G = C10.822- 1+4 order 320 = 26·5
37th non-split extension by C10 of 2- 1+4 acting via 2- 1+4/C4○D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C10.822- 1+4, C5⋊D4⋊7D4, C20⋊Q8⋊32C2, C5⋊6(D4⋊6D4), C4⋊C4.106D10, D10.46(C2×D4), D10⋊D4⋊30C2, C22.13(D4×D5), D10⋊Q8⋊28C2, Dic5⋊6(C4○D4), (C2×D4).164D10, (C2×C20).72C23, C22⋊C4.27D10, Dic5.51(C2×D4), C10.84(C22×D4), C22.D4⋊5D5, Dic5⋊D4⋊21C2, Dic5⋊4D4⋊19C2, (C2×C10).199C24, (C22×C4).257D10, D10.12D4⋊30C2, D10.13D4⋊28C2, (C2×D20).230C22, (D4×C10).137C22, C22.D20⋊20C2, C4⋊Dic5.227C22, (C22×C10).34C23, C22.220(C23×D5), C23.201(C22×D5), Dic5.14D4⋊31C2, C23.D5.42C22, D10⋊C4.32C22, (C22×C20).113C22, (C2×Dic5).103C23, (C4×Dic5).131C22, C10.D4.41C22, (C22×D5).217C23, C2.43(D4.10D10), (C2×Dic10).174C22, (C22×Dic5).129C22, C2.57(C2×D4×D5), (D5×C4⋊C4)⋊32C2, C2.61(D5×C4○D4), (C2×C4○D20)⋊12C2, (C2×C10).60(C2×D4), (C2×D4⋊2D5)⋊16C2, C10.173(C2×C4○D4), (C2×C4×D5).119C22, (C2×C4).62(C22×D5), (C2×C10.D4)⋊27C2, (C5×C4⋊C4).176C22, (C5×C22.D4)⋊7C2, (C2×C5⋊D4).142C22, (C5×C22⋊C4).51C22, SmallGroup(320,1327)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10.822- 1+4
G = < a,b,c,d,e | a10=b4=c2=1, d2=e2=b2, bab-1=cac=dad-1=a-1, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=a5b2d >
Subgroups: 1046 in 292 conjugacy classes, 105 normal (91 characteristic)
C1, C2, C2, C4, C22, 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×D4, C2×Q8, C4○D4, Dic5, Dic5, C20, D10, D10, C2×C10, C2×C10, C2×C10, C2×C4⋊C4, C4×D4, C4⋊D4, C22⋊Q8, C22.D4, C22.D4, C4⋊Q8, C2×C4○D4, Dic10, C4×D5, D20, C2×Dic5, C2×Dic5, C5⋊D4, C5⋊D4, C2×C20, C2×C20, C5×D4, C22×D5, C22×C10, D4⋊6D4, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C2×D20, C4○D20, D4⋊2D5, C22×Dic5, C2×C5⋊D4, C22×C20, D4×C10, Dic5.14D4, Dic5⋊4D4, D10.12D4, D10⋊D4, C22.D20, C20⋊Q8, D5×C4⋊C4, D10.13D4, D10⋊Q8, C2×C10.D4, Dic5⋊D4, C5×C22.D4, C2×C4○D20, C2×D4⋊2D5, C10.822- 1+4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, C24, D10, C22×D4, C2×C4○D4, 2- 1+4, C22×D5, D4⋊6D4, D4×D5, C23×D5, C2×D4×D5, D5×C4○D4, D4.10D10, C10.822- 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 58 23 65)(2 57 24 64)(3 56 25 63)(4 55 26 62)(5 54 27 61)(6 53 28 70)(7 52 29 69)(8 51 30 68)(9 60 21 67)(10 59 22 66)(11 113 158 130)(12 112 159 129)(13 111 160 128)(14 120 151 127)(15 119 152 126)(16 118 153 125)(17 117 154 124)(18 116 155 123)(19 115 156 122)(20 114 157 121)(31 87 48 80)(32 86 49 79)(33 85 50 78)(34 84 41 77)(35 83 42 76)(36 82 43 75)(37 81 44 74)(38 90 45 73)(39 89 46 72)(40 88 47 71)(91 135 108 142)(92 134 109 141)(93 133 110 150)(94 132 101 149)(95 131 102 148)(96 140 103 147)(97 139 104 146)(98 138 105 145)(99 137 106 144)(100 136 107 143)
