?
G = C10.842- (1+4) order 320 = 26·5
39th non-split extension by C10 of 2- (1+4) acting via 2- (1+4)/C4○D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C10.842- (1+4), C10.652+ (1+4), C20⋊Q8⋊33C2, C4⋊C4.199D10, D10⋊Q8⋊32C2, (C2×D4).102D10, C22⋊C4.29D10, Dic5⋊3Q8⋊32C2, C20.17D4⋊22C2, C20.48D4⋊23C2, (C2×C20).181C23, (C2×C10).206C24, Dic5⋊D4.2C2, (C22×C4).260D10, C22.D4⋊11D5, C4⋊Dic5.49C22, D10.12D4⋊35C2, C2.67(D4⋊6D10), C23.30(C22×D5), Dic5.15(C4○D4), Dic5.5D4⋊34C2, (D4×C10).144C22, C23.D10⋊34C2, (C22×C10).38C23, (C22×D5).87C23, C22.227(C23×D5), Dic5.14D4⋊34C2, C23.D5.45C22, D10⋊C4.34C22, C23.11D10⋊15C2, (C22×C20).116C22, C5⋊7(C22.36C24), (C2×Dic5).107C23, (C4×Dic5).133C22, C10.D4.44C22, C2.45(D4.10D10), (C2×Dic10).175C22, (C22×Dic5).132C22, (C4×C5⋊D4)⋊8C2, C2.68(D5×C4○D4), C4⋊C4⋊D5⋊30C2, C10.180(C2×C4○D4), (C2×C4×D5).263C22, (C2×C4).68(C22×D5), (C5×C4⋊C4).179C22, (C2×C5⋊D4).50C22, (C5×C22.D4)⋊14C2, (C5×C22⋊C4).54C22, SmallGroup(320,1334)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 734 in 216 conjugacy classes, 93 normal (91 characteristic)
C1, C2 [×3], C2 [×3], C4 [×13], C22, C22 [×9], C5, C2×C4 [×5], C2×C4 [×11], D4 [×4], Q8 [×4], C23 [×2], C23, D5, C10 [×3], C10 [×2], C42 [×4], C22⋊C4 [×3], C22⋊C4 [×9], C4⋊C4 [×2], C4⋊C4 [×8], C22×C4, C22×C4 [×2], C2×D4, C2×D4 [×2], C2×Q8 [×3], Dic5 [×2], Dic5 [×6], C20 [×5], D10 [×3], C2×C10, C2×C10 [×6], C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8 [×3], C22.D4, C22.D4, C4.4D4 [×3], C42⋊2C2 [×2], C4⋊Q8, Dic10 [×4], C4×D5, C2×Dic5 [×7], C2×Dic5 [×2], C5⋊D4 [×3], C2×C20 [×5], C2×C20, C5×D4, C22×D5, C22×C10 [×2], C22.36C24, C4×Dic5 [×4], C10.D4 [×6], C4⋊Dic5 [×2], D10⋊C4 [×4], C23.D5 [×5], C5×C22⋊C4 [×3], C5×C4⋊C4 [×2], C2×Dic10 [×3], C2×C4×D5, C22×Dic5, C2×C5⋊D4 [×2], C22×C20, D4×C10, C23.11D10, Dic5.14D4, C23.D10, D10.12D4, Dic5.5D4 [×2], Dic5⋊3Q8, C20⋊Q8, D10⋊Q8, C4⋊C4⋊D5, C20.48D4, C4×C5⋊D4, C20.17D4, Dic5⋊D4, C5×C22.D4, C10.842- (1+4)
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×2], C24, D10 [×7], C2×C4○D4, 2+ (1+4), 2- (1+4), C22×D5 [×7], C22.36C24, C23×D5, D4⋊6D10, D5×C4○D4, D4.10D10, C10.842- (1+4)
Generators and relations
G = < a,b,c,d,e | a10=b4=c2=1, d2=b2, e2=a5b2, bab-1=dad-1=a-1, ac=ca, ae=ea, cbc=b-1, bd=db, be=eb, dcd-1=a5c, ce=ec, ede-1=b2d >
►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 38 22 50)(2 37 23 49)(3 36 24 48)(4 35 25 47)(5 34 26 46)(6 33 27 45)(7 32 28 44)(8 31 29 43)(9 40 30 42)(10 39 21 41)(11 150 153 138)(12 149 154 137)(13 148 155 136)(14 147 156 135)(15 146 157 134)(16 145 158 133)(17 144 159 132)(18 143 160 131)(19 142 151 140)(20 141 152 139)(51 82 63 80)(52 81 64 79)(53 90 65 78)(54 89 66 77)(55 88 67 76)(56 87 68 75)(57 86 69 74)(58 85 70 73)(59 84 61 72)(60 83 62 71)(91 122 103 120)(92 121 104 119)(93 130 105 118)(94 129 106 117)(95 128 107 116)(96 127 108 115)(97 126 109 114)(98 125 110 113)(99 124 101 112)(100 123 102 111)
