direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C5×C23.48D4, C2.D8⋊8C10, C2.8(C10×Q16), Q8⋊C4⋊7C10, (C2×C20).339D4, C22⋊C8.4C10, (C2×C10).12Q16, C10.55(C2×Q16), C23.48(C5×D4), C22⋊Q8.6C10, C22.3(C5×Q16), C20.321(C4○D4), (C2×C20).940C23, (C2×C40).263C22, (C22×C10).170D4, C22.105(D4×C10), C10.144(C8⋊C22), (Q8×C10).170C22, (C22×C20).432C22, C10.99(C22.D4), (C5×C2.D8)⋊23C2, C4.33(C5×C4○D4), (C2×C4).40(C5×D4), (C10×C4⋊C4).46C2, (C2×C4⋊C4).17C10, C4⋊C4.61(C2×C10), (C2×C8).10(C2×C10), C2.19(C5×C8⋊C22), (C5×Q8⋊C4)⋊30C2, (C2×C10).661(C2×D4), (C5×C22⋊C8).13C2, (C2×Q8).14(C2×C10), (C5×C22⋊Q8).16C2, (C5×C4⋊C4).384C22, (C22×C4).50(C2×C10), (C2×C4).115(C22×C10), C2.15(C5×C22.D4), SmallGroup(320,985)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C23.48D4
G = < a,b,c,d,e,f | a5=b2=c2=d2=1, e4=f2=d, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ebe-1=fbf-1=bc=cb, bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=cde3 >
Subgroups: 178 in 104 conjugacy classes, 54 normal (30 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, Q8, C23, C10, C10, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×Q8, C20, C20, C2×C10, C2×C10, C2×C10, C22⋊C8, Q8⋊C4, C2.D8, C2×C4⋊C4, C22⋊Q8, C40, C2×C20, C2×C20, C5×Q8, C22×C10, C23.48D4, C5×C22⋊C4, C5×C4⋊C4, C5×C4⋊C4, C5×C4⋊C4, C2×C40, C22×C20, C22×C20, Q8×C10, C5×C22⋊C8, C5×Q8⋊C4, C5×C2.D8, C10×C4⋊C4, C5×C22⋊Q8, C5×C23.48D4
Quotients: C1, C2, C22, C5, D4, C23, C10, Q16, C2×D4, C4○D4, C2×C10, C22.D4, C2×Q16, C8⋊C22, C5×D4, C22×C10, C23.48D4, C5×Q16, D4×C10, C5×C4○D4, C5×C22.D4, C10×Q16, C5×C8⋊C22, C5×C23.48D4
►On 160 points
Generators in S160
(1 122 155 35 147)(2 123 156 36 148)(3 124 157 37 149)(4 125 158 38 150)(5 126 159 39 151)(6 127 160 40 152)(7 128 153 33 145)(8 121 154 34 146)(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 93 56 85 43)(26 94 49 86 44)(27 95 50 87 45)(28 96 51 88 46)(29 89 52 81 47)(30 90 53 82 48)(31 91 54 83 41)(32 92 55 84 42)(57 73 110 65 102)(58 74 111 66 103)(59 75 112 67 104)(60 76 105 68 97)(61 77 106 69 98)(62 78 107 70 99)(63 79 108 71 100)(64 80 109 72 101)
(1 5)(2 88)(3 7)(4 82)(6 84)(8 86)(9 13)(10 97)(11 15)(12 99)(14 101)(16 103)(17 21)(18 105)(19 23)(20 107)(22 109)(24 111)(25 29)(26 154)(27 31)(28 156)(30 158)(32 160)(33 37)(34 94)(35 39)(36 96)(38 90)(40 92)(41 45)(42 127)(43 47)(44 121)(46 123)(48 125)(49 146)(50 54)(51 148)(52 56)(53 150)(55 152)(57 61)(58 120)(59 63)(60 114)(62 116)(64 118)(65 69)(66 136)(67 71)(68 130)(70 132)(72 134)(73 77)(74 144)(75 79)(76 138)(78 140)(80 142)(81 85)(83 87)(89 93)(91 95)(98 102)(100 104)(106 110)(108 112)(113 117)(115 119)(122 126)(124 128)(129 133)(131 135)(137 141)(139 143)(145 149)(147 151)(153 157)(155 159)