(1 138)(2 137)(3 136)(4 135)(5 134)(6 133)(7 132)(8 131)(9 140)(10 139)(11 33)(12 32)(13 31)(14 40)(15 39)(16 38)(17 37)(18 36)(19 35)(20 34)(21 147)(22 146)(23 145)(24 144)(25 143)(26 142)(27 141)(28 150)(29 149)(30 148)(41 157)(42 156)(43 155)(44 154)(45 153)(46 152)(47 151)(48 160)(49 159)(50 158)(51 95)(52 94)(53 93)(54 92)(55 91)(56 100)(57 99)(58 98)(59 97)(60 96)(61 109)(62 108)(63 107)(64 106)(65 105)(66 104)(67 103)(68 102)(69 101)(70 110)(71 120)(72 119)(73 118)(74 117)(75 116)(76 115)(77 114)(78 113)(79 112)(80 111)(81 124)(82 123)(83 122)(84 121)(85 130)(86 129)(87 128)(88 127)(89 126)(90 125)
(1 16 23 153)(2 15 24 152)(3 14 25 151)(4 13 26 160)(5 12 27 159)(6 11 28 158)(7 20 29 157)(8 19 30 156)(9 18 21 155)(10 17 22 154)(31 142 48 135)(32 141 49 134)(33 150 50 133)(34 149 41 132)(35 148 42 131)(36 147 43 140)(37 146 44 139)(38 145 45 138)(39 144 46 137)(40 143 47 136)(51 115 68 122)(52 114 69 121)(53 113 70 130)(54 112 61 129)(55 111 62 128)(56 120 63 127)(57 119 64 126)(58 118 65 125)(59 117 66 124)(60 116 67 123)(71 107 88 100)(72 106 89 99)(73 105 90 98)(74 104 81 97)(75 103 82 96)(76 102 83 95)(77 101 84 94)(78 110 85 93)(79 109 86 92)(80 108 87 91)
(1 58 23 65)(2 59 24 66)(3 60 25 67)(4 51 26 68)(5 52 27 69)(6 53 28 70)(7 54 29 61)(8 55 30 62)(9 56 21 63)(10 57 22 64)(11 125 158 118)(12 126 159 119)(13 127 160 120)(14 128 151 111)(15 129 152 112)(16 130 153 113)(17 121 154 114)(18 122 155 115)(19 123 156 116)(20 124 157 117)(31 88 48 71)(32 89 49 72)(33 90 50 73)(34 81 41 74)(35 82 42 75)(36 83 43 76)(37 84 44 77)(38 85 45 78)(39 86 46 79)(40 87 47 80)(91 148 108 131)(92 149 109 132)(93 150 110 133)(94 141 101 134)(95 142 102 135)(96 143 103 136)(97 144 104 137)(98 145 105 138)(99 146 106 139)(100 147 107 140)
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), (1,58,23,65)(2,57,24,64)(3,56,25,63)(4,55,26,62)(5,54,27,61)(6,53,28,70)(7,52,29,69)(8,51,30,68)(9,60,21,67)(10,59,22,66)(11,113,158,130)(12,112,159,129)(13,111,160,128)(14,120,151,127)(15,119,152,126)(16,118,153,125)(17,117,154,124)(18,116,155,123)(19,115,156,122)(20,114,157,121)(31,87,48,80)(32,86,49,79)(33,85,50,78)(34,84,41,77)(35,83,42,76)(36,82,43,75)(37,81,44,74)(38,90,45,73)(39,89,46,72)(40,88,47,71)(91,135,108,142)(92,134,109,141)(93,133,110,150)(94,132,101,149)(95,131,102,148)(96,140,103,147)(97,139,104,146)(98,138,105,145)(99,137,106,144)(100,136,107,143), (1,138)(2,137)(3,136)(4,135)(5,134)(6,133)(7,132)(8,131)(9,140)(10,139)(11,33)(12,32)(13,31)(14,40)(15,39)(16,38)(17,37)(18,36)(19,35)(20,34)(21,147)(22,146)(23,145)(24,144)(25,143)(26,142)(27,141)(28,150)(29,149)(30,148)(41,157)(42,156)(43,155)(44,154)(45,153)(46,152)(47,151)(48,160)(49,159)(50,158)(51,95)(52,94)(53,93)(54,92)(55,91)(56,100)(57,99)(58,98)(59,97)(60,96)(61,109)(62,108)(63,107)(64,106)(65,105)(66,104)(67,103)(68,102)(69,101)(70,