direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D4.D11, D4.7D22, C22⋊2SD16, C44.16D4, C44.13C23, Dic22⋊6C22, C11⋊C8⋊8C22, C11⋊3(C2×SD16), (C2×D4).4D11, (D4×C22).3C2, (C2×C4).48D22, (C2×C22).40D4, C22.46(C2×D4), (C2×Dic22)⋊9C2, C4.6(C11⋊D4), (C2×C44).31C22, (D4×C11).7C22, C4.13(C22×D11), C22.22(C11⋊D4), (C2×C11⋊C8)⋊5C2, C2.10(C2×C11⋊D4), SmallGroup(352,128)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D4.D11
G = < a,b,c,d,e | a2=b4=c2=d11=1, e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=bc, ede-1=d-1 >
Subgroups: 298 in 68 conjugacy classes, 33 normal (17 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C11, C2×C8, SD16, C2×D4, C2×Q8, C22, C22, C22, C2×SD16, Dic11, C44, C2×C22, C2×C22, C11⋊C8, Dic22, Dic22, C2×Dic11, C2×C44, D4×C11, D4×C11, C22×C22, C2×C11⋊C8, D4.D11, C2×Dic22, D4×C22, C2×D4.D11
Quotients: C1, C2, C22, D4, C23, SD16, C2×D4, D11, C2×SD16, D22, C11⋊D4, C22×D11, D4.D11, C2×C11⋊D4, C2×D4.D11
►On 176 points
Generators in S176
(1 45)(2 46)(3 47)(4 48)(5 49)(6 50)(7 51)(8 52)(9 53)(10 54)(11 55)(12 56)(13 57)(14 58)(15 59)(16 60)(17 61)(18 62)(19 63)(20 64)(21 65)(22 66)(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)(89 133)(90 134)(91 135)(92 136)(93 137)(94 138)(95 139)(96 140)(97 141)(98 142)(99 143)(100 144)(101 145)(102 146)(103 147)(104 148)(105 149)(106 150)(107 151)(108 152)(109 153)(110 154)(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 34 12 23)(2 35 13 24)(3 36 14 25)(4 37 15 26)(5 38 16 27)(6 39 17 28)(7 40 18 29)(8 41 19 30)(9 42 20 31)(10 43 21 32)(11 44 22 33)(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)(133 166 144 155)(134 167 145 156)(135 168 146 157)(136 169 147 158)(137 170 148 159)(138 171 149 160)(139 172 150 161)(140 173 151 162)(141 174 152 163)(142 175 153 164)(143 176 154 165)
(1 23)(2 24)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 31)(10 32)(11 33)(12 34)(13 35)(14 36)(15 37)(16 38)(17 39)(18 40)(19 41)(20 42)(21 43)(22 44)(45 78)(46 79)(47 80)(48 81)(49 82)(50 83)(51 84)(52 85)(53 86)(54 87)(55 88)(56 67)(57 68)(58 69)(59 70)(60 71)(61 72)(62 73)(63 74)(64 75)(65 76)(66 77)(89 100)(90 101)(91 102)(92 103)(93 104)(94 105)(95 106)(96 107)(97 108)(98 109)(99 110)(133 144)(134 145)(135 146)(136 147)(137 148)(138 149)(139 150)(140 151)(141 152)(142 153)(143 154)
(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)
(1 152 12 141)(2 151 13 140)(3 150 14 139)(4 149 15 138)(5 148 16 137)(6 147 17 136)(7 146 18 135)(8 145 19 134)(9 144 20 133)(10 154 21 143)(11 153 22 142)(23 163 34 174)(24 162 35 173)(25 161 36 172)(26 160 37 171)(27 159 38 170)(28 158 39 169)(29 157 40 168)(30 156 41 167)(31 155 42 166)(32 165 43 176)(33 164 44 175)(45 108 56 97)(46 107 57 96)(47 106 58 95)(48 105 59 94)(49 104 60 93)(50 103 61 92)(51 102 62 91)(52 101 63 90)(53 100 64 89)(54 110 65 99)(55 109 66 98)(67 119 78 130)(68 118 79 129)(69 117 80 128)(70 116 81 127)(71 115 82 126)(72 114 83 125)(73 113 84 124)(74 112 85 123)(75 111 86 122)(76 121 87 132)(77 120 88 131)
G:=sub<Sym(176)| (1,45)(2,46)(3,47)(4,48)(5,49)(6,50)(7,51)(8,52)(9,53)(10,54)(11,55)(12,56)(13,57)(14,58)(15,59)(16,60)(17,61)(18,62)(19,63)(20,64)(21,65)(22,66)(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)(89,133)(90,134)(91,135)(92,136)(93,137)(94,138)(95,139)(96,140)(97,141)(98,142)(99,143)(100,144)(101,145)(102,146)(103,147)(104,148)(105,149)(106,150)(107,151)(108,152)(109,153)(110,154)(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), (1,34,12,23)(2,35,13,24)(3,36,14,25)(4,37,15,26)(5,38,16,27)(6,39,17,28)(7,40,18,29)(8,41,19,30)(9,42,20,31)(10,43,21,32)(11,44,22,33)(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)(133,166,144,155)(