direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D4.9D10, C20.36C24, Dic10.31C23, C4○D4.41D10, (C2×C20).219D4, C20.428(C2×D4), C4.36(C23×D5), (C2×D4).233D10, C10⋊5(C8.C22), C5⋊2C8.15C23, D4.D5⋊18C22, (C2×Q8).191D10, (C5×D4).24C23, C5⋊Q16⋊17C22, D4.24(C22×D5), Q8.24(C22×D5), (C5×Q8).24C23, (C2×C20).558C23, (C22×C10).125D4, (C22×C4).283D10, C10.161(C22×D4), C23.69(C5⋊D4), C4.Dic5⋊38C22, (C2×Dic10)⋊70C22, (C22×Dic10)⋊21C2, (D4×C10).273C22, (Q8×C10).238C22, (C22×C20).293C22, C5⋊6(C2×C8.C22), C4.31(C2×C5⋊D4), (C2×D4.D5)⋊31C2, (C2×C4○D4).10D5, (C2×C5⋊Q16)⋊31C2, (C2×C10).77(C2×D4), (C10×C4○D4).11C2, (C2×C4).96(C5⋊D4), (C2×C4.Dic5)⋊32C2, C2.34(C22×C5⋊D4), (C5×C4○D4).50C22, (C2×C4).247(C22×D5), C22.120(C2×C5⋊D4), (C2×C5⋊2C8).184C22, SmallGroup(320,1495)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 734 in 258 conjugacy classes, 111 normal (27 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×2], C4 [×6], C22, C22 [×2], C22 [×6], C5, C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×11], D4 [×2], D4 [×5], Q8 [×2], Q8 [×11], C23, C23, C10, C10 [×2], C10 [×4], C2×C8 [×2], M4(2) [×4], SD16 [×8], Q16 [×8], C22×C4, C22×C4 [×2], C2×D4, C2×D4, C2×Q8, C2×Q8 [×9], C4○D4 [×4], C4○D4 [×2], Dic5 [×4], C20 [×2], C20 [×2], C20 [×2], C2×C10, C2×C10 [×2], C2×C10 [×6], C2×M4(2), C2×SD16 [×2], C2×Q16 [×2], C8.C22 [×8], C22×Q8, C2×C4○D4, C5⋊2C8 [×4], Dic10 [×4], Dic10 [×6], C2×Dic5 [×6], C2×C20 [×2], C2×C20 [×4], C2×C20 [×5], C5×D4 [×2], C5×D4 [×5], C5×Q8 [×2], C5×Q8, C22×C10, C22×C10, C2×C8.C22, C2×C5⋊2C8 [×2], C4.Dic5 [×4], D4.D5 [×8], C5⋊Q16 [×8], C2×Dic10 [×6], C2×Dic10 [×3], C22×Dic5, C22×C20, C22×C20, D4×C10, D4×C10, Q8×C10, C5×C4○D4 [×4], C5×C4○D4 [×2], C2×C4.Dic5, C2×D4.D5 [×2], C2×C5⋊Q16 [×2], D4.9D10 [×8], C22×Dic10, C10×C4○D4, C2×D4.9D10
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C8.C22 [×2], C22×D4, C5⋊D4 [×4], C22×D5 [×7], C2×C8.C22, C2×C5⋊D4 [×6], C23×D5, D4.9D10 [×2], C22×C5⋊D4, C2×D4.9D10
Generators and relations
G = < a,b,c,d,e | a2=b4=c2=d10=1, e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, dcd-1=b2c, ece-1=b-1c, ede-1=d-1 >
►On 160 points
Generators in S160
(1 46)(2 47)(3 48)(4 49)(5 50)(6 41)(7 42)(8 43)(9 44)(10 45)(11 21)(12 22)(13 23)(14 24)(15 25)(16 26)(17 27)(18 28)(19 29)(20 30)(31 110)(32 101)(33 102)(34 103)(35 104)(36 105)(37 106)(38 107)(39 108)(40 109)(51 71)(52 72)(53 73)(54 74)(55 75)(56 76)(57 77)(58 78)(59 79)(60 80)(61 89)(62 90)(63 81)(64 82)(65 83)(66 84)(67 85)(68 86)(69 87)(70 88)(91 116)(92 117)(93 118)(94 119)(95 120)(96 111)(97 112)(98 113)(99 114)(100 115)(121 131)(122 132)(123 133)(124 134)(125 135)(126 136)(127 137)(128 138)(129 139)(130 140)(141 151)(142 152)(143 153)(144 154)(145 155)(146 156)(147 157)(148 158)(149 159)(150 160)
