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## G = C22×C11⋊D4order 352 = 25·11

### Direct product of C22 and C11⋊D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C22×C11⋊D4
 Chief series C1 — C11 — C22 — D22 — C22×D11 — C23×D11 — C22×C11⋊D4
 Lower central C11 — C22 — C22×C11⋊D4
 Upper central C1 — C23 — C24

Generators and relations for C22×C11⋊D4
G = < a,b,c,d,e | a2=b2=c11=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=ece=c-1, ede=d-1 >

Subgroups: 1226 in 236 conjugacy classes, 105 normal (11 characteristic)
C1, C2, C2, C2, C4, C22, C22, C2×C4, D4, C23, C23, C23, C11, C22×C4, C2×D4, C24, C24, D11, C22, C22, C22, C22×D4, Dic11, D22, D22, C2×C22, C2×C22, C2×Dic11, C11⋊D4, C22×D11, C22×D11, C22×C22, C22×C22, C22×C22, C22×Dic11, C2×C11⋊D4, C23×D11, C23×C22, C22×C11⋊D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, D11, C22×D4, D22, C11⋊D4, C22×D11, C2×C11⋊D4, C23×D11, C22×C11⋊D4

Smallest permutation representation of C22×C11⋊D4
On 176 points
Generators in S176
(1 100)(2 101)(3 102)(4 103)(5 104)(6 105)(7 106)(8 107)(9 108)(10 109)(11 110)(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 56)(2 57)(3 58)(4 59)(5 60)(6 61)(7 62)(8 63)(9 64)(10 65)(11 66)(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 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)
(1 111 12 122)(2 121 13 132)(3 120 14 131)(4 119 15 130)(5 118 16 129)(6 117 17 128)(7 116 18 127)(8 115 19 126)(9 114 20 125)(10 113 21 124)(11 112 22 123)(23 100 34 89)(24 110 35 99)(25 109 36 98)(26 108 37 97)(27 107 38 96)(28 106 39 95)(29 105 40 94)(30 104 41 93)(31 103 42 92)(32 102 43 91)(33 101 44 90)(45 155 56 166)(46 165 57 176)(47 164 58 175)(48 163 59 174)(49 162 60 173)(50 161 61 172)(51 160 62 171)(52 159 63 170)(53 158 64 169)(54 157 65 168)(55 156 66 167)(67 144 78 133)(68 154 79 143)(69 153 80 142)(70 152 81 141)(71 151 82 140)(72 150 83 139)(73 149 84 138)(74 148 85 137)(75 147 86 136)(76 146 87 135)(77 145 88 134)
(1 133)(2 143)(3 142)(4 141)(5 140)(6 139)(7 138)(8 137)(9 136)(10 135)(11 134)(12 144)(13 154)(14 153)(15 152)(16 151)(17 150)(18 149)(19 148)(20 147)(21 146)(22 145)(23 166)(24 176)(25 175)(26 174)(27 173)(28 172)(29 171)(30 170)(31 169)(32 168)(33 167)(34 155)(35 165)(36 164)(37 163)(38 162)(39 161)(40 160)(41 159)(42 158)(43 157)(44 156)(45 89)(46 99)(47 98)(48 97)(49 96)(50 95)(51 94)(52 93)(53 92)(54 91)(55 90)(56 100)(57 110)(58 109)(59 108)(60 107)(61 106)(62 105)(63 104)(64 103)(65 102)(66 101)(67 122)(68 132)(69 131)(70 130)(71 129)(72 128)(73 127)(74 126)(75 125)(76 124)(77 123)(78 111)(79 121)(80 120)(81 119)(82 118)(83 117)(84 116)(85 115)(86 114)(87 113)(88 112)