134,167,145,156)(135,168,146,157)(136,169,147,158)(137,170,148,159)(138,171,149,160)(139,172,150,161)(140,173,151,162)(141,174,152,163)(142,175,153,164)(143,176,154,165), (1,23)(2,24)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,31)(10,32)(11,33)(12,34)(13,35)(14,36)(15,37)(16,38)(17,39)(18,40)(19,41)(20,42)(21,43)(22,44)(45,78)(46,79)(47,80)(48,81)(49,82)(50,83)(51,84)(52,85)(53,86)(54,87)(55,88)(56,67)(57,68)(58,69)(59,70)(60,71)(61,72)(62,73)(63,74)(64,75)(65,76)(66,77)(89,100)(90,101)(91,102)(92,103)(93,104)(94,105)(95,106)(96,107)(97,108)(98,109)(99,110)(133,144)(134,145)(135,146)(136,147)(137,148)(138,149)(139,150)(140,151)(141,152)(142,153)(143,154), (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), (1,152,12,141)(2,151,13,140)(3,150,14,139)(4,149,15,138)(5,148,16,137)(6,147,17,136)(7,146,18,135)(8,145,19,134)(9,144,20,133)(10,154,21,143)(11,153,22,142)(23,163,34,174)(24,162,35,173)(25,161,36,172)(26,160,37,171)(27,159,38,170)(28,158,39,169)(29,157,40,168)(30,156,41,167)(31,155,42,166)(32,165,43,176)(33,164,44,175)(45,108,56,97)(46,107,57,96)(47,106,58,95)(48,105,59,94)(49,104,60,93)(50,103,61,92)(51,102,62,91)(52,101,63,90)(53,100,64,89)(54,110,65,99)(55,109,66,98)(67,119,78,130)(68,118,79,129)(69,117,80,128)(70,116,81,127)(71,115,82,126)(72,114,83,125)(73,113,84,124)(74,112,85,123)(75,111,86,122)(76,121,87,132)(77,120,88,131)>;
G:=Group( (1,45)(2,46)(3,47)(4,48)(5,49)(6,50)(7,51)(8,52)(9,53)(10,54)(11,55)(12,56)(13,57)(14,58)(15,59)(16,60)(17,61)(18,62)(19,63)(20,64)(21,65)(22,66)(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)(89,133)(90,134)(91,135)(92,136)(93,137)(94,138)(95,139)(96,140)(97,141)(98,142)(99,143)(100,144)(101,145)(102,146)(103,147)(104,148)(105,149)(106,150)(107,151)(108,152)(109,153)(110,154)(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), (1,34,12,23)(2,35,13,24)(3,36,14,25)(4,37,15,26)(5,38,16,27)(6,39,17,28)(7,40,18,29)(8,41,19,30)(9,42,20,31)(10,43,21,32)(11,44,22,33)(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)(133,166,144,155)(134,167,145,156)(135,168,146,157)(136,169,147,158)(137,170,148,159)(138,171,149,160)(139,172,150,161)(140,173,151,162)(141,174,152,163)(142,175,153,164)(143,176,154,165), (1,23)(2,24)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,31)(10,32)(11,33)(12,34)(13,35)(14,36)(15,37)(16,38)(17,39)(18,40)(19,41)(20,42)(21,43)(22,44)(45,78)(46,79)(47,80)(48,81)(49,82)(50,83)(51,84)(52,85)(53,86)(54,87)(55,88)(56,67)(57,68)(58,69)(59,70)(60,71)(61,72)(62,73)(63,74)(64,75)(65,76)(66,77)(89,100)(90,101)(91,102)(92,103)(93,104)(94,105)(95,106)(96,107)(97,108)(98,109)(99,110)(133,144)(134,145)(135,146)(136,147)(137,148)(138,149)(139,150)(140,151)(141,152)(142,153)(143,154), (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), (1,152,12,141)(2,151,13,140)(3,150,14,139)(4,149,15,138)(5,148,16,137)(6,147,17,136)(7,146,18,135)(8,145,19,134)(9,144,20,133)(10,154,21,143)(11,153,22,142)(23,163,34,174)(24,162,35,173)(25,161,36,172)(26,160,37,171)(27,159,38,170)(28,158,39,169)(29,157,40,168)(30,156,41,167)(31,155,42,166)(32,165,43,176)(33,164,44,175)(45,108,56,97)(46,107,57,96)(47,106,58,95)(48,105,59,94)(49,104,60,93)(50,103,61,92)(51,102,62,91)(52,101,63,90)(53,100,64,89)(54,110,65,99)(55,109,66,98)(67,119,78,130)(68,118,79,129)(69,117,80,128)(70,116,81,127)(71,115,82,126)(72,114,83,125)(73,113,84,124)(74,112,85,123)(75,111,86,122)(76,121,87,132)(77,120,88,131) );