(1 30 77 104)(2 21 78 105)(3 22 79 106)(4 23 80 107)(5 24 71 108)(6 25 72 109)(7 26 73 110)(8 27 74 101)(9 28 75 102)(10 29 76 103)(11 58 36 47)(12 59 37 48)(13 60 38 49)(14 51 39 50)(15 52 40 41)(16 53 31 42)(17 54 32 43)(18 55 33 44)(19 56 34 45)(20 57 35 46)(61 157 114 130)(62 158 115 121)(63 159 116 122)(64 160 117 123)(65 151 118 124)(66 152 119 125)(67 153 120 126)(68 154 111 127)(69 155 112 128)(70 156 113 129)(81 149 91 132)(82 150 92 133)(83 141 93 134)(84 142 94 135)(85 143 95 136)(86 144 96 137)(87 145 97 138)(88 146 98 139)(89 147 99 140)(90 148 100 131)
(1 35)(2 11)(3 37)(4 13)(5 39)(6 15)(7 31)(8 17)(9 33)(10 19)(12 79)(14 71)(16 73)(18 75)(20 77)(21 47)(22 59)(23 49)(24 51)(25 41)(26 53)(27 43)(28 55)(29 45)(30 57)(32 74)(34 76)(36 78)(38 80)(40 72)(42 110)(44 102)(46 104)(48 106)(50 108)(52 109)(54 101)(56 103)(58 105)(60 107)(61 89)(62 100)(63 81)(64 92)(65 83)(66 94)(67 85)(68 96)(69 87)(70 98)(82 117)(84 119)(86 111)(88 113)(90 115)(91 116)(93 118)(95 120)(97 112)(99 114)(121 131)(122 149)(123 133)(124 141)(125 135)(126 143)(127 137)(128 145)(129 139)(130 147)(132 159)(134 151)(136 153)(138 155)(140 157)(142 152)(144 154)(146 156)(148 158)(150 160)
(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)
(1 152 77 125)(2 151 78 124)(3 160 79 123)(4 159 80 122)(5 158 71 121)(6 157 72 130)(7 156 73 129)(8 155 74 128)(9 154 75 127)(10 153 76 126)(11 83 36 93)(12 82 37 92)(13 81 38 91)(14 90 39 100)(15 89 40 99)(16 88 31 98)(17 87 32 97)(18 86 33 96)(19 85 34 95)(20 84 35 94)(21 65 105 118)(22 64 106 117)(23 63 107 116)(24 62 108 115)(25 61 109 114)(26 70 110 113)(27 69 101 112)(28 68 102 111)(29 67 103 120)(30 66 104 119)(41 147 52 140)(42 146 53 139)(43 145 54 138)(44 144 55 137)(45 143 56 136)(46 142 57 135)(47 141 58 134)(48 150 59 133)(49 149 60 132)(50 148 51 131)
G:=sub<Sym(160)| (1,46)(2,47)(3,48)(4,49)(5,50)(6,41)(7,42)(8,43)(9,44)(10,45)(11,21)(12,22)(13,23)(14,24)(15,25)(16,26)(17,27)(18,28)(19,29)(20,30)(31,110)(32,101)(33,102)(34,103)(35,104)(36,105)(37,106)(38,107)(39,108)(40,109)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80)(61,89)(62,90)(63,81)(64,82)(65,83)(66,84)(67,85)(68,86)(69,87)(70,88)(91,116)(92,117)(93,118)(94,119)(95,120)(96,111)(97,112)(98,113)(99,114)(100,115)(121,131)(122,132)(123,133)(124,134)(125,135)(126,136)(127,137)(128,138)(129,139)(130,140)(141,151)(142,152)(143,153)(144,154)(145,155)(146,156)(147,157)(148,158)(149,159)(150,160), (1,30,77,104)(2,21,78,105)(3,22,79,106)(4,23,80,107)(5,24,71,108)(6,25,72,109)(7,26,73,110)(8,27,74,101)(9,28,75,102)(10,29,76,103)(11,58,36,47)(12,59,37,48)(13,60,38,49)(14,51,39,50)(15,52,40,41)(16,53,31,42)(17,54,32,43)(18,55,33,44)(19,56,34,45)(20,57,35,46)(61,157,114,130)(62,158,115,121)(63,159,116,122)(64,160,117,123)(65,151,118,124)(66,152,119,125)(67,153,120,126)(68,154,111,127)(69,155,112,128)(70,156,113,129)(81,149,91,132)(82,150,92,133)(83,141,93,134)(84,142,94,135)(85,143,95,136)(86,144,96,137)(87,145,97,138)(88,146,98,139)(89,147,99,140)(90,148,100,131), (1,35)(2,11)(3,37)(4,13)(5,39)(6,15)(7,31)(8,17)(9,33)(10,19)(12,79)(14,71)(16,73)(18,75)(20,77)(