direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C5×C23.Q8, (C2×C20).309D4, C23.3(C5×Q8), C24.7(C2×C10), (C22×C10).3Q8, C22.72(D4×C10), C22.22(Q8×C10), C2.C42⋊4C10, C10.89(C22⋊Q8), (C23×C10).7C22, C10.139(C4⋊D4), C23.79(C22×C10), (C22×C20).34C22, C10.34(C42⋊2C2), (C22×C10).460C23, (C2×C4⋊C4)⋊6C10, (C10×C4⋊C4)⋊33C2, (C2×C4).16(C5×D4), C2.8(C5×C4⋊D4), C2.8(C5×C22⋊Q8), (C2×C10).612(C2×D4), (C2×C22⋊C4).9C10, (C22×C4).7(C2×C10), C2.4(C5×C42⋊2C2), (C2×C10).110(C2×Q8), C22.39(C5×C4○D4), (C10×C22⋊C4).28C2, (C5×C2.C42)⋊6C2, (C2×C10).220(C4○D4), SmallGroup(320,897)
Series: Derived ►Chief ►Lower central ►Upper central
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, C42⋊2C2, C5×D4, C5×Q8, C22×C10, C23.Q8, D4×C10, Q8×C10, C5×C4○D4, C5×C4⋊D4, C5×C22⋊Q8, C5×C42⋊2C2, 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