(1 45)(2 46)(3 47)(4 48)(5 49)(6 50)(7 41)(8 42)(9 43)(10 44)(11 138)(12 139)(13 140)(14 131)(15 132)(16 133)(17 134)(18 135)(19 136)(20 137)(21 32)(22 33)(23 34)(24 35)(25 36)(26 37)(27 38)(28 39)(29 40)(30 31)(51 88)(52 89)(53 90)(54 81)(55 82)(56 83)(57 84)(58 85)(59 86)(60 87)(61 74)(62 75)(63 76)(64 77)(65 78)(66 79)(67 80)(68 71)(69 72)(70 73)(91 111)(92 112)(93 113)(94 114)(95 115)(96 116)(97 117)(98 118)(99 119)(100 120)(101 121)(102 122)(103 123)(104 124)(105 125)(106 126)(107 127)(108 128)(109 129)(110 130)(141 154)(142 155)(143 156)(144 157)(145 158)(146 159)(147 160)(148 151)(149 152)(150 153)
(1 85 22 73)(2 84 23 72)(3 83 24 71)(4 82 25 80)(5 81 26 79)(6 90 27 78)(7 89 28 77)(8 88 29 76)(9 87 30 75)(10 86 21 74)(11 105 153 93)(12 104 154 92)(13 103 155 91)(14 102 156 100)(15 101 157 99)(16 110 158 98)(17 109 159 97)(18 108 160 96)(19 107 151 95)(20 106 152 94)(31 67 43 55)(32 66 44 54)(33 65 45 53)(34 64 46 52)(35 63 47 51)(36 62 48 60)(37 61 49 59)(38 70 50 58)(39 69 41 57)(40 68 42 56)(111 135 123 147)(112 134 124 146)(113 133 125 145)(114 132 126 144)(115 131 127 143)(116 140 128 142)(117 139 129 141)(118 138 130 150)(119 137 121 149)(120 136 122 148)
(1 98 27 105)(2 99 28 106)(3 100 29 107)(4 91 30 108)(5 92 21 109)(6 93 22 110)(7 94 23 101)(8 95 24 102)(9 96 25 103)(10 97 26 104)(11 85 158 78)(12 86 159 79)(13 87 160 80)(14 88 151 71)(15 89 152 72)(16 90 153 73)(17 81 154 74)(18 82 155 75)(19 83 156 76)(20 84 157 77)(31 128 48 111)(32 129 49 112)(33 130 50 113)(34 121 41 114)(35 122 42 115)(36 123 43 116)(37 124 44 117)(38 125 45 118)(39 126 46 119)(40 127 47 120)(51 148 68 131)(52 149 69 132)(53 150 70 133)(54 141 61 134)(55 142 62 135)(56 143 63 136)(57 144 64 137)(58 145 65 138)(59 146 66 139)(60 147 67 140)
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), (1,38,22,50)(2,37,23,49)(3,36,24,48)(4,35,25,47)(5,34,26,46)(6,33,27,45)(7,32,28,44)(8,31,29,43)(9,40,30,42)(10,39,21,41)(11,150,153,138)(12,149,154,137)(13,148,155,136)(14,147,156,135)(15,146,157,134)(16,145,158,133)(17,144,159,132)(18,143,160,131)(19,142,151,140)(20,141,152,139)(51,82,63,80)(52,81,64,79)(53,90,65,78)(54,89,66,77)(55,88,67,76)(56,87,68,75)(57,86,69,74)(58,85,70,73)(59,84,61,72)(60,83,62,71)(91,122,103,120)(92,121,104,119)(93,130,105,118)(94,129,106,117)(95,128,107,116)(96,127,108,115)(97,126,109,114)(98,125,110,113)(99,124,101,112)(100,123,102,111), (1,45)(2,46)(3,47)(4,48)(5,49)(6,50)(7,41)(8,42)(9,43)(10,44)(11,138)(12,139)(13,140)(14,131)(15,132)(16,133)(17,134)(18,135)(19,136)(20,137)(21,32)(22,33)(23,34)(24,35)(25,36)(26,37)(27,38)(28,39)(29,40)(30,31)(51,88)(52,89)(53,90)(54,81)(55,82)(56,83)(57,84)(58,85)(59,86)(60,87)(61,74)(62,75)(63,76)(64,77)(65,78)(66,79)(67,80)(68,71)(69,72)(70,73)(91,111)(92,112)(93,113)(94,114)(95,115)(96,116)(97,117)(98,118)(99,119)(100,120)(101,121)(102,122)(103,123)(104,124)(105,125)(106,126)(107,127)(108,128)(109,129)(110,130)(141,154)(142,155)(143,156)(144,157)(145,158)(146,159)(147,160)(148,151)(149,152)(150,153), (1,85,22,73)(2,84,23,72)(3,83,24,71)(4,82,25,80)(5,81,26,79)(6,90,27,78)(7,89,28,77)(8,88,29,76)(9,87,30,75)(10,86,21,74)(11,105,153,93)(12,104,154,92)(13,103,155,91)(14,102,156,100)(15,101,157,99)(16,110,158,98)(17,109