(1 83)(2 84)(3 85)(4 86)(5 87)(6 88)(7 81)(8 82)(9 100)(10 101)(11 102)(12 103)(13 104)(14 97)(15 98)(16 99)(17 108)(18 109)(19 110)(20 111)(21 112)(22 105)(23 106)(24 107)(25 157)(26 158)(27 159)(28 160)(29 153)(30 154)(31 155)(32 156)(33 89)(34 90)(35 91)(36 92)(37 93)(38 94)(39 95)(40 96)(41 122)(42 123)(43 124)(44 125)(45 126)(46 127)(47 128)(48 121)(49 150)(50 151)(51 152)(52 145)(53 146)(54 147)(55 148)(56 149)(57 115)(58 116)(59 117)(60 118)(61 119)(62 120)(63 113)(64 114)(65 131)(66 132)(67 133)(68 134)(69 135)(70 136)(71 129)(72 130)(73 139)(74 140)(75 141)(76 142)(77 143)(78 144)(79 137)(80 138)
(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)
(1 114 5 118)(2 63 6 59)(3 120 7 116)(4 61 8 57)(9 51 13 55)(10 151 14 147)(11 49 15 53)(12 149 16 145)(17 28 21 32)(18 159 22 155)(19 26 23 30)(20 157 24 153)(25 107 29 111)(27 105 31 109)(33 132 37 136)(34 65 38 69)(35 130 39 134)(36 71 40 67)(41 80 45 76)(42 137 46 141)(43 78 47 74)(44 143 48 139)(50 97 54 101)(52 103 56 99)(58 85 62 81)(60 83 64 87)(66 93 70 89)(68 91 72 95)(73 125 77 121)(75 123 79 127)(82 115 86 119)(84 113 88 117)(90 131 94 135)(92 129 96 133)(98 146 102 150)(100 152 104 148)(106 154 110 158)(108 160 112 156)(122 138 126 142)(124 144 128 140)
G:=sub<Sym(160)| (1,122,155,35,147)(2,123,156,36,148)(3,124,157,37,149)(4,125,158,38,150)(5,126,159,39,151)(6,127,160,40,152)(7,128,153,33,145)(8,121,154,34,146)(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)(25,93,56,85,43)(26,94,49,86,44)(27,95,50,87,45)(28,96,51,88,46)(29,89,52,81,47)(30,90,53,82,48)(31,91,54,83,41)(32,92,55,84,42)(57,73,110,65,102)(58,74,111,66,103)(59,75,112,67,104)(60,76,105,68,97)(61,77,106,69,98)(62,78,107,70,99)(63,79,108,71,100)(64,80,109,72,101), (1,5)(2,88)(3,7)(4,82)(6,84)(8,86)(9,13)(10,97)(11,15)(12,99)(14,101)(16,103)(17,21)(18,105)(19,23)(20,107)(22,109)(24,111)(25,29)(26,154)(27,31)(28,156)(30,158)(32,160)(33,37)(34,94)(35,39)(36,96)(38,90)(40,92)(41,45)(42,127)(43,47)(44,121)(46,123)(48,125)(49,146)(50,54)(51,148)(52,56)(53,150)(55,152)(57,61)(58,120)(59,63)(60,114)(62,116)(64,118)(65,69)(66,136)(67,71)(68,130)(70,132)(72,134)(73,77)(74,144)(75,79)(76,138)(78,140)(80,142)(81,85)(83,87)(89,93)(91,95)(98,102)(100,104)(106,110)(108,112)(113,117)(115,119)(122,126)(124,128)(129,133)(131,135)(137,141)(139,143)(145,149)(147,151)(153,157)(155,159), (1,83)(2,84)(3,85)(4,86)(5,87)(6,88)(7,81)(8,82)(9,100)(10,101)(11,102)(12,103)(13,104)(14,97)(15,98)(16,99)(17,108)(18,109)(19,110)(20,111)(21,112)(22,105)(23,106)(24,107)(25,157)(26,158)(27,159)(28,160)(29,153