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, C2, C4, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, Q8, Q8, C23, C10, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C2×Q8, C2×Q8, C20, C20, C20, C2×C10, C2×C10, C2×C10, Q8⋊C4, C42⋊C2, C2×M4(2), C22×Q8, C40, C2×C20, C2×C20, C2×C20, C5×Q8, C5×Q8, C22×C10, C23.38D4, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×C40, C5×M4(2), C22×C20, C22×C20, Q8×C10, Q8×C10, C5×Q8⋊C4, C5×C42⋊C2, C10×M4(2), Q8×C2×C10, C5×C23.38D4
Quotients: C1, C2, C4, C22, C5, C2×C4, D4, C23, C10, C22⋊C4, C22×C4, C2×D4, C20, C2×C10, C2×C22⋊C4, C8.C22, C2×C20, C5×D4, C22×C10, C23.38D4, C5×C22⋊C4, C22×C20, D4×C10, C10×C22⋊C4, C5×C8.C22, 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 46)(26 90 49 82 47)(27 91 50 83 48)(28 92 51 84 41)(29 93 52 85 42)(30 94 53 86 43)(31 95 54 87 44)(32 96 55 88 45)(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 87)(2 84)(3 81)(4 86)(5 83)(6 88)(7 85)(8 82)(9 66)(10 71)(11 68)(12 65)(13 70)(14 67)(15 72)(16 69)(17 74)(18 79)(19 76)(20 73)(21 78)(22 75)(23 80)(24 77)(25 153)(26 158)(27 155)(28 160)(29 157)(30 154)(31 159)(32 156)(33 89)(34 94)(35 91)(36 96)(37 93)(38 90)(39 95)(40 92)(41 128)(42 125)(43 122)(44 127)(45 124)(46 121)(47 126)(48 123)(49 150)(50 147)(51 152)(52 149)(53 146)(54 151)(55 148)(56 145)(57 141)(58 138)(59 143)(60 140)(61 137)(62 142)(63 139)(64 144)(97 119)(98 116)(99 113)(100 118)(101 115)(102 120)(103 117)(104 114)(105 135)(106 132)(107 129)(108 134)(109 131)(110 136)(111 133)(112 130)
(1 87)(2 88)(3 81)(4 82)(5 83)(6 84)(7 85)(8 86)(9 66)(10 67)(11 68)(12 69)(13 70)(14 71)(15 72)(16 65)(17 74)(18 75)(19 76)(20 77)(21 78)(22 79)(23 80)(24 73)(25 153)(26 154)(27 155)(28 156)(29 157)(30 158)(31 159)(32 160)(33 89)(34 90)(35 91)(36 92)(37 93)(38 94)(39 95)(40 96)(41 124)(42 125)(43 126)(44 127)(45 128)(46 121)(47 122)(48 123)(49 146)(50 147)(51 148)(52 149)(53 150)(54 151)(55 152)(56 145)(57 141)(58 142)(59 143)(60 144)(61 137)(62 138)(63 139)(64 140)(97 119)(98 120)(99 113)(100 114)(101 115)(102 116)(103 117)(104 118)(105 135)(106 136)(107 129)(108 130)(109 131)(110 132)(111 133)(112 134)
(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)
(1 100 87 114)(2 117 88 103)(3 98 81 120)(4 115 82 101)(5 104 83 118)(6 113 84 99)(7 102 85 116)(8 119 86 97)(9 51 66 148)(10 151 67 54)(11 49 68 146)(12 149 69 52)(13 55 70 152)(14 147 71 50)(15 53 72 150)(16 145 65 56)(17 28 74 156)(18 159 75 31)(19 26 76 154)(20 157 77 29)(21 32 78 160)(22 155 79 27)(23 30 80 158)(24 153 73 25)(33 106 89 136)(34 131 90 109)(35 112 91 134)(36 129 92 107)(37 110 93 132)(38 135 94 105)(39 108 95 130)(40 133 96 111)(41 61 124 137)(42 140 125 64)(43 59 126 143)(44 138 127 62)(45 57 128 141)(46 144 121 60)(47 63 122 139)(48 142 123 