G:=sub<Sym(176)| (1,100)(2,101)(3,102)(4,103)(5,104)(6,105)(7,106)(8,107)(9,108)(10,109)(11,110)(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), (1,56)(2,57)(3,58)(4,59)(5,60)(6,61)(7,62)(8,63)(9,64)(10,65)(11,66)(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), (1,2,3,4,5,6,7,8,9,10,11)(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), (1,111,12,122)(2,121,13,132)(3,120,14,131)(4,119,15,130)(5,118,16,129)(6,117,17,128)(7,116,18,127)(8,115,19,126)(9,114,20,125)(10,113,21,124)(11,112,22,123)(23,100,34,89)(24,110,35,99)(25,109,36,98)(26,108,37,97)(27,107,38,96)(28,106,39,95)(29,105,40,94)(30,104,41,93)(31,103,42,92)(32,102,43,91)(33,101,44,90)(45,155,56,166)(46,165,57,176)(47,164,58,175)(48,163,59,174)(49,162,60,173)(50,161,61,172)(51,160,62,171)(52,159,63,170)(53,158,64,169)(54,157,65,168)(55,156,66,167)(67,144,78,133)(68,154,79,143)(69,153,80,142)(70,152,81,141)(71,151,82,140)(72,150,83,139)(73,149,84,138)(74,148,85,137)(75,147,86,136)(76,146,87,135)(77,145,88,134), (1,133)(2,143)(3,142)(4,141)(5,140)(6,139)(7,138)(8,137)(9,136)(10,135)(11,134)(12,144)(13,154)(14,153)(15,152)(16,151)(17,150)(18,149)(19,148)(20,147)(21,146)(22,145)(23,166)(24,176)(25,175)(26,174)(27,173)(28,172)(29,171)(30,170)(31,169)(32,168)(33,167)(34,155)(35,165)(36,164)(37,163)(38,162)(39,161)(40,160)(41,159)(42,158)(43,157)(44,156)(45,89)(46,99)(47,98)(48,97)(49,96)(50,95)(51,94)(52,93)(53,92)(54,91)(55,90)(56,100)(57,110)(58,109)(59,108)(60,107)(61,106)(62,105)(63,104)(64,103)(65,102)(66,101)(67,122)(68,132)(69,131)(70,130)(71,129)(72,128)(73,127)(74,126)(75,125)(76,124)(77,123)(78,111)(79,121)(80,120)(81,119)(82,118)(83,117)(84,116)(85,115)(86,114)(87,113)(88,112)>;

G:=Group( (1,100)(2,101)(3,102)(4,103)(5,104)(6,105)(7,106)(8,107)(9,108)(10,109)(11,110)(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), (1,56)(2,57)(3,58)(4,59)(5,60)(6,61)(7,62)(8,63)(9,64)(10,65)(11,66)(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), (1,2,3,4,5,6,7,8,9,10,11)(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), (1,111,12,122)(2,121,13,132)(3,120,14,131)(4,119,15,130)(5,118,16,129)(6,117,17,128)(7,116,18,127)(8,115,19,126)(9,114,20,125)(10,113,21,124)(11,112,22,123)(23,100,34,89)(24,110,35,99)(25,109,36,98)(26,108,37,97)(27,107,38,96)(28,106,39,95)(29,105,40,94)(30,104,41,93)(31,103,42,92)(32,102,43,91)(33,101,44,90)(45,155,56,166)(46,165,57,176)(47,164,58,175)(48,163,59,174)(49,162,60,173)(50,161,61,172)(51,160,62,171)(52,159,63,170)(53,158,64,169)(54,157,65,168)(55,156,66,167)(67,144,78,133)(68,154,79,143)(69,153,80,142)(70,152,81,141)(71,151,82,140)(72,150,83,139)(73,149,84,138)(74,148,85,137)(75,147,86,136)(76,146,87,135)(77,145,88,134), (1,133)(2,143)(3,142)(4,141)(5,140)(6,139)(7,138)(8,137)(9,136)(10,135)(11,134)(12,144)(13,154)(14,153)(15,152)(16,151)(17,150)(18,149)(19,148)(20,147)(21,146)(22,145)(23,166)(24,176)(25,175)(26,174)(27,173)(28,172)(29,171)(30,170)(31,169)(32,168)(33,167)(34,155)(35,165)(36,164)(37,163)(38,162)(39,161)(40,160)(41,159)(42,158)(43,157)(44,156)(45,89)(46,99)(47,98)(48,97)(49,96)(50,95)(51,94)(52,93)(53,92)(54,91)(55,90)(56,100)(57,110)(58,109)(59,108)(60,107)(61,106)(62,105)(63,104)(64,103)(65,102)(66,101)(67,122)(68,132)(69,131)(70,130)(71,129)(72,128)(73,127)(74,126)(75,125)(76,124)(77,123)(78,111)(79,121)(80,120)(81,119)(82,118)(83,117)(84,116)(85,115)(86,114)(87,113)(88,112) );

