G = C10.222- 1+4 order 320 = 26·5
22nd non-split extension by C10 of 2- 1+4 acting via 2- 1+4/C2×Q8=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C10.222- 1+4, C10.552+ 1+4, C22⋊Q8⋊17D5, C4⋊C4.195D10, (C2×Q8).77D10, D10⋊Q8⋊23C2, D10⋊D4.2C2, C22⋊C4.20D10, Dic5⋊Q8⋊17C2, Dic5⋊3Q8⋊28C2, C20.23D4⋊15C2, (C2×C10).184C24, (C2×C20).628C23, (C22×C4).246D10, D10.13D4⋊21C2, C2.57(D4⋊6D10), Dic5.13(C4○D4), Dic5.5D4⋊26C2, (C2×D20).158C22, C23.D10⋊24C2, C4⋊Dic5.219C22, (Q8×C10).114C22, (C22×D5).75C23, C22.205(C23×D5), C23.123(C22×D5), C23.D5.35C22, D10⋊C4.26C22, C23.23D10⋊25C2, (C22×C10).212C23, (C22×C20).383C22, C5⋊5(C22.36C24), (C2×Dic5).248C23, (C4×Dic5).120C22, C10.D4.32C22, C2.23(Q8.10D10), (C2×Dic10).168C22, (C4×C5⋊D4)⋊59C2, C2.55(D5×C4○D4), C4⋊C4⋊7D5⋊28C2, C4⋊C4⋊D5⋊19C2, (C5×C22⋊Q8)⋊20C2, C10.167(C2×C4○D4), (C2×C4×D5).262C22, (C2×C4).54(C22×D5), (C5×C4⋊C4).165C22, (C2×C5⋊D4).139C22, (C5×C22⋊C4).39C22, SmallGroup(320,1312)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10.222- 1+4
G = < a,b,c,d,e | a10=b4=e2=1, c2=a5, d2=a5b2, bab-1=dad-1=a-1, ac=ca, ae=ea, cbc-1=a5b-1, bd=db, ebe=a5b, cd=dc, ce=ec, ede=a5b2d >
Subgroups: 766 in 216 conjugacy classes, 93 normal (91 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C2×Q8, Dic5, Dic5, C20, D10, C2×C10, C2×C10, C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22⋊Q8, C22.D4, C4.4D4, C42⋊2C2, C4⋊Q8, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×Q8, C22×D5, C22×C10, C22.36C24, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C2×D20, C2×C5⋊D4, C22×C20, Q8×C10, C23.D10, D10⋊D4, Dic5.5D4, Dic5⋊3Q8, C4⋊C4⋊7D5, D10.13D4, D10⋊Q8, C4⋊C4⋊D5, C4×C5⋊D4, C23.23D10, Dic5⋊Q8, C20.23D4, C5×C22⋊Q8, C10.222- 1+4
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, 2+ 1+4, 2- 1+4, C22×D5, C22.36C24, C23×D5, D4⋊6D10, Q8.10D10, D5×C4○D4, C10.222- 1+4
►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 80 30 90)(2 79 21 89)(3 78 22 88)(4 77 23 87)(5 76 24 86)(6 75 25 85)(7 74 26 84)(8 73 27 83)(9 72 28 82)(10 71 29 81)(11 110 155 100)(12 109 156 99)(13 108 157 98)(14 107 158 97)(15 106 159 96)(16 105 160 95)(17 104 151 94)(18 103 152 93)(19 102 153 92)(20 101 154 91)(31 64 41 54)(32 63 42 53)(33 62 43 52)(34 61 44 51)(35 70 45 60)(36 69 46 59)(37 68 47 58)(38 67 48 57)(39 66 49 56)(40 65 50 55)(111 139 121 149)(112 138 122 148)(113 137 123 147)(114 136 124 146)(115 135 125 145)(116 134 126 144)(117 133 127 143)(118 132 128 142)(119 131 129 141)(120 140 130 150)
(1 95 6 100)(2 96 7 91)(3 97 8 92)(4 98 9 93)(5 99 10 94)(11 85 16 90)(12 86 17 81)(13 87 18 82)(14 88 19 83)(15 89 20 84)(21 106 26 101)(22 107 27 102)(23 108 28 103)(24 109 29 104)(25 110 30 105)(31 116 36 111)(32 117 37 112)(33 118 38 113)(34 119 39 114)(35 120 40 115)(41 126 46 121)(42 127 47 122)(43 128 48 123)(44 129 49 124)(45 130 50 125)(51 136 56 131)(52 137 57 132)(53 138 58 133)(54 139 59 134)(55 140 60 135)(61 146 66 141)(62 147 67 142)(63 148 68 143)(64 149 69 144)(65 150 70 145)(71 156 76 151)(72 157 77 152)(73 158 78 153)(74 159 79 154)(75 160 80 155)