21,47)(22,59)(23,49)(24,51)(25,41)(26,53)(27,43)(28,55)(29,45)(30,57)(32,74)(34,76)(36,78)(38,80)(40,72)(42,110)(44,102)(46,104)(48,106)(50,108)(52,109)(54,101)(56,103)(58,105)(60,107)(61,89)(62,100)(63,81)(64,92)(65,83)(66,94)(67,85)(68,96)(69,87)(70,98)(82,117)(84,119)(86,111)(88,113)(90,115)(91,116)(93,118)(95,120)(97,112)(99,114)(121,131)(122,149)(123,133)(124,141)(125,135)(126,143)(127,137)(128,145)(129,139)(130,147)(132,159)(134,151)(136,153)(138,155)(140,157)(142,152)(144,154)(146,156)(148,158)(150,160), (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), (1,152,77,125)(2,151,78,124)(3,160,79,123)(4,159,80,122)(5,158,71,121)(6,157,72,130)(7,156,73,129)(8,155,74,128)(9,154,75,127)(10,153,76,126)(11,83,36,93)(12,82,37,92)(13,81,38,91)(14,90,39,100)(15,89,40,99)(16,88,31,98)(17,87,32,97)(18,86,33,96)(19,85,34,95)(20,84,35,94)(21,65,105,118)(22,64,106,117)(23,63,107,116)(24,62,108,115)(25,61,109,114)(26,70,110,113)(27,69,101,112)(28,68,102,111)(29,67,103,120)(30,66,104,119)(41,147,52,140)(42,146,53,139)(43,145,54,138)(44,144,55,137)(45,143,56,136)(46,142,57,135)(47,141,58,134)(48,150,59,133)(49,149,60,132)(50,148,51,131)>;
G:=Group( (1,46)(2,47)(3,48)(4,49)(5,50)(6,41)(7,42)(8,43)(9,44)(10,45)(11,21)(12,22)(13,23)(14,24)(15,25)(16,26)(17,27)(18,28)(19,29)(20,30)(31,110)(32,101)(33,102)(34,103)(35,104)(36,105)(37,106)(38,107)(39,108)(40,109)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80)(61,89)(62,90)(63,81)(64,82)(65,83)(66,84)(67,85)(68,86)(69,87)(70,88)(91,116)(92,117)(93,118)(94,119)(95,120)(96,111)(97,112)(98,113)(99,114)(100,115)(121,131)(122,132)(123,133)(124,134)(125,135)(126,136)(127,137)(128,138)(129,139)(130,140)(141,151)(142,152)(143,153)(144,154)(145,155)(146,156)(147,157)(148,158)(149,159)(150,160), (1,30,77,104)(2,21,78,105)(3,22,79,106)(4,23,80,107)(5,24,71,108)(6,25,72,109)(7,26,73,110)(8,27,74,101)(9,28,75,102)(10,29,76,103)(11,58,36,47)(12,59,37,48)(13,60,38,49)(14,51,39,50)(15,52,40,41)(16,53,31,42)(17,54,32,43)(18,55,33,44)(19,56,34,45)(20,57,35,46)(61,157,114,130)(62,158,115,121)(63,159,116,122)(64,160,117,123)(65,151,118,124)(66,152,119,125)(67,153,120,126)(68,154,111,127)(69,155,112,128)(70,156,113,129)(81,149,91,132)(82,150,92,133)(83,141,93,134)(84,142,94,135)(85,143,95,136)(86,144,96,137)(87,145,97,138)(88,146,98,139)(89,147,99,140)(90,148,100,131), (1,35)(2,11)(3,37)(4,13)(5,39)(6,15)(7,31)(8,17)(9,33)(10,19)(12,79)(14,71)(16,73)(18,75)(20,77)(21,47)(22,59)(23,49)(24,51)(25,41)(26,53)(27,43)(28,55)(29,45)(30,57)(32,74)(34,76)(36,78)(38,80)(40,72)(42,110)(44,102)(46,104)(48,106)(50,108)(52,109)(54,101)(56,103)(58,105)(60,107)(61,89)(62,100)(63,81)(64,92)(65,83)(66,94)(67,85)(68,96)(69,87)(70,98)(82,117)(84,119)(86,111)(88,113)(90,115)(91,116)(93,118)(95,120)(97,112)(99,114)(121,131)(122,149)(123,133)(124,141)(125,135)(126,143)(127,137)(128,145)(129,139)(130,147)(132,159)(134,151)(136,153)(138,155)(140,157)(142,152)(144,154)(146