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G = C5×C23.23D4order 320 = 26·5

Direct product of C5 and C23.23D4

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

Aliases: C5×C23.23D4, (C2×D4)⋊3C20, (C2×C20)⋊36D4, C2.5(D4×C20), (D4×C10)⋊27C4, (C23×C4)⋊1C10, C232(C2×C20), (C23×C20)⋊2C2, C10.136(C4×D4), C23.22(C5×D4), C10.89C22≀C2, C24.27(C2×C10), (C22×D4).1C10, C22.35(D4×C10), C2.C428C10, (C22×C10).156D4, C10.133(C4⋊D4), C23.59(C22×C10), C22.35(C22×C20), (C23×C10).87C22, (C22×C10).450C23, (C22×C20).494C22, C10.87(C22.D4), (C2×C4)⋊9(C5×D4), (C2×C4)⋊3(C2×C20), (C2×C20)⋊37(C2×C4), (D4×C2×C10).13C2, C2.2(C5×C4⋊D4), (C2×C22⋊C4)⋊2C10, (C10×C22⋊C4)⋊6C2, C2.3(C5×C22≀C2), C2.7(C10×C22⋊C4), C221(C5×C22⋊C4), (C2×C10)⋊7(C22⋊C4), (C22×C10)⋊11(C2×C4), (C2×C10).602(C2×D4), C22.20(C5×C4○D4), C10.135(C2×C22⋊C4), (C22×C4).87(C2×C10), (C2×C10).210(C4○D4), C2.3(C5×C22.D4), (C5×C2.C42)⋊24C2, (C2×C10).323(C22×C4), SmallGroup(320,887)

Series: Derived Chief Lower central Upper central

C1C22 — C5×C23.23D4
C1C2C22C23C22×C10C22×C20C10×C22⋊C4 — C5×C23.23D4
C1C22 — C5×C23.23D4
C1C22×C10 — C5×C23.23D4

Generators and relations for C5×C23.23D4
 G = < a,b,c,d,e,f | a5=b2=c2=d2=e4=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, fbf=bc=cb, bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=de-1 >

Subgroups: 498 in 286 conjugacy classes, 106 normal (30 characteristic)
C1, C2 [×3], C2 [×4], C2 [×6], C4 [×8], C22 [×3], C22 [×8], C22 [×22], C5, C2×C4 [×6], C2×C4 [×20], D4 [×8], C23, C23 [×8], C23 [×10], C10 [×3], C10 [×4], C10 [×6], C22⋊C4 [×6], C22×C4, C22×C4 [×4], C22×C4 [×6], C2×D4 [×4], C2×D4 [×4], C24 [×2], C20 [×8], C2×C10 [×3], C2×C10 [×8], C2×C10 [×22], C2.C42 [×2], C2×C22⋊C4, C2×C22⋊C4 [×2], C23×C4, C22×D4, C2×C20 [×6], C2×C20 [×20], C5×D4 [×8], C22×C10, C22×C10 [×8], C22×C10 [×10], C23.23D4, C5×C22⋊C4 [×6], C22×C20, C22×C20 [×4], C22×C20 [×6], D4×C10 [×4], D4×C10 [×4], C23×C10 [×2], C5×C2.C42 [×2], C10×C22⋊C4, C10×C22⋊C4 [×2], C23×C20, D4×C2×C10, C5×C23.23D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C5, C2×C4 [×6], D4 [×8], C23, C10 [×7], C22⋊C4 [×4], C22×C4, C2×D4 [×4], C4○D4 [×2], C20 [×4], C2×C10 [×7], C2×C22⋊C4, C4×D4 [×2], C22≀C2, C4⋊D4 [×2], C22.D4, C2×C20 [×6], C5×D4 [×8], C22×C10, C23.23D4, C5×C22⋊C4 [×4], C22×C20, D4×C10 [×4], C5×C4○D4 [×2], C10×C22⋊C4, D4×C20 [×2], C5×C22≀C2, C5×C4⋊D4 [×2], C5×C22.D4, C5×C23.23D4

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

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

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

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

140 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M4A···4H4I···4N5A5B5C5D10A···10AB10AC···10AR10AS···10AZ20A···20AF20AG···20BD
order12···22222224···44···4555510···1010···1010···1020···2020···20
size11···12222442···24···411111···12···24···42···24···4

140 irreducible representations

dim111111111111222222
type+++++++
imageC1C2C2C2C2C4C5C10C10C10C10C20D4D4C4○D4C5×D4C5×D4C5×C4○D4
kernelC5×C23.23D4C5×C2.C42C10×C22⋊C4C23×C20D4×C2×C10D4×C10C23.23D4C2.C42C2×C22⋊C4C23×C4C22×D4C2×D4C2×C20C22×C10C2×C10C2×C4C23C22
# reps12311848124432444161616

Matrix representation of C5×C23.23D4 in GL5(𝔽41)

10000
01000
00100
000100
000010
,
10000
00100
01000
00001
00010
,
10000
040000
004000
000400
000040
,
400000
040000
004000
000400
000040
,
320000
09000
00900
000040
000400
,
400000
01000
004000
000400
00001

G:=sub<GL(5,GF(41))| [1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,10,0,0,0,0,0,10],[1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,1,0],[1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40],[40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40],[32,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,0,40,0,0,0,40,0],[40,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1] >;

C5×C23.23D4 in GAP, Magma, Sage, TeX

C_5\times C_2^3._{23}D_4
% in TeX

G:=Group("C5xC2^3.23D4");
// GroupNames label

G:=SmallGroup(320,887);
// by ID

G=gap.SmallGroup(320,887);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,1120,589,1408,1766]);
// Polycyclic

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

׿
×
𝔽