(1 45 25 40)(2 44 26 39)(3 43 27 38)(4 42 28 37)(5 41 29 36)(6 50 30 35)(7 49 21 34)(8 48 22 33)(9 47 23 32)(10 46 24 31)(11 145 160 140)(12 144 151 139)(13 143 152 138)(14 142 153 137)(15 141 154 136)(16 150 155 135)(17 149 156 134)(18 148 157 133)(19 147 158 132)(20 146 159 131)(51 84 66 79)(52 83 67 78)(53 82 68 77)(54 81 69 76)(55 90 70 75)(56 89 61 74)(57 88 62 73)(58 87 63 72)(59 86 64 71)(60 85 65 80)(91 124 106 119)(92 123 107 118)(93 122 108 117)(94 121 109 116)(95 130 110 115)(96 129 101 114)(97 128 102 113)(98 127 103 112)(99 126 104 111)(100 125 105 120)
(1 40)(2 31)(3 32)(4 33)(5 34)(6 35)(7 36)(8 37)(9 38)(10 39)(11 135)(12 136)(13 137)(14 138)(15 139)(16 140)(17 131)(18 132)(19 133)(20 134)(21 41)(22 42)(23 43)(24 44)(25 45)(26 46)(27 47)(28 48)(29 49)(30 50)(51 81)(52 82)(53 83)(54 84)(55 85)(56 86)(57 87)(58 88)(59 89)(60 90)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)(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)(141 151)(142 152)(143 153)(144 154)(145 155)(146 156)(147 157)(148 158)(149 159)(150 160)
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), (1,80,30,90)(2,79,21,89)(3,78,22,88)(4,77,23,87)(5,76,24,86)(6,75,25,85)(7,74,26,84)(8,73,27,83)(9,72,28,82)(10,71,29,81)(11,110,155,100)(12,109,156,99)(13,108,157,98)(14,107,158,97)(15,106,159,96)(16,105,160,95)(17,104,151,94)(18,103,152,93)(19,102,153,92)(20,101,154,91)(31,64,41,54)(32,63,42,53)(33,62,43,52)(34,61,44,51)(35,70,45,60)(36,69,46,59)(37,68,47,58)(38,67,48,57)(39,66,49,56)(40,65,50,55)(111,139,121,149)(112,138,122,148)(113,137,123,147)(114,136,124,146)(115,135,125,145)(116,134,126,144)(117,133,127,143)(118,132,128,142)(119,131,129,141)(120,140,130,150), (1,95,6,100)(2,96,7,91)(3,97,8,92)(4,98,9,93)(5,99,10,94)(11,85,16,90)(12,86,17,81)(13,87,18,82)(14,88,19,83)(15,89,20,84)(21,106,26,101)(22,107,27,102)(23,108,28,103)(24,109,29,104)(25,110,30,105)(31,116,36,111)(32,117,37,112)(33,118,38,113)(34,119,39,114)(35,120,40,115)(41,126,46,121)(42,127,47,122)(43,128,48,123)(44,129,49,124)(45,130,50,125)(51,136,56,131)(52,137,57,132)(53,138,58,133)(54,139,59,134)(55,140,60,135)(61,146,66,141)(62,147,67,142)(63,148,68,143)(64,149,69,144)(65,150,70,145)(71,156,76,151)(72,157,77,152)(73,158,78,153)(74,159,79,154)(75,160,80,155), (1,45,25,40)(2,44,26,39)(3,43,27,38)(4,42,28,37)(5,41,29,36)(6,50,30,35)(7,49,21,34)(8,48,22,33)(9,47,23,32)(10,46,24,31)(11,145,160,140)(12,144,151,139)(13,143,152,138)(14,142,153,137)(15,141,154,136)(16,150,155,135)(17,149,156,134