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## G = C22×Q8×D5order 320 = 26·5

### Direct product of C22, Q8 and D5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C22×Q8×D5
 Chief series C1 — C5 — C10 — D10 — C22×D5 — C23×D5 — D5×C22×C4 — C22×Q8×D5
 Lower central C5 — C10 — C22×Q8×D5
 Upper central C1 — C23 — C22×Q8

Generators and relations for C22×Q8×D5
G = < a,b,c,d,e,f | a2=b2=c4=e5=f2=1, d2=c2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd-1=c-1, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >

Subgroups: 2238 in 850 conjugacy classes, 503 normal (10 characteristic)
C1, C2, C2 [×6], C2 [×8], C4 [×12], C4 [×12], C22 [×7], C22 [×28], C5, C2×C4 [×18], C2×C4 [×66], Q8 [×16], Q8 [×48], C23, C23 [×14], D5 [×8], C10, C10 [×6], C22×C4 [×3], C22×C4 [×39], C2×Q8 [×12], C2×Q8 [×100], C24, Dic5 [×12], C20 [×12], D10 [×28], C2×C10 [×7], C23×C4 [×3], C22×Q8, C22×Q8 [×27], Dic10 [×48], C4×D5 [×48], C2×Dic5 [×18], C2×C20 [×18], C5×Q8 [×16], C22×D5 [×14], C22×C10, Q8×C23, C2×Dic10 [×36], C2×C4×D5 [×36], Q8×D5 [×64], C22×Dic5 [×3], C22×C20 [×3], Q8×C10 [×12], C23×D5, C22×Dic10 [×3], D5×C22×C4 [×3], C2×Q8×D5 [×24], Q8×C2×C10, C22×Q8×D5
Quotients: C1, C2 [×31], C22 [×155], Q8 [×8], C23 [×155], D5, C2×Q8 [×28], C24 [×31], D10 [×15], C22×Q8 [×14], C25, C22×D5 [×35], Q8×C23, Q8×D5 [×4], C23×D5 [×15], C2×Q8×D5 [×6], D5×C24, C22×Q8×D5

