direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C5×C23.38D4, (C2×Q8)⋊6C20, (Q8×C10)⋊26C4, C4.56(D4×C10), Q8.6(C2×C20), C20.463(C2×D4), (C2×C20).316D4, C4.6(C22×C20), C23.38(C5×D4), Q8⋊C4⋊15C10, C22.46(D4×C10), (C22×Q8).5C10, (C2×C40).323C22, (C2×C20).895C23, C20.210(C22×C4), C42⋊C2.4C10, (C22×C10).160D4, C20.130(C22⋊C4), (C10×M4(2)).30C2, (C2×M4(2)).12C10, (Q8×C10).253C22, C10.127(C8.C22), (C22×C20).412C22, (C2×C4).24(C5×D4), (Q8×C2×C10).15C2, C4⋊C4.40(C2×C10), (C2×C8).48(C2×C10), (C2×C4).22(C2×C20), C4.15(C5×C22⋊C4), (C5×Q8).45(C2×C4), C2.2(C5×C8.C22), (C2×C20).368(C2×C4), (C5×Q8⋊C4)⋊38C2, (C2×C10).622(C2×D4), C2.22(C10×C22⋊C4), (C2×Q8).38(C2×C10), (C5×C4⋊C4).361C22, C10.151(C2×C22⋊C4), (C22×C4).31(C2×C10), (C2×C4).70(C22×C10), C22.21(C5×C22⋊C4), (C5×C42⋊C2).18C2, (C2×C10).146(C22⋊C4), SmallGroup(320,920)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C23.38D4
G = < a,b,c,d,e,f | a5=b2=c2=d2=1, e4=d, f2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, ebe-1=fbf-1=bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=ce3 >
Subgroups: 226 in 150 conjugacy classes, 82 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×2], C4 [×6], C22, C22 [×2], C22 [×2], C5, C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×8], Q8 [×4], Q8 [×6], C23, C10, C10 [×2], C10 [×2], C42, C22⋊C4, C4⋊C4 [×2], C2×C8 [×2], M4(2) [×2], C22×C4, C22×C4, C2×Q8 [×6], C2×Q8 [×3], C20 [×2], C20 [×2], C20 [×6], C2×C10, C2×C10 [×2], C2×C10 [×2], Q8⋊C4 [×4], C42⋊C2, C2×M4(2), C22×Q8, C40 [×2], C2×C20 [×2], C2×C20 [×4], C2×C20 [×8], C5×Q8 [×4], C5×Q8 [×6], C22×C10, C23.38D4, C4×C20, C5×C22⋊C4, C5×C4⋊C4 [×2], C2×C40 [×2], C5×M4(2) [×2], C22×C20, C22×C20, Q8×C10 [×6], Q8×C10 [×3], C5×Q8⋊C4 [×4], C5×C42⋊C2, C10×M4(2), Q8×C2×C10, C5×C23.38D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C5, C2×C4 [×6], D4 [×4], C23, C10 [×7], C22⋊C4 [×4], C22×C4, C2×D4 [×2], C20 [×4], C2×C10 [×7], C2×C22⋊C4, C8.C22 [×2], C2×C20 [×6], C5×D4 [×4], C22×C10, C23.38D4, C5×C22⋊C4 [×4], C22×C20, D4×C10 [×2], C10×C22⋊C4, C5×C8.C22 [×2], C5×C23.38D4
►On 160 points
Generators in S160
(1 127 159 39 151)(2 128 160 40 152)(3 121 153 33 145)(4 122 154 34 146)(5 123 155 35 147)(6 124 156 36 148)(7 125 157 37 149)(8 126 158 38 150)(9 113 137 17 129)(10 114 138 18 130)(11 115 139 19 131)(12 116 140 20 132)(13 117 141 21 133)(14 118 142 22 134)(15 119 143 23 135)(16 120 144 24 136)(25 89 56 81 44)(26 90 49 82 45)(27 91 50 83 46)(28 92 51 84 47)(29 93 52 85 48)(30 94 53 86 41)(31 95 54 87 42)(32 96 55 88 43)(57 78 111 70 103)(58 79 112 71 104)(59 80 105 72 97)(60 73 106 65 98)(61 74 107 66 99)(62 75 108 67 100)(63 76 109 68 101)(64 77 110 69 102)