,156)(148,158)(150,160), (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), (1,152,77,125)(2,151,78,124)(3,160,79,123)(4,159,80,122)(5,158,71,121)(6,157,72,130)(7,156,73,129)(8,155,74,128)(9,154,75,127)(10,153,76,126)(11,83,36,93)(12,82,37,92)(13,81,38,91)(14,90,39,100)(15,89,40,99)(16,88,31,98)(17,87,32,97)(18,86,33,96)(19,85,34,95)(20,84,35,94)(21,65,105,118)(22,64,106,117)(23,63,107,116)(24,62,108,115)(25,61,109,114)(26,70,110,113)(27,69,101,112)(28,68,102,111)(29,67,103,120)(30,66,104,119)(41,147,52,140)(42,146,53,139)(43,145,54,138)(44,144,55,137)(45,143,56,136)(46,142,57,135)(47,141,58,134)(48,150,59,133)(49,149,60,132)(50,148,51,131) );
G=PermutationGroup([(1,46),(2,47),(3,48),(4,49),(5,50),(6,41),(7,42),(8,43),(9,44),(10,45),(11,21),(12,22),(13,23),(14,24),(15,25),(16,26),(17,27),(18,28),(19,29),(20,30),(31,110),(32,101),(33,102),(34,103),(35,104),(36,105),(37,106),(38,107),(39,108),(40,109),(51,71),(52,72),(53,73),(54,74),(55,75),(56,76),(57,77),(58,78),(59,79),(60,80),(61,89),(62,90),(63,81),(64,82),(65,83),(66,84),(67,85),(68,86),(69,87),(70,88),(91,116),(92,117),(93,118),(94,119),(95,120),(96,111),(97,112),(98,113),(99,114),(100,115),(121,131),(122,132),(123,133),(124,134),(125,135),(126,136),(127,137),(128,138),(129,139),(130,140),(141,151),(142,152),(143,153),(144,154),(145,155),(146,156),(147,157),(148,158),(149,159),(150,160)], [(1,30,77,104),(2,21,78,105),(3,22,79,106),(4,23,80,107),(5,24,71,108),(6,25,72,109),(7,26,73,110),(8,27,74,101),(9,28,75,102),(10,29,76,103),(11,58,36,47),(12,59,37,48),(13,60,38,49),(14,51,39,50),(15,52,40,41),(16,53,31,42),(17,54,32,43),(18,55,33,44),(19,56,34,45),(20,57,35,46),(61,157,114,130),(62,158,115,121),(63,159,116,122),(64,160,117,123),(65,151,118,124),(66,152,119,125),(67,153,120,126),(68,154,111,127),(69,155,112,128),(70,156,113,129),(81,149,91,132),(82,150,92,133),(83,141,93,134),(84,142,94,135),(85,143,95,136),(86,144,96,137),(87,145,97,138),(88,146,98,139),(89,147,99,140),(90,148,100,131)], [(1,35),(2,11),(3,37),(4,13),(5,39),(6,15),(7,31),(8,17),(9,33),(10,19),(12,79),(14,71),(16,73),(18,75),(20,77),(21,47),(22,59),(23,49),(24,51),(25,41),(26,53),(27,43),(28,55),(29,45),(30,57),(32,74),(34,76),(36,78),(38,80),(40,72),(42,110),(44,102),(46,104),(48,106),(50,108),(52,109),(54,101),(56,103),(58,105),(60,107),(61,89),(62,100),(63,81),(64,92),(65,83),(66,94),(67,85),(68,96),(69,87),(70,98),(82,117),(84,119),(86,111),(88,113),(90,115),(91,116),(93,118),(95,120),(97,112),(99,114),(121,131),(122,149),(123,133),(124,141),(125,135),(126,143),(127,137),(128,145),(129,139),(130,147),(132,159),(134,151),(136,153),(138,155),(140,157),(142,152),(144,154),(146,156),(148,158),(150,160)], [(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)], [(1,152,77,125),(2,151,78,124),(3,160,79,123),(4,159,80,122),(5,158,71,121),(6,157,72,130),(7,156,73,129),(8,155,74,128),(9,154,75,127),(10,153,76,126),(11,83,36,93),(12,82,37,92),(13,81,38,91),(14,90,39,100),(15,89,40,99),(16,88,31,98),(17,87,32,97),(18,86,33,96),(19,85,34,95),(20,84,35,94),(21,65,105,118),(22,64,106,117),(23,63,107,116),(24,62,108,115),(25,61,109,114),(26,70,110,113),(27,69,101,112),(28,68,102,111),(29,67,103,120),(30,66,104,119),(41,147,52,140),(42,146,53,139),(43,145,54,138),(44,144,55,137),(45,143,56,136),(46,142,57,135),(47,141,58,134),(48,150,59,133),(49,149,60,132),(50,148,51,131)])