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)(25,89,56,81,46)(26,90,49,82,47)(27,91,50,83,48)(28,92,51,84,41)(29,93,52,85,42)(30,94,53,86,43)(31,95,54,87,44)(32,96,55,88,45)(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), (1,87)(2,84)(3,81)(4,86)(5,83)(6,88)(7,85)(8,82)(9,66)(10,71)(11,68)(12,65)(13,70)(14,67)(15,72)(16,69)(17,74)(18,79)(19,76)(20,73)(21,78)(22,75)(23,80)(24,77)(25,153)(26,158)(27,155)(28,160)(29,157)(30,154)(31,159)(32,156)(33,89)(34,94)(35,91)(36,96)(37,93)(38,90)(39,95)(40,92)(41,128)(42,125)(43,122)(44,127)(45,124)(46,121)(47,126)(48,123)(49,150)(50,147)(51,152)(52,149)(53,146)(54,151)(55,148)(56,145)(57,141)(58,138)(59,143)(60,140)(61,137)(62,142)(63,139)(64,144)(97,119)(98,116)(99,113)(100,118)(101,115)(102,120)(103,117)(104,114)(105,135)(106,132)(107,129)(108,134)(109,131)(110,136)(111,133)(112,130), (1,87)(2,88)(3,81)(4,82)(5,83)(6,84)(7,85)(8,86)(9,66)(10,67)(11,68)(12,69)(13,70)(14,71)(15,72)(16,65)(17,74)(18,75)(19,76)(20,77)(21,78)(22,79)(23,80)(24,73)(25,153)(26,154)(27,155)(28,156)(29,157)(30,158)(31,159)(32,160)(33,89)(34,90)(35,91)(36,92)(37,93)(38,94)(39,95)(40,96)(41,124)(42,125)(43,126)(44,127)(45,128)(46,121)(47,122)(48,123)(49,146)(50,147)(51,148)(52,149)(53,150)(54,151)(55,152)(56,145)(57,141)(58,142)(59,143)(60,144)(61,137)(62,138)(63,139)(64,140)(97,119)(98,120)(99,113)(100,114)(101,115)(102,116)(103,117)(104,118)(105,135)(106,136)(107,129)(108,130)(109,131)(110,132)(111,133)(112,134), (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), (1,100,87,114)(2,117,88,103)(3,98,81,120)(4,115,82,101)(5,104,83,118)(6,113,84,99)(7,102,85,116)(8,119,86,97)(9,51,66,148)(10,151,67,54)(11,49,68,146)(12,149,69,52)(13,55,70,152)(14,147,71,50)(15,53,72,150)(16,145,65,56)(17,28,74,156)(18,159,75,31)(19,26,76,154)(20,157,77,29)(21,32,78,160)(22,155,79,27)(23,30,80,158)(24,153,73,25)(33,106,89,136)(34,131,90,109)(35,112,91,134)(36,129,92,107)(37,110,93,132)(38,135,94,105)(39,108,95,130)(40,133,96,111)(41,61,124,137)(42,140,125,64)(43,59,126,143)(44,138,127,62)(45,57,128,141)(46,144,121,60)(47,63,122,139)(48,142,123,58)>;
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)(25,89,56,81,46)(26,90,49,82,47)(27,91,50,83,48)(28,92,51,84,41)(29,93,52,85,42)(30,94,53,86,43)(31,95,54,87,44)(32,96,55,88,45)(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), (1,87)(2,84)(3,81)(4,86)(5,83)(6,88)(7,85)(8,82)(9,66)(10,71)(11,68)(12,65)(13,70)(14,67)(15,72)(16,69)(17,74)(18,79)(19,76)(20,73)(21,78)(22,75)(23,80)(24,77)(25,153)(26,158)(27,155)(28,160)(29,157)(30,154)(31,159)(32,156)(33,89)(34,94)(35,91)(36,96)(37,93)(38,90)(39,95)(40,92)(41,128)(42,125)(43,122)(44,127)(45,124)(46,121)(47,126)(48,123)(49,150)(50,147)(51,152)(52,149)(53,146)(54,151)(55,148)(56,145)(57,141)(58,138)(59,143)(60,140)(61,137)(62,142)(63,139)(64,144)(97,119)(98,116)(99,113)(100,118)(101,115)(102,120)(103,117)(104,114)(105,135)(106,132)(107,129)(108,134)(109,131)(110,136)(111,133)(112,130), (1,87)(2,88)(3,81)(4,82)(5,83)(6,84)(7,85)(8,86)(9,66)(10,67)(11,68)(12,69)(13,70)(14,71)(15,72)(16,65)(17,74)(18,75)(19,76)(20,77)(21,78)(22,79)(23,80)(24,73)(25,153)(26,154)(27,155)(28,156)(29,157)(30,158)(31,159)(32,160)(33,89)(34,90)(35,91)(36,92)(37,93)(38,94)(39,95)(40,96)(41,124)(42,125)(43,126)(44,127)(45,128)(46,121)(47,122)(48,123)(49,146)(50,147)(51,148)(52,149)(53,150)(54,151)(55,152)(56,145)(57,141)(58,142)(59,143)(60,144)(61,137)(62,138)(63,139)(64,140)(97,119)(98,120)(99,113)(100,114)(101