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## G = C2×Q8⋊D11order 352 = 25·11

### Direct product of C2 and Q8⋊D11

Series: Derived Chief Lower central Upper central

 Derived series C1 — C44 — C2×Q8⋊D11
 Chief series C1 — C11 — C22 — C44 — D44 — C2×D44 — C2×Q8⋊D11
 Lower central C11 — C22 — C44 — C2×Q8⋊D11
 Upper central C1 — C22 — C2×C4 — C2×Q8

Generators and relations for C2×Q8⋊D11
G = < a,b,c,d,e | a2=b4=d11=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe=b-1, bd=db, cd=dc, ece=b-1c, ede=d-1 >

Subgroups: 458 in 68 conjugacy classes, 33 normal (17 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, C11, C2×C8, SD16, C2×D4, C2×Q8, D11, C22, C22, C2×SD16, C44, C44, D22, C2×C22, C11⋊C8, D44, D44, C2×C44, C2×C44, Q8×C11, Q8×C11, C22×D11, C2×C11⋊C8, Q8⋊D11, C2×D44, Q8×C22, C2×Q8⋊D11
Quotients: C1, C2, C22, D4, C23, SD16, C2×D4, D11, C2×SD16, D22, C11⋊D4, C22×D11, Q8⋊D11, C2×C11⋊D4, C2×Q8⋊D11

Smallest permutation representation of C2×Q8⋊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)
(1 144 12 133)(2 145 13 134)(3 146 14 135)(4 147 15 136)(5 148 16 137)(6 149 17 138)(7 150 18 139)(8 151 19 140)(9 152 20 141)(10 153 21 142)(11 154 22 143)(23 166 34 155)(24 167 35 156)(25 168 36 157)(26 169 37 158)(27 170 38 159)(28 171 39 160)(29 172 40 161)(30 173 41 162)(31 174 42 163)(32 175 43 164)(33 176 44 165)(45 89 56 100)(46 90 57 101)(47 91 58 102)(48 92 59 103)(49 93 60 104)(50 94 61 105)(51 95 62 106)(52 96 63 107)(53 97 64 108)(54 98 65 109)(55 99 66 110)(67 111 78 122)(68 112 79 123)(69 113 80 124)(70 114 81 125)(71 115 82 126)(72 116 83 127)(73 117 84 128)(74 118 85 129)(75 119 86 130)(76 120 87 131)(77 121 88 132)
(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 11)(2 10)(3 9)(4 8)(5 7)(12 22)(13 21)(14 20)(15 19)(16 18)(23 44)(24 43)(25 42)(26 41)(27 40)(28 39)(29 38)(30 37)(31 36)(32 35)(33 34)(45 77)(46 76)(47 75)(48 74)(49 73)(50 72)(51 71)(52 70)(53 69)(54 68)(55 67)(56 88)(57 87)(58 86)(59 85)(60 84)(61 83)(62 82)(63 81)(64 80)(65 79)(66 78)(89 99)(90 98)(91 97)(92 96)(93 95)(100 110)(101 109)(102 108)(103 107)(104 106)(111 132)(112 131)(113 130)(114 129)(115 128)(116 127)(117 126)(118 125)(119 124)(120 123)(121 122)(133 165)(134 164)(135 163)(136 162)(137 161)(138 160)(139 159)(140 158)(141 157)(142 156)(143 155)(144 176)(145 175)(146 174)(147 173)(148 172)(149 171)(150 170)(151 169)(152 168)(153 167)(154 166)

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

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

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

64 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 8A 8B 8C 8D 11A ··· 11E 22A ··· 22O 44A ··· 44AD order 1 2 2 2 2 2 4 4 4 4 8 8 8 8 11 ··· 11 22 ··· 22 44 ··· 44 size 1 1 1 1 44 44 2 2 4 4 22 22 22 22 2 ··· 2 2 ··· 2 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 Q8⋊D11 kernel C2×Q8⋊D11 C2×C11⋊C8 Q8⋊D11 C2×D44 Q8×C22 C44 C2×C22 C22 C2×Q8 C2×C4 Q8 C4 C22 C2 # reps 1 1 4 1 1 1 1 4 5 5 10 10 10 10

Matrix representation of C2×Q8⋊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 36 0 0 84 88
,
 1 0 0 0 0 1 0 0 0 0 49 81 0 0 11 40
,
 55 88 0 0 60 41 0 0 0 0 1 0 0 0 0 1
,
 20 62 0 0 84 69 0 0 0 0 1 0 0 0 84 88
G:=sub<GL(4,GF(89))| [88,0,0,0,0,88,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,1,84,0,0,36,88],[1,0,0,0,0,1,0,0,0,0,49,11,0,0,81,40],[55,60,0,0,88,41,0,0,0,0,1,0,0,0,0,1],[20,84,0,0,62,69,0,0,0,0,1,84,0,0,0,88] >;

C2×Q8⋊D11 in GAP, Magma, Sage, TeX

C_2\times Q_8\rtimes D_{11}
% in TeX

G:=Group("C2xQ8:D11");
// GroupNames label

G:=SmallGroup(352,136);
// by ID

G=gap.SmallGroup(352,136);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-11,218,86,579,159,69,11525]);
// Polycyclic

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

׿
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𝔽