(1 85)(2 82)(3 87)(4 84)(5 81)(6 86)(7 83)(8 88)(9 70)(10 67)(11 72)(12 69)(13 66)(14 71)(15 68)(16 65)(17 78)(18 75)(19 80)(20 77)(21 74)(22 79)(23 76)(24 73)(25 155)(26 160)(27 157)(28 154)(29 159)(30 156)(31 153)(32 158)(33 95)(34 92)(35 89)(36 94)(37 91)(38 96)(39 93)(40 90)(41 124)(42 121)(43 126)(44 123)(45 128)(46 125)(47 122)(48 127)(49 152)(50 149)(51 146)(52 151)(53 148)(54 145)(55 150)(56 147)(57 137)(58 142)(59 139)(60 144)(61 141)(62 138)(63 143)(64 140)(97 115)(98 120)(99 117)(100 114)(101 119)(102 116)(103 113)(104 118)(105 131)(106 136)(107 133)(108 130)(109 135)(110 132)(111 129)(112 134)
(1 85)(2 86)(3 87)(4 88)(5 81)(6 82)(7 83)(8 84)(9 70)(10 71)(11 72)(12 65)(13 66)(14 67)(15 68)(16 69)(17 78)(18 79)(19 80)(20 73)(21 74)(22 75)(23 76)(24 77)(25 155)(26 156)(27 157)(28 158)(29 159)(30 160)(31 153)(32 154)(33 95)(34 96)(35 89)(36 90)(37 91)(38 92)(39 93)(40 94)(41 128)(42 121)(43 122)(44 123)(45 124)(46 125)(47 126)(48 127)(49 148)(50 149)(51 150)(52 151)(53 152)(54 145)(55 146)(56 147)(57 137)(58 138)(59 139)(60 140)(61 141)(62 142)(63 143)(64 144)(97 115)(98 116)(99 117)(100 118)(101 119)(102 120)(103 113)(104 114)(105 131)(106 132)(107 133)(108 134)(109 135)(110 136)(111 129)(112 130)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 64)(65 69)(66 70)(67 71)(68 72)(73 77)(74 78)(75 79)(76 80)(81 85)(82 86)(83 87)(84 88)(89 93)(90 94)(91 95)(92 96)(97 101)(98 102)(99 103)(100 104)(105 109)(106 110)(107 111)(108 112)(113 117)(114 118)(115 119)(116 120)(121 125)(122 126)(123 127)(124 128)(129 133)(130 134)(131 135)(132 136)(137 141)(138 142)(139 143)(140 144)(145 149)(146 150)(147 151)(148 152)(153 157)(154 158)(155 159)(156 160)
(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)
(1 104 85 114)(2 117 86 99)(3 102 87 120)(4 115 88 97)(5 100 81 118)(6 113 82 103)(7 98 83 116)(8 119 84 101)(9 49 70 148)(10 151 71 52)(11 55 72 146)(12 149 65 50)(13 53 66 152)(14 147 67 56)(15 51 68 150)(16 145 69 54)(17 26 78 156)(18 159 79 29)(19 32 80 154)(20 157 73 27)(21 30 74 160)(22 155 75 25)(23 28 76 158)(24 153 77 31)(33 110 95 136)(34 131 96 105)(35 108 89 134)(36 129 90 111)(37 106 91 132)(38 135 92 109)(39 112 93 130)(40 133 94 107)(41 61 128 141)(42 144 121 64)(43 59 122 139)(44 142 123 62)(45 57 124 137)(46 140 125 60)(47 63 126 143)(48 138 127 58)
G:=sub<Sym(160)| (1,127,159,39,151)(2,128,160,40,152)(3,121,153,33,145)(4,122,154,34,146)(5,123,155,35,147)(6,124,156,36,148)(7,125,157,37,149)(8,126,158,38,150)(9,113,137,17,129)(10,114,138,18,130)(11,115,139,19,131)(12,116,140,20,132)(13,117,141,21,133)(14,118,142,22,134)(15,119,143,23,135)(16,120,144,24,136)(25,89,56,81,44)(26,90,49,82,45)(27,91,50,83,46)(28,92,51,84,47)(29,93,52,85,48)(30,94,53,86,41)(31,95,54,87,42)(32,96,55,88,43)(57,78,111,70,103)(58,79,112,71,104)(59,80,105,72,97)(60,73,106,65