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## G = C2×D4⋊D11order 352 = 25·11

### Direct product of C2 and D4⋊D11

Series: Derived Chief Lower central Upper central

 Derived series C1 — C44 — C2×D4⋊D11
 Chief series C1 — C11 — C22 — C44 — D44 — C2×D44 — C2×D4⋊D11
 Lower central C11 — C22 — C44 — C2×D4⋊D11
 Upper central C1 — C22 — C2×C4 — C2×D4

Generators and relations for C2×D4⋊D11
G = < a,b,c,d,e | a2=b4=c2=d11=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe=b-1, bd=db, cd=dc, ece=bc, ede=d-1 >

Subgroups: 490 in 76 conjugacy classes, 33 normal (17 characteristic)
C1, C2, C2, C2, C4, C22, C22, C8, C2×C4, D4, D4, C23, C11, C2×C8, D8, C2×D4, C2×D4, D11, C22, C22, C22, C2×D8, C44, D22, C2×C22, C2×C22, C11⋊C8, D44, D44, C2×C44, D4×C11, D4×C11, C22×D11, C22×C22, C2×C11⋊C8, D4⋊D11, C2×D44, D4×C22, C2×D4⋊D11
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, D11, C2×D8, D22, C11⋊D4, C22×D11, D4⋊D11, C2×C11⋊D4, C2×D4⋊D11

Smallest permutation representation of C2×D4⋊D11
On 176 points
Generators in S176
(1 89)(2 90)(3 91)(4 92)(5 93)(6 94)(7 95)(8 96)(9 97)(10 98)(11 99)(12 100)(13 101)(14 102)(15 103)(16 104)(17 105)(18 106)(19 107)(20 108)(21 109)(22 110)(23 111)(24 112)(25 113)(26 114)(27 115)(28 116)(29 117)(30 118)(31 119)(32 120)(33 121)(34 122)(35 123)(36 124)(37 125)(38 126)(39 127)(40 128)(41 129)(42 130)(43 131)(44 132)(45 133)(46 134)(47 135)(48 136)(49 137)(50 138)(51 139)(52 140)(53 141)(54 142)(55 143)(56 144)(57 145)(58 146)(59 147)(60 148)(61 149)(62 150)(63 151)(64 152)(65 153)(66 154)(67 155)(68 156)(69 157)(70 158)(71 159)(72 160)(73 161)(74 162)(75 163)(76 164)(77 165)(78 166)(79 167)(80 168)(81 169)(82 170)(83 171)(84 172)(85 173)(86 174)(87 175)(88 176)
(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)(89 122 100 111)(90 123 101 112)(91 124 102 113)(92 125 103 114)(93 126 104 115)(94 127 105 116)(95 128 106 117)(96 129 107 118)(97 130 108 119)(98 131 109 120)(99 132 110 121)(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)
(1 166)(2 167)(3 168)(4 169)(5 170)(6 171)(7 172)(8 173)(9 174)(10 175)(11 176)(12 155)(13 156)(14 157)(15 158)(16 159)(17 160)(18 161)(19 162)(20 163)(21 164)(22 165)(23 133)(24 134)(25 135)(26 136)(27 137)(28 138)(29 139)(30 140)(31 141)(32 142)(33 143)(34 144)(35 145)(36 146)(37 147)(38 148)(39 149)(40 150)(41 151)(42 152)(43 153)(44 154)(45 111)(46 112)(47 113)(48 114)(49 115)(50 116)(51 117)(52 118)(53 119)(54 120)(55 121)(56 122)(57 123)(58 124)(59 125)(60 126)(61 127)(62 128)(63 129)(64 130)(65 131)(66 132)(67 100)(68 101)(69 102)(70 103)(71 104)(72 105)(73 106)(74 107)(75 108)(76 109)(77 110)(78 89)(79 90)(80 91)(81 92)(82 93)(83 94)(84 95)(85 96)(86 97)(87 98)(88 99)
(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 99)(2 98)(3 97)(4 96)(5 95)(6 94)(7 93)(8 92)(9 91)(10 90)(11 89)(12 110)(13 109)(14 108)(15 107)(16 106)(17 105)(18 104)(19 103)(20 102)(21 101)(22 100)(23 132)(24 131)(25 130)(26 129)(27 128)(28 127)(29 126)(30 125)(31 124)(32 123)(33 122)(34 121)(35 120)(36 119)(37 118)(38 117)(39 116)(40 115)(41 114)(42 113)(43 112)(44 111)(45 165)(46 164)(47 163)(48 162)(49 161)(50 160)(51 159)(52 158)(53 157)(54 156)(55 155)(56 176)(57 175)(58 174)(59 173)(60 172)(61 171)(62 170)(63 169)(64 168)(65 167)(66 166)(67 143)(68 142)(69 141)(70 140)(71 139)(72 138)(73 137)(74 136)(75 135)(76 134)(77 133)(78 154)(79 153)(80 152)(81 151)(82 150)(83 149)(84 148)(85 147)(86 146)(87 145)(88 144)

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

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

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

64 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 8A 8B 8C 8D 11A ··· 11E 22A ··· 22O 22P ··· 22AI 44A ··· 44J order 1 2 2 2 2 2 2 2 4 4 8 8 8 8 11 ··· 11 22 ··· 22 22 ··· 22 44 ··· 44 size 1 1 1 1 4 4 44 44 2 2 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 D8 D11 D22 D22 C11⋊D4 C11⋊D4 D4⋊D11 kernel C2×D4⋊D11 C2×C11⋊C8 D4⋊D11 C2×D44 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 1 84 0 0 36 88
,
 88 0 0 0 0 88 0 0 0 0 0 71 0 0 84 0
,
 55 88 0 0 1 0 0 0 0 0 1 0 0 0 0 1
,
 34 1 0 0 2 55 0 0 0 0 1 0 0 0 36 88
G:=sub<GL(4,GF(89))| [88,0,0,0,0,88,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,1,36,0,0,84,88],[88,0,0,0,0,88,0,0,0,0,0,84,0,0,71,0],[55,1,0,0,88,0,0,0,0,0,1,0,0,0,0,1],[34,2,0,0,1,55,0,0,0,0,1,36,0,0,0,88] >;

C2×D4⋊D11 in GAP, Magma, Sage, TeX

C_2\times D_4\rtimes D_{11}
% in TeX

G:=Group("C2xD4:D11");
// GroupNames label

G:=SmallGroup(352,126);
// by ID

G=gap.SmallGroup(352,126);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-11,218,579,159,69,11525]);
// Polycyclic

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

׿
×
𝔽