)(18,148,157,133)(19,147,158,132)(20,146,159,131)(51,84,66,79)(52,83,67,78)(53,82,68,77)(54,81,69,76)(55,90,70,75)(56,89,61,74)(57,88,62,73)(58,87,63,72)(59,86,64,71)(60,85,65,80)(91,124,106,119)(92,123,107,118)(93,122,108,117)(94,121,109,116)(95,130,110,115)(96,129,101,114)(97,128,102,113)(98,127,103,112)(99,126,104,111)(100,125,105,120), (1,40)(2,31)(3,32)(4,33)(5,34)(6,35)(7,36)(8,37)(9,38)(10,39)(11,135)(12,136)(13,137)(14,138)(15,139)(16,140)(17,131)(18,132)(19,133)(20,134)(21,41)(22,42)(23,43)(24,44)(25,45)(26,46)(27,47)(28,48)(29,49)(30,50)(51,81)(52,82)(53,83)(54,84)(55,85)(56,86)(57,87)(58,88)(59,89)(60,90)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80)(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)(141,151)(142,152)(143,153)(144,154)(145,155)(146,156)(147,157)(148,158)(149,159)(150,160)>;
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), (1,80,30,90)(2,79,21,89)(3,78,22,88)(4,77,23,87)(5,76,24,86)(6,75,25,85)(7,74,26,84)(8,73,27,83)(9,72,28,82)(10,71,29,81)(11,110,155,100)(12,109,156,99)(13,108,157,98)(14,107,158,97)(15,106,159,96)(16,105,160,95)(17,104,151,94)(18,103,152,93)(19,102,153,92)(20,101,154,91)(31,64,41,54)(32,63,42,53)(33,62,43,52)(34,61,44,51)(35,70,45,60)(36,69,46,59)(37,68,47,58)(38,67,48,57)(39,66,49,56)(40,65,50,55)(111,139,121,149)(112,138,122,148)(113,137,123,147)(114,136,124,146)(115,135,125,145)(116,134,126,144)(117,133,127,143)(118,132,128,142)(119,131,129,141)(120,140,130,150), (1,95,6,100)(2,96,7,91)(3,97,8,92)(4,98,9,93)(5,99,10,94)(11,85,16,90)(12,86,17,81)(13,87,18,82)(14,88,19,83)(15,89,20,84)(21,106,26,101)(22,107,27,102)(23,108,28,103)(24,109,29,104)(25,110,30,105)(31,116,36,111)(32,117,37,112)(33,118,38,113)(34,119,39,114)(35,120,40,115)(41,126,46,121)(42,127,47,122)(43,128,48,123)(44,129,49,124)(45,130,50,125)(51,136,56,131)(52,137,57,132)(53,138,58,133)(54,139,59,134)(55,140,60,135)(61,146,66,141)(62,147,67,142)(63,148,68,143)(64,149,69,144)(65,150,70,145)(71,156,76,151)(72,157,77,152)(73,158,78,153)(74,159,79,154)(75,160,80,155), (1,45,25,40)(2,44,26,39)(3,43,27,38)(4,42,28,37)(5,41,29,36)(6,50,30,35)(7,49,21,34)(8,48,22,33)(9,47,23,32)(10,46,24,31)(11,145,160,140)(12,144,151,139)(13,143,152,138)(14,142,153,137)(15,141,154,136)(16,150,155,135)(17,149,156,134)(18,148,157,133)(19,147,158,132)(20,146,159,131)(51,84,66,79)(52,83,67,78)(53,82,68,77)(54,81,69,76)(55,90,70,75)(56,89,61,74)(57,88,62,73)(58,87,63,72)(59,86,64,71)(60,85,65,80)(91,124,106,119)(92,123,107,118)(93,122,108,117)(94,121,109,116)(95,130,110,115)(96,129,101,114)(97,128,102,113)(98,127,103,112)(99,126,104,111)(100,125,105,120), (1,40)(2,31)(3,32)(4,33)(5,34)(6,35)(7,36)(8,37)(9,38)(10,39)(11,135)(12,136)(13,137)(14,138)(15,139)(16,140)(17,131)(18,132)(19,133)(20,134)(21,41)(22,42)(23,43)(24,44)(25,45)(26,46)(27,47)(28,48)(29,49)(30,50)(51,81)(52,82)(53,83)(54,84)(55,85)(56,86)(57,87)(58,88)(59,89)(60,90)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80)(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)(141,151)(142,152)(143,153)(144,154)(145,155)(146,156)(147,157)(148,158)(149,159)(150,160) );
