,166)(112,167)(113,168)(114,169)(115,170)(116,171)(117,172)(118,173)(119,174)(120,175)(121,176)(122,155)(123,156)(124,157)(125,158)(126,159)(127,160)(128,161)(129,162)(130,163)(131,164)(132,165), (1,32,21,43)(2,33,22,44)(3,23,12,34)(4,24,13,35)(5,25,14,36)(6,26,15,37)(7,27,16,38)(8,28,17,39)(9,29,18,40)(10,30,19,41)(11,31,20,42)(45,67,56,78)(46,68,57,79)(47,69,58,80)(48,70,59,81)(49,71,60,82)(50,72,61,83)(51,73,62,84)(52,74,63,85)(53,75,64,86)(54,76,65,87)(55,77,66,88)(89,111,100,122)(90,112,101,123)(91,113,102,124)(92,114,103,125)(93,115,104,126)(94,116,105,127)(95,117,106,128)(96,118,107,129)(97,119,108,130)(98,120,109,131)(99,121,110,132)(133,155,144,166)(134,156,145,167)(135,157,146,168)(136,158,147,169)(137,159,148,170)(138,160,149,171)(139,161,150,172)(140,162,151,173)(141,163,152,174)(142,164,153,175)(143,165,154,176), (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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,22)(11,21)(23,40)(24,39)(25,38)(26,37)(27,36)(28,35)(29,34)(30,44)(31,43)(32,42)(33,41)(45,62)(46,61)(47,60)(48,59)(49,58)(50,57)(51,56)(52,66)(53,65)(54,64)(55,63)(67,84)(68,83)(69,82)(70,81)(71,80)(72,79)(73,78)(74,88)(75,87)(76,86)(77,85)(89,106)(90,105)(91,104)(92,103)(93,102)(94,101)(95,100)(96,110)(97,109)(98,108)(99,107)(111,128)(112,127)(113,126)(114,125)(115,124)(116,123)(117,122)(118,132)(119,131)(120,130)(121,129)(133,150)(134,149)(135,148)(136,147)(137,146)(138,145)(139,144)(140,154)(141,153)(142,152)(143,151)(155,172)(156,171)(157,170)(158,169)(159,168)(160,167)(161,166)(162,176)(163,175)(164,174)(165,173) );
G=PermutationGroup([[(1,109),(2,110),(3,100),(4,101),(5,102),(6,103),(7,104),(8,105),(9,106),(10,107),(11,108),(12,89),(13,90),(14,91),(15,92),(16,93),(17,94),(18,95),(19,96),(20,97),(21,98),(22,99),(23,122),(24,123),(25,124),(26,125),(27,126),(28,127),(29,128),(30,129),(31,130),(32,131),(33,132),(34,111),(35,112),(36,113),(37,114),(38,115),(39,116),(40,117),(41,118),(42,119),(43,120),(44,121),(45,144),(46,145),(47,146),(48,147),(49,148),(50,149),(51,150),(52,151),(53,152),(54,153),(55,154),(56,133),(57,134),(58,135),(59,136),(60,137),(61,138),(62,139),(63,140),(64,141),(65,142),(66,143),(67,166),(68,167),(69,168),(70,169),(71,170),(72,171),(73,172),(74,173),(75,174),(76,175),(77,176),(78,155),(79,156),(80,157),(81,158),(82,159),(83,160),(84,161),(85,162),(86,163),(87,164),(88,165)], [(1,65),(2,66),(3,56),(4,57),(5,58),(6,59),(7,60),(8,61),(9,62),(10,63),(11,64),(12,45),(13,46),(14,47),(15,48),(16,49),(17,50),(18,51),(19,52),(20,53),(21,54),(22,55),(23,78),(24,79),(25,80),(26,81),(27,82),(28,83),(29,84),(30,85),(31,86),(32,87),(33,88),(34,67),(35,68),(36,69),(37,70),(38,71),(39,72),(40,73),(41,74),(42,75),(43,76),(44,77),(89,144),(90,145),(91,146),(92,147),(93,148),(94,149),(95,150),(96,151),(97,152),(98,153),(99,154),(100,133),(101,134),(102,135),(103,136),(104,137),(105,138),(106,139),(107,140),(108,141),(109,142),(110,143),(111,166),(112,167),(113,168),(114,169),(115,170),(116,171),(117,172),(118,173),(119,174),(120,175),(121,176),(122,155),(123,156),(124,157),(125,158),(126,159),(127,160),(128,161),(129,162),(130