G=PermutationGroup([[(1,100),(2,101),(3,102),(4,103),(5,104),(6,105),(7,106),(8,107),(9,108),(10,109),(11,110),(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)], [(1,56),(2,57),(3,58),(4,59),(5,60),(6,61),(7,62),(8,63),(9,64),(10,65),(11,66),(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)], [(1,2,3,4,5,6,7,8,9,10,11),(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)], [(1,111,12,122),(2,121,13,132),(3,120,14,131),(4,119,15,130),(5,118,16,129),(6,117,17,128),(7,116,18,127),(8,115,19,126),(9,114,20,125),(10,113,21,124),(11,112,22,123),(23,100,34,89),(24,110,35,99),(25,109,36,98),(26,108,37,97),(27,107,38,96),(28,106,39,95),(29,105,40,94),(30,104,41,93),(31,103,42,92),(32,102,43,91),(33,101,44,90),(45,155,56,166),(46,165,57,176),(47,164,58,175),(48,163,59,174),(49,162,60,173),(50,161,61,172),(51,160,62,171),(52,159,63,170),(53,158,64,169),(54,157,65,168),(55,156,66,167),(67,144,78,133),(68,154,79,143),(69,153,80,142),(70,152,81,141),(71,151,82,140),(72,150,83,139),(73,149,84,138),(74,148,85,137),(75,147,86,136),(76,146,87,135),(77,145,88,134)], [(1,133),(2,143),(3,142),(4,141),(5,140),(6,139),(7,138),(8,137),(9,136),(10,135),(11,134),(12,144),(13,154),(14,153),(15,152),(16,151),(17,150),(18,149),(19,148),(20,147),(21,146),(22,145),(23,166),(24,176),(25,175),(26,174),(27,173),(28,172),(29,171),(30,170),(31,169),(32,168),(33,167),(34,155),(35,165),(36,164),(37,163),(38,162),(39,161),(40,160),(41,159),(42,158),(43,157),(44,156),(45,89),(46,99),(47,98),(48,97),(49,96),(50,95),(51,94),(52,93),(53,92),(54,91),(55,90),(56,100),(57,110),(58,109),(59,108),(60,107),(61,106),(62,105),(63,104),(64,103),(65,102),(66,101),(67,122),(68,132),(69,131),(70,130),(71,129),(72,128),(73,127),(74,126),(75,125),(76,124),(77,123),(78,111),(79,121),(80,120),(81,119),(82,118),(83,117),(84,116),(85,115),(86,114),(87,113),(88,112)]])

100 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 2L 2M 2N 2O 4A 4B 4C 4D 11A ··· 11E 22A ··· 22BW order 1 2 ··· 2 2 2 2 2 2 2 2 2 4 4 4 4 11 ··· 11 22 ··· 22 size 1 1 ··· 1 2 2 2 2 22 22 22 22 22 22 22 22 2 ··· 2 2 ··· 2

100 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 D4 D11 D22 C11⋊D4 kernel C22×C11⋊D4 C22×Dic11 C2×C11⋊D4 C23×D11 C23×C22 C2×C22 C24 C23 C22 # reps 1 1 12 1 1 4 5 35 40

Matrix representation of C22×C11⋊D4 in GL4(𝔽89) generated by

 88 0 0 0 0 88 0 0 0 0 88 0 0 0 0 88
,
 1 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 86
,
 1 0 0 0 0 88 0 0 0 0 42 80 0 0 28 47
,
 1 0 0 0 0 88 0 0 0 0 88 3 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],[1,0,0,0,0,88,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,88,86],[1,0,0,0,0,88,0,0,0,0,42,28,0,0,80,47],[1,0,0,0,0,88,0,0,0,0,88,0,0,0,3,1] >;

C22×C11⋊D4 in GAP, Magma, Sage, TeX

C_2^2\times C_{11}\rtimes D_4
% in TeX

G:=Group("C2^2xC11:D4");
// GroupNames label

G:=SmallGroup(352,187);
// by ID

G=gap.SmallGroup(352,187);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-11,579,11525]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^11=d^4=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,d*c*d^-1=e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations

׿
×
𝔽