,115)(102,116)(103,117)(104,118)(105,135)(106,136)(107,129)(108,130)(109,131)(110,132)(111,133)(112,134), (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), (1,100,87,114)(2,117,88,103)(3,98,81,120)(4,115,82,101)(5,104,83,118)(6,113,84,99)(7,102,85,116)(8,119,86,97)(9,51,66,148)(10,151,67,54)(11,49,68,146)(12,149,69,52)(13,55,70,152)(14,147,71,50)(15,53,72,150)(16,145,65,56)(17,28,74,156)(18,159,75,31)(19,26,76,154)(20,157,77,29)(21,32,78,160)(22,155,79,27)(23,30,80,158)(24,153,73,25)(33,106,89,136)(34,131,90,109)(35,112,91,134)(36,129,92,107)(37,110,93,132)(38,135,94,105)(39,108,95,130)(40,133,96,111)(41,61,124,137)(42,140,125,64)(43,59,126,143)(44,138,127,62)(45,57,128,141)(46,144,121,60)(47,63,122,139)(48,142,123,58) );
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),(25,89,56,81,46),(26,90,49,82,47),(27,91,50,83,48),(28,92,51,84,41),(29,93,52,85,42),(30,94,53,86,43),(31,95,54,87,44),(32,96,55,88,45),(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)], [(1,87),(2,84),(3,81),(4,86),(5,83),(6,88),(7,85),(8,82),(9,66),(10,71),(11,68),(12,65),(13,70),(14,67),(15,72),(16,69),(17,74),(18,79),(19,76),(20,73),(21,78),(22,75),(23,80),(24,77),(25,153),(26,158),(27,155),(28,160),(29,157),(30,154),(31,159),(32,156),(33,89),(34,94),(35,91),(36,96),(37,93),(38,90),(39,95),(40,92),(41,128),(42,125),(43,122),(44,127),(45,124),(46,121),(47,126),(48,123),(49,150),(50,147),(51,152),(52,149),(53,146),(54,151),(55,148),(56,145),(57,141),(58,138),(59,143),(60,140),(61,137),(62,142),(63,139),(64,144),(97,119),(98,116),(99,113),(100,118),(101,115),(102,120),(103,117),(104,114),(105,135),(106,132),(107,129),(108,134),(109,131),(110,136),(111,133),(112,130)], [(1,87),(2,88),(3,81),(4,82),(5,83),(6,84),(7,85),(8,86),(9,66),(10,67),(11,68),(12,69),(13,70),(14,71),(15,72),(16,65),(17,74),(18,75),(19,76),(20,77),(21,78),(22,79),(23,80),(24,73),(25,153),(26,154),(27,155),(28,156),(29,157),(30,158),(31,159),(32,160),(33,89),(34,90),(35,91),(36,92),(37,93),(38,94),(39,95),(40,96),(41,124),(42,125),(43,126),(44,127),(45,128),(46,121),(47,122),(48,123),(49,146),(50,147),(51,148),(52,149),(53,150),(54,151),(55,152),(56,145),(57,141),(58,142),(59,143),(60,144),(61,137),(62,138),(63,139),(64,140),(97,119),(98,120),(99,113),(100,114),(101,115),(102,116),(103,117),(104,118),(105,135),(106,136),(107,129),(108,130),(109,131),(110,132),(111,133),(112,134)], [(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)], [(1,100,87,114),(2,117,88,103),(3,98,81,120),(4,115,82,101),(5,104,83,118),(6,113,84,99),(7,102,85,116),(8,119,86,97),(9,51,66,148),(10,151,67,54),(11,49,68,146),(12,149,69,52),(13,55,70,152),(14,147,71,50),(15,53,72,150),(16,145,65,56),(17,28,74,156),(18,159,75,31),(19,26,76,154),(20,157,77,29),(21,32,78,160),(22,155,79,27),(23,30,80,158),(24,153,73,25),(33,106,89,136),(34,131,90,109),(35,112,91,134),(36,129,92,107),(37,110,93,132),(38,135,94,105),(39,108,95,130),(40,133,96,111),(41,61,124,137),(42,140,125,64),(43,59,126,143),(44,138,127,62),(45,57,128,141),(46,144,121,60),(47,63,122,139),(48,142,123,58)]])
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