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)], [(1,80,30,90),(2,79,21,89),(3,78,22,88),(4,77,23,87),(5,76,24,86),(6,75,25,85),(7,74,26,84),(8,73,27,83),(9,72,28,82),(10,71,29,81),(11,110,155,100),(12,109,156,99),(13,108,157,98),(14,107,158,97),(15,106,159,96),(16,105,160,95),(17,104,151,94),(18,103,152,93),(19,102,153,92),(20,101,154,91),(31,64,41,54),(32,63,42,53),(33,62,43,52),(34,61,44,51),(35,70,45,60),(36,69,46,59),(37,68,47,58),(38,67,48,57),(39,66,49,56),(40,65,50,55),(111,139,121,149),(112,138,122,148),(113,137,123,147),(114,136,124,146),(115,135,125,145),(116,134,126,144),(117,133,127,143),(118,132,128,142),(119,131,129,141),(120,140,130,150)], [(1,95,6,100),(2,96,7,91),(3,97,8,92),(4,98,9,93),(5,99,10,94),(11,85,16,90),(12,86,17,81),(13,87,18,82),(14,88,19,83),(15,89,20,84),(21,106,26,101),(22,107,27,102),(23,108,28,103),(24,109,29,104),(25,110,30,105),(31,116,36,111),(32,117,37,112),(33,118,38,113),(34,119,39,114),(35,120,40,115),(41,126,46,121),(42,127,47,122),(43,128,48,123),(44,129,49,124),(45,130,50,125),(51,136,56,131),(52,137,57,132),(53,138,58,133),(54,139,59,134),(55,140,60,135),(61,146,66,141),(62,147,67,142),(63,148,68,143),(64,149,69,144),(65,150,70,145),(71,156,76,151),(72,157,77,152),(73,158,78,153),(74,159,79,154),(75,160,80,155)], [(1,45,25,40),(2,44,26,39),(3,43,27,38),(4,42,28,37),(5,41,29,36),(6,50,30,35),(7,49,21,34),(8,48,22,33),(9,47,23,32),(10,46,24,31),(11,145,160,140),(12,144,151,139),(13,143,152,138),(14,142,153,137),(15,141,154,136),(16,150,155,135),(17,149,156,134),(18,148,157,133),(19,147,158,132),(20,146,159,131),(51,84,66,79),(52,83,67,78),(53,82,68,77),(54,81,69,76),(55,90,70,75),(56,89,61,74),(57,88,62,73),(58,87,63,72),(59,86,64,71),(60,85,65,80),(91,124,106,119),(92,123,107,118),(93,122,108,117),(94,121,109,116),(95,130,110,115),(96,129,101,114),(97,128,102,113),(98,127,103,112),(99,126,104,111),(100,125,105,120)], [(1,40),(2,31),(3,32),(4,33),(5,34),(6,35),(7,36),(8,37),(9,38),(10,39),(11,135),(12,136),(13,137),(14,138),(15,139),(16,140),(17,131),(18,132),(19,133),(20,134),(21,41),(22,42),(23,43),(24,44),(25,45),(26,46),(27,47),(28,48),(29,49),(30,50),(51,81),(52,82),(53,83),(54,84),(55,85),(56,86),(57,87),(58,88),(59,89),(60,90),(61,71),(62,72),(63,73),(64,74),(65,75),(66,76),(67,77),(68,78),(69,79),(70,80),(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),(141,151),(142,152),(143,153),(144,154),(145,155),(146,156),(147,157),(148,158),(149,159),(150,160)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | ··· | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | ··· | 20P |