,159,97)(18,108,160,96)(19,107,151,95)(20,106,152,94)(31,67,43,55)(32,66,44,54)(33,65,45,53)(34,64,46,52)(35,63,47,51)(36,62,48,60)(37,61,49,59)(38,70,50,58)(39,69,41,57)(40,68,42,56)(111,135,123,147)(112,134,124,146)(113,133,125,145)(114,132,126,144)(115,131,127,143)(116,140,128,142)(117,139,129,141)(118,138,130,150)(119,137,121,149)(120,136,122,148), (1,98,27,105)(2,99,28,106)(3,100,29,107)(4,91,30,108)(5,92,21,109)(6,93,22,110)(7,94,23,101)(8,95,24,102)(9,96,25,103)(10,97,26,104)(11,85,158,78)(12,86,159,79)(13,87,160,80)(14,88,151,71)(15,89,152,72)(16,90,153,73)(17,81,154,74)(18,82,155,75)(19,83,156,76)(20,84,157,77)(31,128,48,111)(32,129,49,112)(33,130,50,113)(34,121,41,114)(35,122,42,115)(36,123,43,116)(37,124,44,117)(38,125,45,118)(39,126,46,119)(40,127,47,120)(51,148,68,131)(52,149,69,132)(53,150,70,133)(54,141,61,134)(55,142,62,135)(56,143,63,136)(57,144,64,137)(58,145,65,138)(59,146,66,139)(60,147,67,140)>;
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), (1,38,22,50)(2,37,23,49)(3,36,24,48)(4,35,25,47)(5,34,26,46)(6,33,27,45)(7,32,28,44)(8,31,29,43)(9,40,30,42)(10,39,21,41)(11,150,153,138)(12,149,154,137)(13,148,155,136)(14,147,156,135)(15,146,157,134)(16,145,158,133)(17,144,159,132)(18,143,160,131)(19,142,151,140)(20,141,152,139)(51,82,63,80)(52,81,64,79)(53,90,65,78)(54,89,66,77)(55,88,67,76)(56,87,68,75)(57,86,69,74)(58,85,70,73)(59,84,61,72)(60,83,62,71)(91,122,103,120)(92,121,104,119)(93,130,105,118)(94,129,106,117)(95,128,107,116)(96,127,108,115)(97,126,109,114)(98,125,110,113)(99,124,101,112)(100,123,102,111), (1,45)(2,46)(3,47)(4,48)(5,49)(6,50)(7,41)(8,42)(9,43)(10,44)(11,138)(12,139)(13,140)(14,131)(15,132)(16,133)(17,134)(18,135)(19,136)(20,137)(21,32)(22,33)(23,34)(24,35)(25,36)(26,37)(27,38)(28,39)(29,40)(30,31)(51,88)(52,89)(53,90)(54,81)(55,82)(56,83)(57,84)(58,85)(59,86)(60,87)(61,74)(62,75)(63,76)(64,77)(65,78)(66,79)(67,80)(68,71)(69,72)(70,73)(91,111)(92,112)(93,113)(94,114)(95,115)(96,116)(97,117)(98,118)(99,119)(100,120)(101,121)(102,122)(103,123)(104,124)(105,125)(106,126)(107,127)(108,128)(109,129)(110,130)(141,154)(142,155)(143,156)(144,157)(145,158)(146,159)(147,160)(148,151)(149,152)(150,153), (1,85,22,73)(2,84,23,72)(3,83,24,71)(4,82,25,80)(5,81,26,79)(6,90,27,78)(7,89,28,77)(8,88,29,76)(9,87,30,75)(10,86,21,74)(11,105,153,93)(12,104,154,92)(13,103,155,91)(14,102,156,100)(15,101,157,99)(16,110,158,98)(17,109,159,97)(18,108,160,96)(19,107,151,95)(20,106,152,94)(31,67,43,55)(32,66,44,54)(33,65,45,53)(34,64,46,52)(35,63,47,51)(36,62,48,60)(37,61,49,59)(38,70,50,58)(39,69,41,57)(40,68,42,56)(111,135,123,147)(112,134,124,146)(113,133,125,145)(114,132,126,144)(115,131,127,143)(116,140,128,142)(117,139,129,141)(118,138,130,150)(119,137,121,149)(120,136,122,148), (1,98,27,105)(2,99,28,106)(3,100,29,107)(4,91,30,108)(5,92,21,109)(6,93,22,110)(7,94,23,101)(8,95,24,102)(9,96,25,103)(10,97,26,104)(11,85,158,78)(12,86,159,79)(13,87,160,80)(14,88,151,71)(15,89,152,72)(16,90,153,73)(17,81,154,74)(18,82,155,75)(19,83,156,76)(20,84,157,77)(31,128,48,111)(32,129,49,112)(33,130,50,113)(34,121,41,114)(35,122,42,115)(36,123,43,116)(37,124,44,117)(38,125,45,118)(39,126,46,119)(40,127,47,120)(51,148,68,131)(52,149,69,132)(53,150,70,133)(54,141,61,134)(55,142,62,135)(56,143,63,136)(57,144,64,137)(58,145,65,138)(59,146,66,139)(60,147,67,140) );