,98)(61,74,107,66,99)(62,75,108,67,100)(63,76,109,68,101)(64,77,110,69,102), (1,85)(2,82)(3,87)(4,84)(5,81)(6,86)(7,83)(8,88)(9,70)(10,67)(11,72)(12,69)(13,66)(14,71)(15,68)(16,65)(17,78)(18,75)(19,80)(20,77)(21,74)(22,79)(23,76)(24,73)(25,155)(26,160)(27,157)(28,154)(29,159)(30,156)(31,153)(32,158)(33,95)(34,92)(35,89)(36,94)(37,91)(38,96)(39,93)(40,90)(41,124)(42,121)(43,126)(44,123)(45,128)(46,125)(47,122)(48,127)(49,152)(50,149)(51,146)(52,151)(53,148)(54,145)(55,150)(56,147)(57,137)(58,142)(59,139)(60,144)(61,141)(62,138)(63,143)(64,140)(97,115)(98,120)(99,117)(100,114)(101,119)(102,116)(103,113)(104,118)(105,131)(106,136)(107,133)(108,130)(109,135)(110,132)(111,129)(112,134), (1,85)(2,86)(3,87)(4,88)(5,81)(6,82)(7,83)(8,84)(9,70)(10,71)(11,72)(12,65)(13,66)(14,67)(15,68)(16,69)(17,78)(18,79)(19,80)(20,73)(21,74)(22,75)(23,76)(24,77)(25,155)(26,156)(27,157)(28,158)(29,159)(30,160)(31,153)(32,154)(33,95)(34,96)(35,89)(36,90)(37,91)(38,92)(39,93)(40,94)(41,128)(42,121)(43,122)(44,123)(45,124)(46,125)(47,126)(48,127)(49,148)(50,149)(51,150)(52,151)(53,152)(54,145)(55,146)(56,147)(57,137)(58,138)(59,139)(60,140)(61,141)(62,142)(63,143)(64,144)(97,115)(98,116)(99,117)(100,118)(101,119)(102,120)(103,113)(104,114)(105,131)(106,132)(107,133)(108,134)(109,135)(110,136)(111,129)(112,130), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64)(65,69)(66,70)(67,71)(68,72)(73,77)(74,78)(75,79)(76,80)(81,85)(82,86)(83,87)(84,88)(89,93)(90,94)(91,95)(92,96)(97,101)(98,102)(99,103)(100,104)(105,109)(106,110)(107,111)(108,112)(113,117)(114,118)(115,119)(116,120)(121,125)(122,126)(123,127)(124,128)(129,133)(130,134)(131,135)(132,136)(137,141)(138,142)(139,143)(140,144)(145,149)(146,150)(147,151)(148,152)(153,157)(154,158)(155,159)(156,160), (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), (1,104,85,114)(2,117,86,99)(3,102,87,120)(4,115,88,97)(5,100,81,118)(6,113,82,103)(7,98,83,116)(8,119,84,101)(9,49,70,148)(10,151,71,52)(11,55,72,146)(12,149,65,50)(13,53,66,152)(14,147,67,56)(15,51,68,150)(16,145,69,54)(17,26,78,156)(18,159,79,29)(19,32,80,154)(20,157,73,27)(21,30,74,160)(22,155,75,25)(23,28,76,158)(24,153,77,31)(33,110,95,136)(34,131,96,105)(35,108,89,134)(36,129,90,111)(37,106,91,132)(38,135,92,109)(39,112,93,130)(40,133,94,107)(41,61,128,141)(42,144,121,64)(43,59,122,139)(44,142,123,62)(45,57,124,137)(46,140,125,60)(47,63,126,143)(48,138,127,58)>;