Smallest permutation representation of C22×Q8×D5
On 160 points
Generators in S160
(1 89)(2 90)(3 86)(4 87)(5 88)(6 81)(7 82)(8 83)(9 84)(10 85)(11 96)(12 97)(13 98)(14 99)(15 100)(16 91)(17 92)(18 93)(19 94)(20 95)(21 106)(22 107)(23 108)(24 109)(25 110)(26 101)(27 102)(28 103)(29 104)(30 105)(31 116)(32 117)(33 118)(34 119)(35 120)(36 111)(37 112)(38 113)(39 114)(40 115)(41 126)(42 127)(43 128)(44 129)(45 130)(46 121)(47 122)(48 123)(49 124)(50 125)(51 136)(52 137)(53 138)(54 139)(55 140)(56 131)(57 132)(58 133)(59 134)(60 135)(61 146)(62 147)(63 148)(64 149)(65 150)(66 141)(67 142)(68 143)(69 144)(70 145)(71 156)(72 157)(73 158)(74 159)(75 160)(76 151)(77 152)(78 153)(79 154)(80 155)
(1 49)(2 50)(3 46)(4 47)(5 48)(6 41)(7 42)(8 43)(9 44)(10 45)(11 56)(12 57)(13 58)(14 59)(15 60)(16 51)(17 52)(18 53)(19 54)(20 55)(21 66)(22 67)(23 68)(24 69)(25 70)(26 61)(27 62)(28 63)(29 64)(30 65)(31 76)(32 77)(33 78)(34 79)(35 80)(36 71)(37 72)(38 73)(39 74)(40 75)(81 126)(82 127)(83 128)(84 129)(85 130)(86 121)(87 122)(88 123)(89 124)(90 125)(91 136)(92 137)(93 138)(94 139)(95 140)(96 131)(97 132)(98 133)(99 134)(100 135)(101 146)(102 147)(103 148)(104 149)(105 150)(106 141)(107 142)(108 143)(109 144)(110 145)(111 156)(112 157)(113 158)(114 159)(115 160)(116 151)(117 152)(118 153)(119 154)(120 155)
(1 29 9 24)(2 30 10 25)(3 26 6 21)(4 27 7 22)(5 28 8 23)(11 36 16 31)(12 37 17 32)(13 38 18 33)(14 39 19 34)(15 40 20 35)(41 66 46 61)(42 67 47 62)(43 68 48 63)(44 69 49 64)(45 70 50 65)(51 76 56 71)(52 77 57 72)(53 78 58 73)(54 79 59 74)(55 80 60 75)(81 106 86 101)(82 107 87 102)(83 108 88 103)(84 109 89 104)(85 110 90 105)(91 116 96 111)(92 117 97 112)(93 118 98 113)(94 119 99 114)(95 120 100 115)(121 146 126 141)(122 147 127 142)(123 148 128 143)(124 149 129 144)(125 150 130 145)(131 156 136 151)(132 157 137 152)(133 158 138 153)(134 159 139 154)(135 160 140 155)
(1 99 9 94)(2 100 10 95)(3 96 6 91)(4 97 7 92)(5 98 8 93)(11 81 16 86)(12 82 17 87)(13 83 18 88)(14 84 19 89)(15 85 20 90)(21 111 26 116)(22 112 27 117)(23 113 28 118)(24 114 29 119)(25 115 30 120)(31 106 36 101)(32 107 37 102)(33 108 38 103)(34 109 39 104)(35 110 40 105)(41 136 46 131)(42 137 47 132)(43 138 48 133)(44 139 49 134)(45 140 50 135)(51 121 56 126)(52 122 57 127)(53 123 58 128)(54 124 59 129)(55 125 60 130)(61 151 66 156)(62 152 67 157)(63 153 68 158)(64 154 69 159)(65 155 70 160)(71 146 76 141)(72 147 77 142)(73 148 78 143)(74 149 79 144)(75 150 80 145)
(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)
(1 48)(2 47)(3 46)(4 50)(5 49)(6 41)(7 45)(8 44)(9 43)(10 42)(11 56)(12 60)(13 59)(14 58)(15 57)(16 51)(17 55)(18 54)(19 53)(20 52)(21 66)(22 70)(23 69)(24 68)(25 67)(26 61)(27 65)(28 64)(29 63)(30 62)(31 76)(32 80)(33 79)(34 78)(35 77)(36 71)(37 75)(38 74)(39 73)(40 72)(81 126)(82 130)(83 129)(84 128)(85 127)(86 121)(87 125)(88 124)(89 123)(90 122)(91 136)(92 140)(93 139)(94 138)(95 137)(96 131)(97 135)(98 134)(99 133)(100 132)(101 146)(102 150)(103 149)(104 148)(105 147)(106 141)(107 145)(108 144)(109 143)(110 142)(111 156)(112 160)(113 159)(114 158)(115 157)(116 151)(117 155)(118 154)(119 153)(120 152)

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

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

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

80 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 4A ··· 4L 4M ··· 4X 5A 5B 10A ··· 10N 20A ··· 20X order 1 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4 5 5 10 ··· 10 20 ··· 20 size 1 1 ··· 1 5 ··· 5 2 ··· 2 10 ··· 10 2 2 2 ··· 2 4 ··· 4

80 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 4 type + + + + + - + + + - image C1 C2 C2 C2 C2 Q8 D5 D10 D10 Q8×D5 kernel C22×Q8×D5 C22×Dic10 D5×C22×C4 C2×Q8×D5 Q8×C2×C10 C22×D5 C22×Q8 C22×C4 C2×Q8 C22 # reps 1 3 3 24 1 8 2 6 24 8

Matrix representation of C22×Q8×D5 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 40 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 9 0 0 0 0 0 0 32
,
 40 1 0 0 0 0 5 35 0 0 0 0 0 0 34 1 0 0 0 0 40 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 40 0 0 0 0 0 5 1 0 0 0 0 0 0 40 7 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,9,0,0,0,0,0,0,32],[40,5,0,0,0,0,1,35,0,0,0,0,0,0,34,40,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,5,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,7,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C22×Q8×D5 in GAP, Magma, Sage, TeX

C_2^2\times Q_8\times D_5
% in TeX

G:=Group("C2^2xQ8xD5");
// GroupNames label

G:=SmallGroup(320,1615);
// by ID

G=gap.SmallGroup(320,1615);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,136,235,102,12550]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^4=e^5=f^2=1,d^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,d*c*d^-1=c^-1,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f=e^-1>;
// generators/relations

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