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)], [(1,38,22,50),(2,37,23,49),(3,36,24,48),(4,35,25,47),(5,34,26,46),(6,33,27,45),(7,32,28,44),(8,31,29,43),(9,40,30,42),(10,39,21,41),(11,150,153,138),(12,149,154,137),(13,148,155,136),(14,147,156,135),(15,146,157,134),(16,145,158,133),(17,144,159,132),(18,143,160,131),(19,142,151,140),(20,141,152,139),(51,82,63,80),(52,81,64,79),(53,90,65,78),(54,89,66,77),(55,88,67,76),(56,87,68,75),(57,86,69,74),(58,85,70,73),(59,84,61,72),(60,83,62,71),(91,122,103,120),(92,121,104,119),(93,130,105,118),(94,129,106,117),(95,128,107,116),(96,127,108,115),(97,126,109,114),(98,125,110,113),(99,124,101,112),(100,123,102,111)], [(1,45),(2,46),(3,47),(4,48),(5,49),(6,50),(7,41),(8,42),(9,43),(10,44),(11,138),(12,139),(13,140),(14,131),(15,132),(16,133),(17,134),(18,135),(19,136),(20,137),(21,32),(22,33),(23,34),(24,35),(25,36),(26,37),(27,38),(28,39),(29,40),(30,31),(51,88),(52,89),(53,90),(54,81),(55,82),(56,83),(57,84),(58,85),(59,86),(60,87),(61,74),(62,75),(63,76),(64,77),(65,78),(66,79),(67,80),(68,71),(69,72),(70,73),(91,111),(92,112),(93,113),(94,114),(95,115),(96,116),(97,117),(98,118),(99,119),(100,120),(101,121),(102,122),(103,123),(104,124),(105,125),(106,126),(107,127),(108,128),(109,129),(110,130),(141,154),(142,155),(143,156),(144,157),(145,158),(146,159),(147,160),(148,151),(149,152),(150,153)], [(1,85,22,73),(2,84,23,72),(3,83,24,71),(4,82,25,80),(5,81,26,79),(6,90,27,78),(7,89,28,77),(8,88,29,76),(9,87,30,75),(10,86,21,74),(11,105,153,93),(12,104,154,92),(13,103,155,91),(14,102,156,100),(15,101,157,99),(16,110,158,98),(17,109,159,97),(18,108,160,96),(19,107,151,95),(20,106,152,94),(31,67,43,55),(32,66,44,54),(33,65,45,53),(34,64,46,52),(35,63,47,51),(36,62,48,60),(37,61,49,59),(38,70,50,58),(39,69,41,57),(40,68,42,56),(111,135,123,147),(112,134,124,146),(113,133,125,145),(114,132,126,144),(115,131,127,143),(116,140,128,142),(117,139,129,141),(118,138,130,150),(119,137,121,149),(120,136,122,148)], [(1,98,27,105),(2,99,28,106),(3,100,29,107),(4,91,30,108),(5,92,21,109),(6,93,22,110),(7,94,23,101),(8,95,24,102),(9,96,25,103),(10,97,26,104),(11,85,158,78),(12,86,159,79),(13,87,160,80),(14,88,151,71),(15,89,152,72),(16,90,153,73),(17,81,154,74),(18,82,155,75),(19,83,156,76),(20,84,157,77),(31,128,48,111),(32,129,49,112),(33,130,50,113),(34,121,41,114),(35,122,42,115),(36,123,43,116),(37,124,44,117),(38,125,45,118),(39,126,46,119),(40,127,47,120),(51,148,68,131),(52,149,69,132),(53,150,70,133),(54,141,61,134),(55,142,62,135),(56,143,63,136),(57,144,64,137),(58,145,65,138),(59,146,66,139),(60,147,67,140)])
Matrix representation ►G ⊆ GL8(𝔽41)
| 0 | 6 | 0 | 0 | 0 | 0 | 0 | 0 |
| 34 | 7 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 35 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 34 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 35 |