G:=Group( (1,127,159,39,151)(2,128,160,40,152)(3,121,153,33,145)(4,122,154,34,146)(5,123,155,35,147)(6,124,156,36,148)(7,125,157,37,149)(8,126,158,38,150)(9,113,137,17,129)(10,114,138,18,130)(11,115,139,19,131)(12,116,140,20,132)(13,117,141,21,133)(14,118,142,22,134)(15,119,143,23,135)(16,120,144,24,136)(25,89,56,81,44)(26,90,49,82,45)(27,91,50,83,46)(28,92,51,84,47)(29,93,52,85,48)(30,94,53,86,41)(31,95,54,87,42)(32,96,55,88,43)(57,78,111,70,103)(58,79,112,71,104)(59,80,105,72,97)(60,73,106,65,98)(61,74,107,66,99)(62,75,108,67,100)(63,76,109,68,101)(64,77,110,69,102), (1,85)(2,82)(3,87)(4,84)(5,81)(6,86)(7,83)(8,88)(9,70)(10,67)(11,72)(12,69)(13,66)(14,71)(15,68)(16,65)(17,78)(18,75)(19,80)(20,77)(21,74)(22,79)(23,76)(24,73)(25,155)(26,160)(27,157)(28,154)(29,159)(30,156)(31,153)(32,158)(33,95)(34,92)(35,89)(36,94)(37,91)(38,96)(39,93)(40,90)(41,124)(42,121)(43,126)(44,123)(45,128)(46,125)(47,122)(48,127)(49,152)(50,149)(51,146)(52,151)(53,148)(54,145)(55,150)(56,147)(57,137)(58,142)(59,139)(60,144)(61,141)(62,138)(63,143)(64,140)(97,115)(98,120)(99,117)(100,114)(101,119)(102,116)(103,113)(104,118)(105,131)(106,136)(107,133)(108,130)(109,135)(110,132)(111,129)(112,134), (1,85)(2,86)(3,87)(4,88)(5,81)(6,82)(7,83)(8,84)(9,70)(10,71)(11,72)(12,65)(13,66)(14,67)(15,68)(16,69)(17,78)(18,79)(19,80)(20,73)(21,74)(22,75)(23,76)(24,77)(25,155)(26,156)(27,157)(28,158)(29,159)(30,160)(31,153)(32,154)(33,95)(34,96)(35,89)(36,90)(37,91)(38,92)(39,93)(40,94)(41,128)(42,121)(43,122)(44,123)(45,124)(46,125)(47,126)(48,127)(49,148)(50,149)(51,150)(52,151)(53,152)(54,145)(55,146)(56,147)(57,137)(58,138)(59,139)(60,140)(61,141)(62,142)(63,143)(64,144)(97,115)(98,116)(99,117)(100,118)(101,119)(102,120)(103,113)(104,114)(105,131)(106,132)(107,133)(108,134)(109,135)(110,136)(111,129)(112,130), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64)(65,69)(66,70)(67,71)(68,72)(73,77)(74,78)(75,79)(76,80)(81,85)(82,86)(83,87)(84,88)(89,93)(90,94)(91,95)(92,96)(97,101)(98,102)(99,103)(100,104)(105,109)(106,110)(107,111)(108,112)(113,117)(114,118)(115,119)(116,120)(121,125)(122,126)(123,127)(124,128)(129,133)(130,134)(131,135)(132,136)(137,141)(138,142)(139,143)(140,144)(145,149)(146,150)(147,151)(148,152)(153,157)(154,158)(155,159)(156,160), (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), (1,104,85,114)(2,117,86,99)(3,102,87,120)(4,115,88,97)(5,100,81,118)(6,113,82,103)(7,98,83,116)(8,119,84,101)(9,49,70,148)(10,151,71,52)(11,55,72,146)(12,149,65,50)(13,53,66,152)(14,147,67,56)(15,51,68,150)(16,145,69,54)(17,26,78,156)(18,159,79,29)(19,32,80,154)(20,157,73,27)(21,30,74,160)(22,155,75,25)(23,28,76,158)(24,153,77,31)(33,110,95,136)(34,131,96,105)(35,108,89,134)(36,129,90,111)(37,106,91,132)(38,135,92,109)(39,112,93,130)(40,133,94,107)(41,61,128,141)(42,144,121,64)(43,59,122,139)(44,142,123,62)(45,57,124,137)(46,140,125,60)(47,63,126,143)(48,138,127,58) );