)(30,154)(31,155)(32,156)(33,89)(34,90)(35,91)(36,92)(37,93)(38,94)(39,95)(40,96)(41,122)(42,123)(43,124)(44,125)(45,126)(46,127)(47,128)(48,121)(49,150)(50,151)(51,152)(52,145)(53,146)(54,147)(55,148)(56,149)(57,115)(58,116)(59,117)(60,118)(61,119)(62,120)(63,113)(64,114)(65,131)(66,132)(67,133)(68,134)(69,135)(70,136)(71,129)(72,130)(73,139)(74,140)(75,141)(76,142)(77,143)(78,144)(79,137)(80,138), (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), (1,114,5,118)(2,63,6,59)(3,120,7,116)(4,61,8,57)(9,51,13,55)(10,151,14,147)(11,49,15,53)(12,149,16,145)(17,28,21,32)(18,159,22,155)(19,26,23,30)(20,157,24,153)(25,107,29,111)(27,105,31,109)(33,132,37,136)(34,65,38,69)(35,130,39,134)(36,71,40,67)(41,80,45,76)(42,137,46,141)(43,78,47,74)(44,143,48,139)(50,97,54,101)(52,103,56,99)(58,85,62,81)(60,83,64,87)(66,93,70,89)(68,91,72,95)(73,125,77,121)(75,123,79,127)(82,115,86,119)(84,113,88,117)(90,131,94,135)(92,129,96,133)(98,146,102,150)(100,152,104,148)(106,154,110,158)(108,160,112,156)(122,138,126,142)(124,144,128,140)>;
G:=Group( (1,122,155,35,147)(2,123,156,36,148)(3,124,157,37,149)(4,125,158,38,150)(5,126,159,39,151)(6,127,160,40,152)(7,128,153,33,145)(8,121,154,34,146)(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)(25,93,56,85,43)(26,94,49,86,44)(27,95,50,87,45)(28,96,51,88,46)(29,89,52,81,47)(30,90,53,82,48)(31,91,54,83,41)(32,92,55,84,42)(57,73,110,65,102)(58,74,111,66,103)(59,75,112,67,104)(60,76,105,68,97)(61,77,106,69,98)(62,78,107,70,99)(63,79,108,71,100)(64,80,109,72,101), (1,5)(2,88)(3,7)(4,82)(6,84)(8,86)(9,13)(10,97)(11,15)(12,99)(14,101)(16,103)(17,21)(18,105)(19,23)(20,107)(22,109)(24,111)(25,29)(26,154)(27,31)(28,156)(30,158)(32,160)(33,37)(34,94)(35,39)(36,96)(38,90)(40,92)(41,45)(42,127)(43,47)(44,121)(46,123)(48,125)(49,146)(50,54)(51,148)(52,56)(53,150)(55,152)(57,61)(58,120)(59,63)(60,114)(62,116)(64,118)(65,69)(66,136)(67,71)(68,130)(70,132)(72,134)(73,77)(74,144)(75,79)(76,138)(78,140)(80,142)(81,85)(83,87)(89,93)(91,95)(98,102)(100,104)(106,110)(108,112)(113,117)(115,119)(122,126)(124,128)(129,133)(131,135)(137,141)(139,143)(145,149)(147,151)(153,157)(155,159), (1,83)(2,84)(3,85)(4,86)(5,87)(6,88)(7,81)(8,82)(9,100)(10,101)(11,102)(12,103)(13,104)(14,97)(15,98)(16,99)(17,108)(18,109)(19,110)(20,111)(21,112)(22,105)(23,106)(24,107)(25,157)(26,158)(27,159)(28,160)(29,153)(30,154)(31,155)(32,156)(33,89)(34,90)(35,91)(36,92)(37,93)(38,94)(39,95)(40,96)(41,122)(42,123)(43,124)(44,125)(45,126)(46,127)(47,128)(48,121)(49,150)(50,151)(51,152)(52,145)(53,146)(54,147)(55,148)(56,149)(57,115)(58,116)(59,117)(60,118)(61,119)(62,120)(63,113)(64,114)(65,131)