| 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 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 20 | 20 | 2 | 2 | 4 | ··· | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
50 irreducible representations
| dim | 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 | D5 | C4○D4 | D10 | D10 | D10 | D10 | 2+ 1+4 | 2- 1+4 | D4⋊6D10 | Q8.10D10 | D5×C4○D4 |
| kernel | C10.222- 1+4 | C23.D10 | D10⋊D4 | Dic5.5D4 | Dic5⋊3Q8 | C4⋊C4⋊7D5 | D10.13D4 | D10⋊Q8 | C4⋊C4⋊D5 | C4×C5⋊D4 | C23.23D10 | Dic5⋊Q8 | C20.23D4 | C5×C22⋊Q8 | C22⋊Q8 | Dic5 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×Q8 | C10 | C10 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 6 | 2 | 2 | 1 | 1 | 4 | 4 | 4 |
Matrix representation of C10.222- 1+4 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 34 | 0 | 0 |
| 0 | 0 | 7 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 34 | 34 |
| 0 | 0 | 0 | 0 | 7 | 1 |
| 3 | 37 | 0 | 0 | 0 | 0 |
| 23 | 38 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 40 |
| 0 | 0 | 0 | 0 | 3 | 24 |
| 0 | 0 | 24 | 1 | 0 | 0 |
| 0 | 0 | 38 | 17 | 0 | 0 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 27 | 0 | 34 | 0 |
| 0 | 0 | 0 | 27 | 0 | 34 |
| 0 | 0 | 34 | 0 | 14 | 0 |
| 0 | 0 | 0 | 34 | 0 | 14 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 40 | 0 | 0 |
| 0 | 0 | 3 | 24 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 40 |
| 0 | 0 | 0 | 0 | 3 | 24 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 19 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 40 | 0 | 0 |
| 0 | 0 | 1 | 24 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 40 |
| 0 | 0 | 0 | 0 | 1 | 24 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,34,7,0,0,0,0,34,1,0,0,0,0,0,0,34,7,0,0,0,0,34,1],[3,23,0,0,0,0,37,38,0,0,0,0,0,0,0,0,24,38,0,0,0,0,1,17,0,0,17,3,0,0,0,0,40,24,0,0],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,27,0,34,0,0,0,0,27,0,34,0,0,34,0,14,0,0,0,0,34,0,14],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,17,3,0,0,0,0,40,24,0,0,0,0,0,0,17,3,0,0,0,0,40,24],[40,19,0,0,0,0,0,1,0,0,0,0,0,0,17,1,0,0,0,0,40,24,0,0,0,0,0,0,17,1,0,0,0,0,40,24] >;
C10.222- 1+4 in GAP, Magma, Sage, TeX
C_{10}._{22}2_-^{1+4} % in TeX
G:=Group("C10.22ES-(2,2)"); // GroupNames label
G:=SmallGroup(320,1312);
// by ID
G=gap.SmallGroup(320,1312);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,758,555,100,1571,297,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^10=b^4=e^2=1,c^2=a^5,d^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^-1=a^5*b^-1,b*d=d*b,e*b*e=a^5*b,c*d=d*c,c*e=e*c,e*d*e=a^5*b^2*d>;
// generators/relations