| 0 | 0 | 0 | 0 | 0 | 0 | 7 | 34 |
| 0 | 0 | 7 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 34 | 0 | 0 | 0 | 0 |
| 7 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 7 | 34 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 18 | 14 | 39 |
| 0 | 0 | 0 | 0 | 4 | 38 | 14 | 27 |
| 0 | 0 | 0 | 0 | 0 | 0 | 3 | 18 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 38 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 6 | 2 | 0 |
| 0 | 0 | 0 | 0 | 34 | 24 | 0 | 2 |
| 0 | 0 | 0 | 0 | 0 | 0 | 24 | 35 |
| 0 | 0 | 0 | 0 | 0 | 0 | 7 | 17 |
| 37 | 17 | 5 | 30 | 0 | 0 | 0 | 0 |
| 22 | 4 | 34 | 36 | 0 | 0 | 0 | 0 |
| 36 | 11 | 4 | 24 | 0 | 0 | 0 | 0 |
| 7 | 5 | 19 | 37 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 13 | 27 | 39 |
| 0 | 0 | 0 | 0 | 32 | 9 | 36 | 14 |
| 0 | 0 | 0 | 0 | 2 | 12 | 32 | 13 |
| 0 | 0 | 0 | 0 | 30 | 39 | 32 | 9 |
| 32 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 0 | 22 | 15 |
| 0 | 0 | 0 | 0 | 0 | 32 | 3 | 19 |
| 0 | 0 | 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 9 |
G:=sub<GL(8,GF(41))| [0,34,0,0,0,0,0,0,6,7,0,0,0,0,0,0,0,0,0,34,0,0,0,0,0,0,6,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,35,34,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,35,34],[0,0,7,7,0,0,0,0,0,0,40,34,0,0,0,0,7,7,0,0,0,0,0,0,40,34,0,0,0,0,0,0,0,0,0,0,3,4,0,0,0,0,0,0,18,38,0,0,0,0,0,0,14,14,3,4,0,0,0,0,39,27,18,38],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,17,34,0,0,0,0,0,0,6,24,0,0,0,0,0,0,2,0,24,7,0,0,0,0,0,2,35,17],[37,22,36,7,0,0,0,0,17,4,11,5,0,0,0,0,5,34,4,19,0,0,0,0,30,36,24,37,0,0,0,0,0,0,0,0,32,32,2,30,0,0,0,0,13,9,12,39,0,0,0,0,27,36,32,32,0,0,0,0,39,14,13,9],[32,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,22,3,9,0,0,0,0,0,15,19,0,9] >;
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | ··· | 4O | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 20A | ··· | 20H | 20I | ··· | 20N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 20 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | D10 | D10 | D10 | D10 | 2+ (1+4) | 2- (1+4) | D4⋊6D10 | D5×C4○D4 | D4.10D10 |
| kernel | C10.842- (1+4) | C23.11D10 | Dic5.14D4 | C23.D10 | D10.12D4 | Dic5.5D4 | Dic5⋊3Q8 | C20⋊Q8 | D10⋊Q8 | C4⋊C4⋊D5 | C20.48D4 | C4×C5⋊D4 | C20.17D4 | Dic5⋊D4 | C5×C22.D4 | C22.D4 | Dic5 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C10 | C10 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 6 | 4 | 2 | 2 | 1 | 1 | 4 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_{10}._{84}2_-^{(1+4)} % in TeX
G:=Group("C10.84ES-(2,2)"); // GroupNames label
G:=SmallGroup(320,1334);
// by ID
G=gap.SmallGroup(320,1334);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,219,675,297,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^10=b^4=c^2=1,d^2=b^2,e^2=a^5*b^2,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,d*c*d^-1=a^5*c,c*e=e*c,e*d*e^-1=b^2*d>;
// generators/relations
×
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ℤ
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