G=PermutationGroup([(1,127,159,39,151),(2,128,160,40,152),(3,121,153,33,145),(4,122,154,34,146),(5,123,155,35,147),(6,124,156,36,148),(7,125,157,37,149),(8,126,158,38,150),(9,113,137,17,129),(10,114,138,18,130),(11,115,139,19,131),(12,116,140,20,132),(13,117,141,21,133),(14,118,142,22,134),(15,119,143,23,135),(16,120,144,24,136),(25,89,56,81,44),(26,90,49,82,45),(27,91,50,83,46),(28,92,51,84,47),(29,93,52,85,48),(30,94,53,86,41),(31,95,54,87,42),(32,96,55,88,43),(57,78,111,70,103),(58,79,112,71,104),(59,80,105,72,97),(60,73,106,65,98),(61,74,107,66,99),(62,75,108,67,100),(63,76,109,68,101),(64,77,110,69,102)], [(1,85),(2,82),(3,87),(4,84),(5,81),(6,86),(7,83),(8,88),(9,70),(10,67),(11,72),(12,69),(13,66),(14,71),(15,68),(16,65),(17,78),(18,75),(19,80),(20,77),(21,74),(22,79),(23,76),(24,73),(25,155),(26,160),(27,157),(28,154),(29,159),(30,156),(31,153),(32,158),(33,95),(34,92),(35,89),(36,94),(37,91),(38,96),(39,93),(40,90),(41,124),(42,121),(43,126),(44,123),(45,128),(46,125),(47,122),(48,127),(49,152),(50,149),(51,146),(52,151),(53,148),(54,145),(55,150),(56,147),(57,137),(58,142),(59,139),(60,144),(61,141),(62,138),(63,143),(64,140),(97,115),(98,120),(99,117),(100,114),(101,119),(102,116),(103,113),(104,118),(105,131),(106,136),(107,133),(108,130),(109,135),(110,132),(111,129),(112,134)], [(1,85),(2,86),(3,87),(4,88),(5,81),(6,82),(7,83),(8,84),(9,70),(10,71),(11,72),(12,65),(13,66),(14,67),(15,68),(16,69),(17,78),(18,79),(19,80),(20,73),(21,74),(22,75),(23,76),(24,77),(25,155),(26,156),(27,157),(28,158),(29,159),(30,160),(31,153),(32,154),(33,95),(34,96),(35,89),(36,90),(37,91),(38,92),(39,93),(40,94),(41,128),(42,121),(43,122),(44,123),(45,124),(46,125),(47,126),(48,127),(49,148),(50,149),(51,150),(52,151),(53,152),(54,145),(55,146),(56,147),(57,137),(58,138),(59,139),(60,140),(61,141),(62,142),(63,143),(64,144),(97,115),(98,116),(99,117),(100,118),(101,119),(102,120),(103,113),(104,114),(105,131),(106,132),(107,133),(108,134),(109,135),(110,136),(111,129),(112,130)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24),(25,29),(26,30),(27,31),(28,32),(33,37),(34,38),(35,39),(36,40),(41,45),(42,46),(43,47),(44,48),(49,53),(50,54),(51,55),(52,56),(57,61),(58,62),(59,63),(60,64),(65,69),(66,70),(67,71),(68,72),(73,77),(74,78),(75,79),(76,80),(81,85),(82,86),(83,87),(84,88),(89,93),(90,94),(91,95),(92,96),(97,101),(98,102),(99,103),(100,104),(105,109),(106,110),(107,111),(108,112),(113,117),(114,118),(115,119),(116,120),(121,125),(122,126),(123,127),(124,128),(129,133),(130,134),(131,135),(132,136),(137,141),(138,142),(139,143),(140,144),(145,149),(146,150),(147,151),(148,152),(153,157),(154,158),(155,159),(156,160)], [(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)], [(1,104,85,114),(2,117,86,99),(3,102,87,120),(4,115,88,97),(5,100,81,118),(6,113,82,103),(7,98,83,116),(8,119,84,101),(9,49,70,148),(10,151,71,52),(11,55,72,146),(12,149,65,50),(13