(66,132)(67,133)(68,134)(69,135)(70,136)(71,129)(72,130)(73,139)(74,140)(75,141)(76,142)(77,143)(78,144)(79,137)(80,138), (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), (1,114,5,118)(2,63,6,59)(3,120,7,116)(4,61,8,57)(9,51,13,55)(10,151,14,147)(11,49,15,53)(12,149,16,145)(17,28,21,32)(18,159,22,155)(19,26,23,30)(20,157,24,153)(25,107,29,111)(27,105,31,109)(33,132,37,136)(34,65,38,69)(35,130,39,134)(36,71,40,67)(41,80,45,76)(42,137,46,141)(43,78,47,74)(44,143,48,139)(50,97,54,101)(52,103,56,99)(58,85,62,81)(60,83,64,87)(66,93,70,89)(68,91,72,95)(73,125,77,121)(75,123,79,127)(82,115,86,119)(84,113,88,117)(90,131,94,135)(92,129,96,133)(98,146,102,150)(100,152,104,148)(106,154,110,158)(108,160,112,156)(122,138,126,142)(124,144,128,140) );
G=PermutationGroup([[(1,122,155,35,147),(2,123,156,36,148),(3,124,157,37,149),(4,125,158,38,150),(5,126,159,39,151),(6,127,160,40,152),(7,128,153,33,145),(8,121,154,34,146),(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),(25,93,56,85,43),(26,94,49,86,44),(27,95,50,87,45),(28,96,51,88,46),(29,89,52,81,47),(30,90,53,82,48),(31,91,54,83,41),(32,92,55,84,42),(57,73,110,65,102),(58,74,111,66,103),(59,75,112,67,104),(60,76,105,68,97),(61,77,106,69,98),(62,78,107,70,99),(63,79,108,71,100),(64,80,109,72,101)], [(1,5),(2,88),(3,7),(4,82),(6,84),(8,86),(9,13),(10,97),(11,15),(12,99),(14,101),(16,103),(17,21),(18,105),(19,23),(20,107),(22,109),(24,111),(25,29),(26,154),(27,31),(28,156),(30,158),(32,160),(33,37),(34,94),(35,39),(36,96),(38,90),(40,92),(41,45),(42,127),(43,47),(44,121),(46,123),(48,125),(49,146),(50,54),(51,148),(52,56),(53,150),(55,152),(57,61),(58,120),(59,63),(60,114),(62,116),(64,118),(65,69),(66,136),(67,71),(68,130),(70,132),(72,134),(73,77),(74,144),(75,79),(76,138),(78,140),(80,142),(81,85),(83,87),(89,93),(91,95),(98,102),(100,104),(106,110),(108,112),(113,117),(115,119),(122,126),(124,128),(129,133),(131,135),(137,141),(139,143),(145,149),(147,151),(153,157),(155,159)], [(1,83),(2,84),(3,85),(4,86),(5,87),(6,88),(7,81),(8,82),(9,100),(10,101),(11,102),(12,103),(13,104),(14,97),(15,98),(16,99),(17,108),(18,109),(19,110),(20,111),(21,112),(22,105),(23,106),(24,107),(25,157),(26,158),(27,159),(28,160),(29,153),(30,154),(31,155),(32,156),(33,89),(34,90),(35,91),(36,92),(37,93),(38,94),(39,95),(40,96),(41,122),(42,123),(43,124),(44,125),(45,126),(46,127),(47,128),(48,121),(49,150),(50,151),(51,152),(52,145),(53,146),(54,147),(55,148),(56,149),(57,115),(58,116),(59,117),(60,118),(61,119),(62,120),(63,113),(64,114),(65,131),(66,132),(67,133),(68,134),(69,135),(70,136),(71,129),(72,130),(73,139),(74,140),(75,141),(76,142),