Matrix representation ►G ⊆ GL8(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 32 | 0 | 0 |
| 0 | 0 | 0 | 0 | 23 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 26 | 37 | 1 | 37 |
| 0 | 0 | 0 | 0 | 31 | 3 | 21 | 40 |
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 25 | 5 | 1 | 37 |
| 0 | 0 | 0 | 0 | 0 | 28 | 0 | 40 |
| 1 | 35 | 0 | 0 | 0 | 0 | 0 | 0 |
| 6 | 6 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 17 | 9 | 0 | 10 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 11 | 33 | 13 | 32 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 6 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 22 | 36 | 0 | 0 | 0 | 0 |
| 0 | 0 | 31 | 19 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 19 | 0 | 32 |
| 0 | 0 | 0 | 0 | 36 | 37 | 20 | 0 |
| 0 | 0 | 0 | 0 | 30 | 8 | 4 | 8 |
| 0 | 0 | 0 | 0 | 17 | 21 | 24 | 40 |
G:=sub<GL(8,GF(41))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,23,26,31,0,0,0,0,32,40,37,3,0,0,0,0,0,0,1,21,0,0,0,0,0,0,37,40],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,17,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,25,0,0,0,0,0,9,1,5,28,0,0,0,0,0,0,1,0,0,0,0,0,0,0,37,40],[1,6,0,0,0,0,0,0,35,6,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,17,0,11,0,0,0,0,0,9,0,33,0,0,0,0,2,0,1,13,0,0,0,0,0,10,0,32],[1,6,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,22,31,0,0,0,0,0,0,36,19,0,0,0,0,0,0,0,0,1,36,30,17,0,0,0,0,19,37,8,21,0,0,0,0,0,20,4,24,0,0,0,0,32,0,8,40] >;
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | ··· | 10R | 20A | ··· | 20H | 20I | ··· | 20T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 20 | 20 | 20 | 20 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | D10 | D10 | D10 | D10 | C5⋊D4 | C5⋊D4 | C8.C22 | D4.9D10 |
| kernel | C2×D4.9D10 | C2×C4.Dic5 | C2×D4.D5 | C2×C5⋊Q16 | D4.9D10 | C22×Dic10 | C10×C4○D4 | C2×C20 | C22×C10 | C2×C4○D4 | C22×C4 | C2×D4 | C2×Q8 | C4○D4 | C2×C4 | C23 | C10 | C2 |
| # reps | 1 | 1 | 2 | 2 | 8 | 1 | 1 | 3 | 1 | 2 | 2 | 2 | 2 | 8 | 12 | 4 | 2 | 8 |
In GAP, Magma, Sage, TeX
C_2\times D_4._9D_{10} % in TeX
G:=Group("C2xD4.9D10"); // GroupNames label
G:=SmallGroup(320,1495);
// by ID
G=gap.SmallGroup(320,1495);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,675,297,1684,235,102,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=c^2=d^10=1,e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e^-1=b^-1,b*d=d*b,d*c*d^-1=b^2*c,e*c*e^-1=b^-1*c,e*d*e^-1=d^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