,163),(131,164),(132,165)], [(1,32,21,43),(2,33,22,44),(3,23,12,34),(4,24,13,35),(5,25,14,36),(6,26,15,37),(7,27,16,38),(8,28,17,39),(9,29,18,40),(10,30,19,41),(11,31,20,42),(45,67,56,78),(46,68,57,79),(47,69,58,80),(48,70,59,81),(49,71,60,82),(50,72,61,83),(51,73,62,84),(52,74,63,85),(53,75,64,86),(54,76,65,87),(55,77,66,88),(89,111,100,122),(90,112,101,123),(91,113,102,124),(92,114,103,125),(93,115,104,126),(94,116,105,127),(95,117,106,128),(96,118,107,129),(97,119,108,130),(98,120,109,131),(99,121,110,132),(133,155,144,166),(134,156,145,167),(135,157,146,168),(136,158,147,169),(137,159,148,170),(138,160,149,171),(139,161,150,172),(140,162,151,173),(141,163,152,174),(142,164,153,175),(143,165,154,176)], [(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)], [(1,20),(2,19),(3,18),(4,17),(5,16),(6,15),(7,14),(8,13),(9,12),(10,22),(11,21),(23,40),(24,39),(25,38),(26,37),(27,36),(28,35),(29,34),(30,44),(31,43),(32,42),(33,41),(45,62),(46,61),(47,60),(48,59),(49,58),(50,57),(51,56),(52,66),(53,65),(54,64),(55,63),(67,84),(68,83),(69,82),(70,81),(71,80),(72,79),(73,78),(74,88),(75,87),(76,86),(77,85),(89,106),(90,105),(91,104),(92,103),(93,102),(94,101),(95,100),(96,110),(97,109),(98,108),(99,107),(111,128),(112,127),(113,126),(114,125),(115,124),(116,123),(117,122),(118,132),(119,131),(120,130),(121,129),(133,150),(134,149),(135,148),(136,147),(137,146),(138,145),(139,144),(140,154),(141,153),(142,152),(143,151),(155,172),(156,171),(157,170),(158,169),(159,168),(160,167),(161,166),(162,176),(163,175),(164,174),(165,173)]])
112 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2O | 4A | ··· | 4H | 4I | ··· | 4P | 11A | ··· | 11E | 22A | ··· | 22AI | 44A | ··· | 44AN |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 11 | ··· | 11 | 22 | ··· | 22 | 44 | ··· | 44 |
| size | 1 | 1 | ··· | 1 | 11 | ··· | 11 | 1 | ··· | 1 | 11 | ··· | 11 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
112 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D11 | D22 | D22 | C4×D11 |
| kernel | C22×C4×D11 | C2×C4×D11 | C22×Dic11 | C22×C44 | C23×D11 | C22×D11 | C22×C4 | C2×C4 | C23 | C22 |
| # reps | 1 | 12 | 1 | 1 | 1 | 16 | 5 | 30 | 5 | 40 |
Matrix representation of C22×C4×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 | 1 | 0 | 0 |
| 0 | 0 | 88 | 0 |
| 0 | 0 | 0 | 88 |
| 1 | 0 | 0 | 0 |
| 0 | 88 | 0 | 0 |
| 0 | 0 | 55 | 0 |
| 0 | 0 | 0 | 55 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 47 | 1 |
| 0 | 0 | 88 | 0 |
| 88 | 0 | 0 | 0 |
| 0 | 88 | 0 | 0 |
| 0 | 0 | 0 | 88 |
| 0 | 0 | 88 | 0 |
G:=sub<GL(4,GF(89))| [88,0,0,0,0,88,0,0,0,0,88,0,0,0,0,88],[88,0,0,0,0,1,0,0,0,0,88,0,0,0,0,88],[1,0,0,0,0,88,0,0,0,0,55,0,0,0,0,55],[1,0,0,0,0,1,0,0,0,0,47,88,0,0,1,0],[88,0,0,0,0,88,0,0,0,0,0,88,0,0,88,0] >;
C22×C4×D11 in GAP, Magma, Sage, TeX
C_2^2\times C_4\times D_{11} % in TeX
G:=Group("C2^2xC4xD11"); // GroupNames label
G:=SmallGroup(352,174);
// by ID
G=gap.SmallGroup(352,174);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-11,69,11525]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^4=d^11=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations