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## G = C5×C23.Q8order 320 = 26·5

### Direct product of C5 and C23.Q8

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C23 — C5×C23.Q8
 Chief series C1 — C2 — C22 — C23 — C22×C10 — C22×C20 — C10×C4⋊C4 — C5×C23.Q8
 Lower central C1 — C23 — C5×C23.Q8
 Upper central C1 — C22×C10 — C5×C23.Q8

Generators and relations for C5×C23.Q8
G = < a,b,c,d,e,f | a5=b2=c2=d2=e4=1, f2=ce2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, fbf-1=bc=cb, ebe-1=bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e-1 >

Subgroups: 330 in 186 conjugacy classes, 78 normal (14 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, C23, C23, C23, C10, C10, C10, C22⋊C4, C4⋊C4, C22×C4, C24, C20, C2×C10, C2×C10, C2×C10, C2.C42, C2×C22⋊C4, C2×C4⋊C4, C2×C20, C2×C20, C22×C10, C22×C10, C22×C10, C23.Q8, C5×C22⋊C4, C5×C4⋊C4, C22×C20, C23×C10, C5×C2.C42, C10×C22⋊C4, C10×C4⋊C4, C5×C23.Q8
Quotients: C1, C2, C22, C5, D4, Q8, C23, C10, C2×D4, C2×Q8, C4○D4, C2×C10, C4⋊D4, C22⋊Q8, C422C2, C5×D4, C5×Q8, C22×C10, C23.Q8, D4×C10, Q8×C10, C5×C4○D4, C5×C4⋊D4, C5×C22⋊Q8, C5×C422C2, C5×C23.Q8

Smallest permutation representation of C5×C23.Q8
On 160 points
Generators in S160
(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 42)(2 43)(3 44)(4 45)(5 41)(6 22)(7 23)(8 24)(9 25)(10 21)(11 70)(12 66)(13 67)(14 68)(15 69)(16 32)(17 33)(18 34)(19 35)(20 31)(26 49)(27 50)(28 46)(29 47)(30 48)(36 53)(37 54)(38 55)(39 51)(40 52)(56 72)(57 73)(58 74)(59 75)(60 71)(61 79)(62 80)(63 76)(64 77)(65 78)(81 87)(82 88)(83 89)(84 90)(85 86)(91 108)(92 109)(93 110)(94 106)(95 107)(96 133)(97 134)(98 135)(99 131)(100 132)(101 123)(102 124)(103 125)(104 121)(105 122)(111 149)(112 150)(113 146)(114 147)(115 148)(116 126)(117 127)(118 128)(119 129)(120 130)(136 143)(137 144)(138 145)(139 141)(140 142)(151 160)(152 156)(153 157)(154 158)(155 159)
(1 66)(2 67)(3 68)(4 69)(5 70)(6 20)(7 16)(8 17)(9 18)(10 19)(11 41)(12 42)(13 43)(14 44)(15 45)(21 35)(22 31)(23 32)(24 33)(25 34)(26 39)(27 40)(28 36)(29 37)(30 38)(46 53)(47 54)(48 55)(49 51)(50 52)(56 76)(57 77)(58 78)(59 79)(60 80)(61 75)(62 71)(63 72)(64 73)(65 74)(81 106)(82 107)(83 108)(84 109)(85 110)(86 93)(87 94)(88 95)(89 91)(90 92)(96 116)(97 117)(98 118)(99 119)(100 120)(101 115)(102 111)(103 112)(104 113)(105 114)(121 146)(122 147)(123 148)(124 149)(125 150)(126 133)(127 134)(128 135)(129 131)(130 132)(136 156)(137 157)(138 158)(139 159)(140 160)(141 155)(142 151)(143 152)(144 153)(145 154)
(1 29)(2 30)(3 26)(4 27)(5 28)(6 136)(7 137)(8 138)(9 139)(10 140)(11 53)(12 54)(13 55)(14 51)(15 52)(16 157)(17 158)(18 159)(19 160)(20 156)(21 142)(22 143)(23 144)(24 145)(25 141)(31 152)(32 153)(33 154)(34 155)(35 151)(36 70)(37 66)(38 67)(39 68)(40 69)(41 46)(42 47)(43 48)(44 49)(45 50)(56 85)(57 81)(58 82)(59 83)(60 84)(61 91)(62 92)(63 93)(64 94)(65 95)(71 90)(72 86)(73 87)(74 88)(75 89)(76 110)(77 106)(78 107)(79 108)(80 109)(96 125)(97 121)(98 122)(99 123)(100 124)(101 131)(102 132)(103 133)(104 134)(105 135)(111 130)(112 126)(113 127)(114 128)(115 129)(116 150)(117 146)(118 147)(119 148)(120 149)
(1 64 12 81)(2 65 13 82)(3 61 14 83)(4 62 15 84)(5 63 11 85)(6 103 143 125)(7 104 144 121)(8 105 145 122)(9 101 141 123)(10 102 142 124)(16 113 153 146)(17 114 154 147)(18 115 155 148)(19 111 151 149)(20 112 152 150)(21 100 140 132)(22 96 136 133)(23 97 137 134)(24 98 138 135)(25 99 139 131)(26 91 51 59)(27 92 52 60)(28 93 53 56)(29 94 54 57)(30 95 55 58)(31 116 156 126)(32 117 157 127)(33 118 158 128)(34 119 159 129)(35 120 160 130)(36 86 46 76)(37 87 47 77)(38 88 48 78)(39 89 49 79)(40 90 50 80)(41 110 70 72)(42 106 66 73)(43 107 67 74)(44 108 68 75)(45 109 69 71)
(1 113 42 121)(2 114 43 122)(3 115 44 123)(4 111 45 124)(5 112 41 125)(6 85 152 72)(7 81 153 73)(8 82 154 74)(9 83 155 75)(10 84 151 71)(11 150 70 103)(12 146 66 104)(13 147 67 105)(14 148 68 101)(15 149 69 102)(16 106 144 64)(17 107 145 65)(18 108 141 61)(19 109 142 62)(20 110 143 63)(21 92 160 80)(22 93 156 76)(23 94 157 77)(24 95 158 78)(25 91 159 79)(26 129 49 99)(27 130 50 100)(28 126 46 96)(29 127 47 97)(30 128 48 98)(31 86 136 56)(32 87 137 57)(33 88 138 58)(34 89 139 59)(35 90 140 60)(36 133 53 116)(37 134 54 117)(38 135 55 118)(39 131 51 119)(40 132 52 120)

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

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

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

110 conjugacy classes

 class 1 2A ··· 2G 2H 2I 4A ··· 4L 5A 5B 5C 5D 10A ··· 10AB 10AC ··· 10AJ 20A ··· 20AV order 1 2 ··· 2 2 2 4 ··· 4 5 5 5 5 10 ··· 10 10 ··· 10 20 ··· 20 size 1 1 ··· 1 4 4 4 ··· 4 1 1 1 1 1 ··· 1 4 ··· 4 4 ··· 4

110 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + - image C1 C2 C2 C2 C5 C10 C10 C10 D4 Q8 C4○D4 C5×D4 C5×Q8 C5×C4○D4 kernel C5×C23.Q8 C5×C2.C42 C10×C22⋊C4 C10×C4⋊C4 C23.Q8 C2.C42 C2×C22⋊C4 C2×C4⋊C4 C2×C20 C22×C10 C2×C10 C2×C4 C23 C22 # reps 1 1 3 3 4 4 12 12 6 2 6 24 8 24

Matrix representation of C5×C23.Q8 in GL6(𝔽41)

 18 0 0 0 0 0 0 18 0 0 0 0 0 0 10 0 0 0 0 0 0 10 0 0 0 0 0 0 18 0 0 0 0 0 0 18
,
 40 21 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 9 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 40 0 0 0 0 0 0 40 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
,
 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
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 9 39 0 0 0 0 0 32 0 0 0 0 0 0 9 39 0 0 0 0 0 32
,
 10 19 0 0 0 0 40 31 0 0 0 0 0 0 32 2 0 0 0 0 1 9 0 0 0 0 0 0 17 4 0 0 0 0 30 24

G:=sub<GL(6,GF(41))| [18,0,0,0,0,0,0,18,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,18,0,0,0,0,0,0,18],[40,0,0,0,0,0,21,1,0,0,0,0,0,0,1,9,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,0,0,0,0,0,39,32,0,0,0,0,0,0,9,0,0,0,0,0,39,32],[10,40,0,0,0,0,19,31,0,0,0,0,0,0,32,1,0,0,0,0,2,9,0,0,0,0,0,0,17,30,0,0,0,0,4,24] >;

C5×C23.Q8 in GAP, Magma, Sage, TeX

C_5\times C_2^3.Q_8
% in TeX

G:=Group("C5xC2^3.Q8");
// GroupNames label

G:=SmallGroup(320,897);
// by ID

G=gap.SmallGroup(320,897);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,840,589,288,1766,1731]);
// Polycyclic

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

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