G=PermutationGroup([[(1,45),(2,46),(3,47),(4,48),(5,49),(6,50),(7,51),(8,52),(9,53),(10,54),(11,55),(12,56),(13,57),(14,58),(15,59),(16,60),(17,61),(18,62),(19,63),(20,64),(21,65),(22,66),(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),(89,133),(90,134),(91,135),(92,136),(93,137),(94,138),(95,139),(96,140),(97,141),(98,142),(99,143),(100,144),(101,145),(102,146),(103,147),(104,148),(105,149),(106,150),(107,151),(108,152),(109,153),(110,154),(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)], [(1,34,12,23),(2,35,13,24),(3,36,14,25),(4,37,15,26),(5,38,16,27),(6,39,17,28),(7,40,18,29),(8,41,19,30),(9,42,20,31),(10,43,21,32),(11,44,22,33),(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),(133,166,144,155),(134,167,145,156),(135,168,146,157),(136,169,147,158),(137,170,148,159),(138,171,149,160),(139,172,150,161),(140,173,151,162),(141,174,152,163),(142,175,153,164),(143,176,154,165)], [(1,23),(2,24),(3,25),(4,26),(5,27),(6,28),(7,29),(8,30),(9,31),(10,32),(11,33),(12,34),(13,35),(14,36),(15,37),(16,38),(17,39),(18,40),(19,41),(20,42),(21,43),(22,44),(45,78),(46,79),(47,80),(48,81),(49,82),(50,83),(51,84),(52,85),(53,86),(54,87),(55,88),(56,67),(57,68),(58,69),(59,70),(60,71),(61,72),(62,73),(63,74),(64,75),(65,76),(66,77),(89,100),(90,101),(91,102),(92,103),(93,104),(94,105),(95,106),(96,107),(97,108),(98,109),(99,110),(133,144),(134,145),(135,146),(136,147),(137,148),(138,149),(139,150),(140,151),(141,152),(142,153),(143,154)], [(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)], [(1,152,12,141),(2,151,13,140),(3,150,14,139),(4,149,15,138),(5,148,16,137),(6,147,17,136),(7,146,18,135),(8,145,19,134),(9,144,20,133),(10,154,21,143),(11,153,22,142),(23,163,34,174),(24,162,35,173),(25,161,36,172),(26,160,37,171),(27,159,38,170),(28,158,39,169),(29,157,40,168),(30,156,41,167),(31,155,42,166),(32,165,43,176),(33,164,44,175),(45,108,56,97),(46,107,57,96),(47,106,58,95),(48,105,59,94),(49,104,60,93),(50,103,61,92),(51,102,62,91),(52,101,63,90),(53,100,64,89),(54,110,65,99),(55,109,66,98),(67,119,78,130),(68,118,79,129),(69,117,80,128),(70,116,81,127),(71,115,82,126),(72,114,83,125),(73,113,84,124),(74,112,85,123),(75,111,86,122),(76,121,87,132),(77,120,88,131)]])
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 8A | 8B | 8C | 8D | 11A | ··· | 11E | 22A | ··· | 22O | 22P | ··· | 22AI | 44A | ··· | 44J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 11 | ··· | 11 | 22 | ··· | 22 | 22 | ··· | 22 | 44 | ··· | 44 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 44 | 44 | 22 | 22 | 22 | 22 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | D4 | D4 | SD16 | D11 | D22 | D22 | C11⋊D4 | C11⋊D4 | D4.D11 |
| kernel | C2×D4.D11 | C2×C11⋊C8 | D4.D11 | C2×Dic22 | D4×C22 | C44 | C2×C22 | C22 | C2×D4 | C2×C4 | D4 | C4 | C22 | C2 |
| # reps | 1 | 1 | 4 | 1 | 1 | 1 | 1 | 4 | 5 | 5 | 10 | 10 | 10 | 10 |
Matrix representation of C2×D4.D11 ►in GL4(𝔽89) generated by
| 88 | 0 | 0 | 0 |
| 0 | 88 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 88 |
| 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 88 |
| 0 | 0 | 88 | 0 |
| 36 | 88 | 0 | 0 |
| 70 | 50 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 77 | 17 | 0 | 0 |
| 23 | 12 | 0 | 0 |
| 0 | 0 | 69 | 69 |
| 0 | 0 | 69 | 20 |
G:=sub<GL(4,GF(89))| [88,0,0,0,0,88,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,88,0],[1,0,0,0,0,1,0,0,0,0,0,88,0,0,88,0],[36,70,0,0,88,50,0,0,0,0,1,0,0,0,0,1],[77,23,0,0,17,12,0,0,0,0,69,69,0,0,69,20] >;
C2×D4.D11 in GAP, Magma, Sage, TeX
C_2\times D_4.D_{11} % in TeX
G:=Group("C2xD4.D11"); // GroupNames label
G:=SmallGroup(352,128);
// by ID
G=gap.SmallGroup(352,128);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-11,96,218,579,159,69,11525]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=c^2=d^11=1,e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e^-1=b^-1,b*d=d*b,c*d=d*c,e*c*e^-1=b*c,e*d*e^-1=d^-1>;
// generators/relations