(77,143),(78,144),(79,137),(80,138)], [(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)], [(1,114,5,118),(2,63,6,59),(3,120,7,116),(4,61,8,57),(9,51,13,55),(10,151,14,147),(11,49,15,53),(12,149,16,145),(17,28,21,32),(18,159,22,155),(19,26,23,30),(20,157,24,153),(25,107,29,111),(27,105,31,109),(33,132,37,136),(34,65,38,69),(35,130,39,134),(36,71,40,67),(41,80,45,76),(42,137,46,141),(43,78,47,74),(44,143,48,139),(50,97,54,101),(52,103,56,99),(58,85,62,81),(60,83,64,87),(66,93,70,89),(68,91,72,95),(73,125,77,121),(75,123,79,127),(82,115,86,119),(84,113,88,117),(90,131,94,135),(92,129,96,133),(98,146,102,150),(100,152,104,148),(106,154,110,158),(108,160,112,156),(122,138,126,142),(124,144,128,140)]])
95 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | ··· | 4G | 4H | 4I | 5A | 5B | 5C | 5D | 8A | 8B | 8C | 8D | 10A | ··· | 10L | 10M | ··· | 10T | 20A | ··· | 20H | 20I | ··· | 20AB | 20AC | ··· | 20AJ | 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 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
95 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | D4 | D4 | C4○D4 | Q16 | C5×D4 | C5×D4 | C5×C4○D4 | C5×Q16 | C8⋊C22 | C5×C8⋊C22 |
| kernel | C5×C23.48D4 | C5×C22⋊C8 | C5×Q8⋊C4 | C5×C2.D8 | C10×C4⋊C4 | C5×C22⋊Q8 | C23.48D4 | C22⋊C8 | Q8⋊C4 | C2.D8 | C2×C4⋊C4 | C22⋊Q8 | C2×C20 | C22×C10 | C20 | C2×C10 | C2×C4 | C23 | C4 | C22 | C10 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 4 | 4 | 8 | 8 | 4 | 4 | 1 | 1 | 4 | 4 | 4 | 4 | 16 | 16 | 1 | 4 |
Matrix representation of C5×C23.48D4 ►in GL4(𝔽41) generated by
| 18 | 0 | 0 | 0 |
| 0 | 18 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 22 | 40 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 29 | 12 | 0 | 0 |
| 29 | 29 | 0 | 0 |
| 0 | 0 | 7 | 18 |
| 0 | 0 | 11 | 34 |
| 20 | 38 | 0 | 0 |
| 38 | 21 | 0 | 0 |
| 0 | 0 | 19 | 2 |
| 0 | 0 | 25 | 22 |
G:=sub<GL(4,GF(41))| [18,0,0,0,0,18,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,40,0,0,0,0,1,22,0,0,0,40],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[29,29,0,0,12,29,0,0,0,0,7,11,0,0,18,34],[20,38,0,0,38,21,0,0,0,0,19,25,0,0,2,22] >;
C5×C23.48D4 in GAP, Magma, Sage, TeX
C_5\times C_2^3._{48}D_4 % in TeX
G:=Group("C5xC2^3.48D4"); // GroupNames label
G:=SmallGroup(320,985);
// by ID
G=gap.SmallGroup(320,985);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,1120,589,1766,1066,10085,2539,124]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^5=b^2=c^2=d^2=1,e^4=f^2=d,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,e*b*e^-1=f*b*f^-1=b*c=c*b,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*d*e^3>;
// generators/relations