110)(71,120)(72,119)(73,118)(74,117)(75,116)(76,115)(77,114)(78,113)(79,112)(80,111)(81,124)(82,123)(83,122)(84,121)(85,130)(86,129)(87,128)(88,127)(89,126)(90,125), (1,16,23,153)(2,15,24,152)(3,14,25,151)(4,13,26,160)(5,12,27,159)(6,11,28,158)(7,20,29,157)(8,19,30,156)(9,18,21,155)(10,17,22,154)(31,142,48,135)(32,141,49,134)(33,150,50,133)(34,149,41,132)(35,148,42,131)(36,147,43,140)(37,146,44,139)(38,145,45,138)(39,144,46,137)(40,143,47,136)(51,115,68,122)(52,114,69,121)(53,113,70,130)(54,112,61,129)(55,111,62,128)(56,120,63,127)(57,119,64,126)(58,118,65,125)(59,117,66,124)(60,116,67,123)(71,107,88,100)(72,106,89,99)(73,105,90,98)(74,104,81,97)(75,103,82,96)(76,102,83,95)(77,101,84,94)(78,110,85,93)(79,109,86,92)(80,108,87,91), (1,58,23,65)(2,59,24,66)(3,60,25,67)(4,51,26,68)(5,52,27,69)(6,53,28,70)(7,54,29,61)(8,55,30,62)(9,56,21,63)(10,57,22,64)(11,125,158,118)(12,126,159,119)(13,127,160,120)(14,128,151,111)(15,129,152,112)(16,130,153,113)(17,121,154,114)(18,122,155,115)(19,123,156,116)(20,124,157,117)(31,88,48,71)(32,89,49,72)(33,90,50,73)(34,81,41,74)(35,82,42,75)(36,83,43,76)(37,84,44,77)(38,85,45,78)(39,86,46,79)(40,87,47,80)(91,148,108,131)(92,149,109,132)(93,150,110,133)(94,141,101,134)(95,142,102,135)(96,143,103,136)(97,144,104,137)(98,145,105,138)(99,146,106,139)(100,147,107,140)>;
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), (1,58,23,65)(2,57,24,64)(3,56,25,63)(4,55,26,62)(5,54,27,61)(6,53,28,70)(7,52,29,69)(8,51,30,68)(9,60,21,67)(10,59,22,66)(11,113,158,130)(12,112,159,129)(13,111,160,128)(14,120,151,127)(15,119,152,126)(16,118,153,125)(17,117,154,124)(18,116,155,123)(19,115,156,122)(20,114,157,121)(31,87,48,80)(32,86,49,79)(33,85,50,78)(34,84,41,77)(35,83,42,76)(36,82,43,75)(37,81,44,74)(38,90,45,73)(39,89,46,72)(40,88,47,71)(91,135,108,142)(92,134,109,141)(93,133,110,150)(94,132,101,149)(95,131,102,148)(96,140,103,147)(97,139,104,146)(98,138,105,145)(99,137,106,144)(100,136,107,143), (1,138)(2,137)(3,136)(4,135)(5,134)(6,133)(7,132)(8,131)(9,140)(10,139)(11,33)(12,32)(13,31)(14,40)(15,39)(16,38)(17,37)(18,36)(19,35)(20,34)(21,147)(22,146)(23,145)(24,144)(25,143)(26,142)(27,141)(28,150)(29,149)(30,148)(41,157)(42,156)(43,155)(44,154)(45,153)(46,152)(47,151)(48,160)(49,159)(50,158)(51,95)(52,94)(53,93)(54,92)(55,91)(56,100)(57,99)(58,98)(59,97)(60,96)(61,109)(62,108)(63,107)(64,106)(65,105)(66,104)(67,103)(68,102)(69,101)(70,110)(71,120)(72,119)(73,118)(74,117)(75,116)(76,115)(77,114)(78,113)(79,112)(80,111)(81,124)(82,123)(83,122)(84,121)(85,130)(86,129)(87,128)(88,127)(89,126)(90,125), (1,16,23,153)(2,15,24,152)(3,14,25,151)(4,13,26,160)(5,12,27,159)(6,11,28,158)(7,20,29,157)(8,19,30,156)(9,18,21,155)(10,17,22,154)(31,142,48,135)(32,141,49,134)(33,150,50,133)(34,149,41,132