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## G = C11×2- 1+4order 352 = 25·11

### Direct product of C11 and 2- 1+4

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C11×2- 1+4
 Chief series C1 — C2 — C22 — C2×C22 — D4×C11 — C11×C4○D4 — C11×2- 1+4
 Lower central C1 — C2 — C11×2- 1+4
 Upper central C1 — C22 — C11×2- 1+4

Generators and relations for C11×2- 1+4
G = < a,b,c,d,e | a11=b4=c2=1, d2=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=b2d >

Subgroups: 156 in 146 conjugacy classes, 136 normal (6 characteristic)
C1, C2, C2, C4, C22, C2×C4, D4, Q8, C11, C2×Q8, C4○D4, C22, C22, 2- 1+4, C44, C2×C22, C2×C44, D4×C11, Q8×C11, Q8×C22, C11×C4○D4, C11×2- 1+4
Quotients: C1, C2, C22, C23, C11, C24, C22, 2- 1+4, C2×C22, C22×C22, C23×C22, C11×2- 1+4

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

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

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

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

187 conjugacy classes

 class 1 2A 2B ··· 2F 4A ··· 4J 11A ··· 11J 22A ··· 22J 22K ··· 22BH 44A ··· 44CV order 1 2 2 ··· 2 4 ··· 4 11 ··· 11 22 ··· 22 22 ··· 22 44 ··· 44 size 1 1 2 ··· 2 2 ··· 2 1 ··· 1 1 ··· 1 2 ··· 2 2 ··· 2

187 irreducible representations

 dim 1 1 1 1 1 1 4 4 type + + + - image C1 C2 C2 C11 C22 C22 2- 1+4 C11×2- 1+4 kernel C11×2- 1+4 Q8×C22 C11×C4○D4 2- 1+4 C2×Q8 C4○D4 C11 C1 # reps 1 5 10 10 50 100 1 10

Matrix representation of C11×2- 1+4 in GL4(𝔽89) generated by

 16 0 0 0 0 16 0 0 0 0 16 0 0 0 0 16
,
 0 1 0 0 88 0 0 0 0 0 0 88 0 0 1 0
,
 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0
,
 0 0 0 1 0 0 1 0 0 88 0 0 88 0 0 0
,
 35 0 0 51 0 35 51 0 0 51 54 0 51 0 0 54
G:=sub<GL(4,GF(89))| [16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[0,88,0,0,1,0,0,0,0,0,0,1,0,0,88,0],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0],[0,0,0,88,0,0,88,0,0,1,0,0,1,0,0,0],[35,0,0,51,0,35,51,0,0,51,54,0,51,0,0,54] >;

C11×2- 1+4 in GAP, Magma, Sage, TeX

C_{11}\times 2_-^{1+4}
% in TeX

G:=Group("C11xES-(2,2)");
// GroupNames label

G:=SmallGroup(352,193);
// by ID

G=gap.SmallGroup(352,193);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-11,-2,2137,1063,1628,806,4419]);
// Polycyclic

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

׿
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𝔽