Copied to
clipboard

G = C5×C23⋊Q8order 320 = 26·5

Direct product of C5 and C23⋊Q8

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

Aliases: C5×C23⋊Q8, C231(C5×Q8), (C22×C10)⋊1Q8, (C2×C20).307D4, C24.5(C2×C10), (C22×Q8)⋊1C10, C10.91C22≀C2, C22.69(D4×C10), C22.20(Q8×C10), C10.87(C22⋊Q8), (C23×C10).5C22, C2.C4210C10, C10.67(C4.4D4), C23.76(C22×C10), (C22×C20).401C22, (C22×C10).457C23, (Q8×C2×C10)⋊13C2, (C2×C4).14(C5×D4), C2.6(C5×C22⋊Q8), C2.5(C5×C22≀C2), C2.5(C5×C4.4D4), (C2×C10).609(C2×D4), (C2×C22⋊C4).8C10, (C22×C4).5(C2×C10), (C2×C10).108(C2×Q8), C22.36(C5×C4○D4), (C10×C22⋊C4).27C2, (C2×C10).217(C4○D4), (C5×C2.C42)⋊26C2, SmallGroup(320,894)

Series: Derived Chief Lower central Upper central

C1C23 — C5×C23⋊Q8
C1C2C22C23C22×C10C22×C20Q8×C2×C10 — C5×C23⋊Q8
C1C23 — C5×C23⋊Q8
C1C22×C10 — C5×C23⋊Q8

Generators and relations for C5×C23⋊Q8
 G = < a,b,c,d,e,f | a5=b2=c2=d2=e4=1, f2=e2, 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: 362 in 202 conjugacy classes, 78 normal (14 characteristic)
C1, C2, C2 [×6], C2 [×2], C4 [×9], C22, C22 [×6], C22 [×10], C5, C2×C4 [×6], C2×C4 [×15], Q8 [×8], C23, C23 [×2], C23 [×6], C10, C10 [×6], C10 [×2], C22⋊C4 [×6], C22×C4 [×6], C2×Q8 [×6], C24, C20 [×9], C2×C10, C2×C10 [×6], C2×C10 [×10], C2.C42 [×3], C2×C22⋊C4 [×3], C22×Q8, C2×C20 [×6], C2×C20 [×15], C5×Q8 [×8], C22×C10, C22×C10 [×2], C22×C10 [×6], C23⋊Q8, C5×C22⋊C4 [×6], C22×C20 [×6], Q8×C10 [×6], C23×C10, C5×C2.C42 [×3], C10×C22⋊C4 [×3], Q8×C2×C10, C5×C23⋊Q8
Quotients: C1, C2 [×7], C22 [×7], C5, D4 [×6], Q8 [×2], C23, C10 [×7], C2×D4 [×3], C2×Q8, C4○D4 [×3], C2×C10 [×7], C22≀C2, C22⋊Q8 [×3], C4.4D4 [×3], C5×D4 [×6], C5×Q8 [×2], C22×C10, C23⋊Q8, D4×C10 [×3], Q8×C10, C5×C4○D4 [×3], C5×C22≀C2, C5×C22⋊Q8 [×3], C5×C4.4D4 [×3], C5×C23⋊Q8

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

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

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

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

110 conjugacy classes

class 1 2A···2G2H2I4A···4L5A5B5C5D10A···10AB10AC···10AJ20A···20AV
order12···2224···4555510···1010···1020···20
size11···1444···411111···14···44···4

110 irreducible representations

dim11111111222222
type+++++-
imageC1C2C2C2C5C10C10C10D4Q8C4○D4C5×D4C5×Q8C5×C4○D4
kernelC5×C23⋊Q8C5×C2.C42C10×C22⋊C4Q8×C2×C10C23⋊Q8C2.C42C2×C22⋊C4C22×Q8C2×C20C22×C10C2×C10C2×C4C23C22
# reps133141212462624824

Matrix representation of C5×C23⋊Q8 in GL6(𝔽41)

1000000
0100000
0010000
0001000
0000160
0000016
,
40320000
010000
001000
0094000
000010
00002840
,
4000000
0400000
001000
000100
0000400
0000040
,
4000000
0400000
0040000
0004000
000010
000001
,
3200000
290000
0093900
00403200
0000400
0000040
,
190000
18400000
0040000
0004000
0000271
00001014

G:=sub<GL(6,GF(41))| [10,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[40,0,0,0,0,0,32,1,0,0,0,0,0,0,1,9,0,0,0,0,0,40,0,0,0,0,0,0,1,28,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[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],[32,2,0,0,0,0,0,9,0,0,0,0,0,0,9,40,0,0,0,0,39,32,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,18,0,0,0,0,9,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,27,10,0,0,0,0,1,14] >;

C5×C23⋊Q8 in GAP, Magma, Sage, TeX

C_5\times C_2^3\rtimes Q_8
% in TeX

G:=Group("C5xC2^3:Q8");
// GroupNames label

G:=SmallGroup(320,894);
// by ID

G=gap.SmallGroup(320,894);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,280,589,288,1766,1731]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^5=b^2=c^2=d^2=e^4=1,f^2=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

׿
×
𝔽