)(35,148,42,131)(36,147,43,140)(37,146,44,139)(38,145,45,138)(39,144,46,137)(40,143,47,136)(51,115,68,122)(52,114,69,121)(53,113,70,130)(54,112,61,129)(55,111,62,128)(56,120,63,127)(57,119,64,126)(58,118,65,125)(59,117,66,124)(60,116,67,123)(71,107,88,100)(72,106,89,99)(73,105,90,98)(74,104,81,97)(75,103,82,96)(76,102,83,95)(77,101,84,94)(78,110,85,93)(79,109,86,92)(80,108,87,91), (1,58,23,65)(2,59,24,66)(3,60,25,67)(4,51,26,68)(5,52,27,69)(6,53,28,70)(7,54,29,61)(8,55,30,62)(9,56,21,63)(10,57,22,64)(11,125,158,118)(12,126,159,119)(13,127,160,120)(14,128,151,111)(15,129,152,112)(16,130,153,113)(17,121,154,114)(18,122,155,115)(19,123,156,116)(20,124,157,117)(31,88,48,71)(32,89,49,72)(33,90,50,73)(34,81,41,74)(35,82,42,75)(36,83,43,76)(37,84,44,77)(38,85,45,78)(39,86,46,79)(40,87,47,80)(91,148,108,131)(92,149,109,132)(93,150,110,133)(94,141,101,134)(95,142,102,135)(96,143,103,136)(97,144,104,137)(98,145,105,138)(99,146,106,139)(100,147,107,140) );
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)], [(1,58,23,65),(2,57,24,64),(3,56,25,63),(4,55,26,62),(5,54,27,61),(6,53,28,70),(7,52,29,69),(8,51,30,68),(9,60,21,67),(10,59,22,66),(11,113,158,130),(12,112,159,129),(13,111,160,128),(14,120,151,127),(15,119,152,126),(16,118,153,125),(17,117,154,124),(18,116,155,123),(19,115,156,122),(20,114,157,121),(31,87,48,80),(32,86,49,79),(33,85,50,78),(34,84,41,77),(35,83,42,76),(36,82,43,75),(37,81,44,74),(38,90,45,73),(39,89,46,72),(40,88,47,71),(91,135,108,142),(92,134,109,141),(93,133,110,150),(94,132,101,149),(95,131,102,148),(96,140,103,147),(97,139,104,146),(98,138,105,145),(99,137,106,144),(100,136,107,143)], [(1,138),(2,137),(3,136),(4,135),(5,134),(6,133),(7,132),(8,131),(9,140),(10,139),(11,33),(12,32),(13,31),(14,40),(15,39),(16,38),(17,37),(18,36),(19,35),(20,34),(21,147),(22,146),(23,145),(24,144),(25,143),(26,142),(27,141),(28,150),(29,149),(30,148),(41,157),(42,156),(43,155),(44,154),(45,153),(46,152),(47,151),(48,160),(49,159),(50,158),(51,95),(52,94),(53,93),(54,92),(55,91),(56,100),(57,99),(58,98),(59,97),(60,96),(61,109),(62,108),(63,107),(64,106),(65,105),(66,104),(67,103),(68,102),(69,101),(70,110),(71,120),(72,119),(73,118),(74,117),(75,116),(76,115),(77,114),(78,113),(79,112),(80,111),(81,124),(82,123),(83,122),(84,121),(85,130),(86,129),(87,128),(88,127),(89,126),(90,125)], [(1,16,23,153),(2,15,24,152),(3,14,25,151),(4,13,26,160),(5,12,27,159),(6,11,28,158),(7,20,29,157),(8,19,30,156),(9,18,21,155),(10,17,22,154),(31,142,48,135),(32,141,49,134),(33,150,50,133),(34,149,41,132),(35,148,42,131),(36,147,43,140),(37,146,44,139),(38,145,45,138),(39,144,46,137),(40,143,47,136),(51,115,68,122),(52,114,69,121),(53,113,70,130),(54,112,61,129),(55,111,62,128),(56,120,63,127),(57,119,64,126),(58,118,65