,53,66,152),(14,147,67,56),(15,51,68,150),(16,145,69,54),(17,26,78,156),(18,159,79,29),(19,32,80,154),(20,157,73,27),(21,30,74,160),(22,155,75,25),(23,28,76,158),(24,153,77,31),(33,110,95,136),(34,131,96,105),(35,108,89,134),(36,129,90,111),(37,106,91,132),(38,135,92,109),(39,112,93,130),(40,133,94,107),(41,61,128,141),(42,144,121,64),(43,59,122,139),(44,142,123,62),(45,57,124,137),(46,140,125,60),(47,63,126,143),(48,138,127,58)])
110 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 5A | 5B | 5C | 5D | 8A | 8B | 8C | 8D | 10A | ··· | 10L | 10M | ··· | 10T | 20A | ··· | 20P | 20Q | ··· | 20AV | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
110 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C5 | C10 | C10 | C10 | C10 | C20 | D4 | D4 | C5×D4 | C5×D4 | C8.C22 | C5×C8.C22 |
| kernel | C5×C23.38D4 | C5×Q8⋊C4 | C5×C42⋊C2 | C10×M4(2) | Q8×C2×C10 | Q8×C10 | C23.38D4 | Q8⋊C4 | C42⋊C2 | C2×M4(2) | C22×Q8 | C2×Q8 | C2×C20 | C22×C10 | C2×C4 | C23 | C10 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 4 | 16 | 4 | 4 | 4 | 32 | 3 | 1 | 12 | 4 | 2 | 8 |
Matrix representation of C5×C23.38D4 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 0 | 0 | 0 |
| 0 | 0 | 0 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 0 | 0 | 10 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 1 | 1 |
| 0 | 0 | 0 | 1 | 7 | 7 |
| 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 | 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 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 9 | 30 | 0 | 0 | 0 | 0 |
| 11 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 21 | 14 | 15 | 8 |
| 0 | 0 | 24 | 16 | 0 | 34 |
| 0 | 0 | 13 | 34 | 23 | 23 |
| 0 | 0 | 27 | 20 | 22 | 22 |
| 30 | 9 | 0 | 0 | 0 | 0 |
| 32 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 7 | 0 | 24 | 25 |
| 0 | 0 | 39 | 40 | 16 | 16 |
| 0 | 0 | 0 | 1 | 24 | 24 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,10],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,1,7,40,0,0,0,1,7,0,40],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[9,11,0,0,0,0,30,32,0,0,0,0,0,0,21,24,13,27,0,0,14,16,34,20,0,0,15,0,23,22,0,0,8,34,23,22],[30,32,0,0,0,0,9,11,0,0,0,0,0,0,1,7,39,0,0,0,0,0,40,1,0,0,0,24,16,24,0,0,0,25,16,24] >;
C5×C23.38D4 in GAP, Magma, Sage, TeX
C_5\times C_2^3._{38}D_4 % in TeX
G:=Group("C5xC2^3.38D4"); // GroupNames label
G:=SmallGroup(320,920);
// by ID
G=gap.SmallGroup(320,920);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,560,589,1128,1731,856,7004,3511,172]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^5=b^2=c^2=d^2=1,e^4=d,f^2=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,e*b*e^-1=f*b*f^-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=c*e^3>;
// generators/relations