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## G = C22×C4×D11order 352 = 25·11

### Direct product of C22×C4 and D11

Series: Derived Chief Lower central Upper central

 Derived series C1 — C11 — C22×C4×D11
 Chief series C1 — C11 — C22 — D22 — C22×D11 — C23×D11 — C22×C4×D11
 Lower central C11 — C22×C4×D11
 Upper central C1 — C22×C4

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

Smallest permutation representation of 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

׿
×
𝔽