,125),(59,117,66,124),(60,116,67,123),(71,107,88,100),(72,106,89,99),(73,105,90,98),(74,104,81,97),(75,103,82,96),(76,102,83,95),(77,101,84,94),(78,110,85,93),(79,109,86,92),(80,108,87,91)], [(1,58,23,65),(2,59,24,66),(3,60,25,67),(4,51,26,68),(5,52,27,69),(6,53,28,70),(7,54,29,61),(8,55,30,62),(9,56,21,63),(10,57,22,64),(11,125,158,118),(12,126,159,119),(13,127,160,120),(14,128,151,111),(15,129,152,112),(16,130,153,113),(17,121,154,114),(18,122,155,115),(19,123,156,116),(20,124,157,117),(31,88,48,71),(32,89,49,72),(33,90,50,73),(34,81,41,74),(35,82,42,75),(36,83,43,76),(37,84,44,77),(38,85,45,78),(39,86,46,79),(40,87,47,80),(91,148,108,131),(92,149,109,132),(93,150,110,133),(94,141,101,134),(95,142,102,135),(96,143,103,136),(97,144,104,137),(98,145,105,138),(99,146,106,139),(100,147,107,140)]])
53 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4L | 4M | 4N | 4O | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 20A | ··· | 20H | 20I | ··· | 20N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 2 | 2 | 4 | 10 | 10 | 20 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | ··· | 10 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
53 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | C4○D4 | D10 | D10 | D10 | D10 | 2- 1+4 | D4×D5 | D5×C4○D4 | D4.10D10 |
| kernel | C10.822- 1+4 | Dic5.14D4 | Dic5⋊4D4 | D10.12D4 | D10⋊D4 | C22.D20 | C20⋊Q8 | D5×C4⋊C4 | D10.13D4 | D10⋊Q8 | C2×C10.D4 | Dic5⋊D4 | C5×C22.D4 | C2×C4○D20 | C2×D4⋊2D5 | C5⋊D4 | C22.D4 | Dic5 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C10 | C22 | C2 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 2 | 4 | 6 | 4 | 2 | 2 | 1 | 4 | 4 | 4 |
Matrix representation of C10.822- 1+4 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 35 | 0 | 0 |
| 0 | 0 | 6 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 6 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 6 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 35 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,35,6,0,0,0,0,35,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[32,0,0,0,0,0,0,9,0,0,0,0,0,0,1,6,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,32,0,0,0,0,9,0,0,0,0,0,0,0,1,6,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,40,35,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40] >;
C10.822- 1+4 in GAP, Magma, Sage, TeX
C_{10}._{82}2_-^{1+4} % in TeX
G:=Group("C10.82ES-(2,2)"); // GroupNames label
G:=SmallGroup(320,1327);
// by ID
G=gap.SmallGroup(320,1327);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,184,570,185,136,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^10=b^4=c^2=1,d^2=e^2=b^2,b*a*b^-1=c*a*c=d*a*d^-1=a^-1,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=a^5*b^2*d>;
// generators/relations