G = C10.852- 1+4 order 320 = 26·5
40th non-split extension by C10 of 2- 1+4 acting via 2- 1+4/C4○D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C10.852- 1+4, C10.1232+ 1+4, C4⋊C4.108D10, (D4×Dic5)⋊29C2, D10⋊2Q8⋊33C2, (C2×D4).167D10, C22⋊C4.69D10, C4.Dic10⋊30C2, Dic5⋊4D4⋊26C2, (C2×C20).183C23, (C2×C10).209C24, Dic5⋊D4.3C2, (C22×C4).262D10, C22.D4⋊14D5, D10.12D4⋊37C2, C2.45(D4⋊8D10), C23.33(C22×D5), Dic5.Q8⋊30C2, (D4×C10).147C22, D10⋊C4.9C22, C23.23D10⋊8C2, C22.D20⋊23C2, C23.D10⋊35C2, C4⋊Dic5.313C22, (C22×C10).41C23, (C22×C20).87C22, (C22×D5).90C23, C22.230(C23×D5), Dic5.14D4⋊35C2, C23.D5.47C22, C22.12(D4⋊2D5), C5⋊7(C22.33C24), (C4×Dic5).136C22, (C2×Dic5).108C23, C10.D4.10C22, C2.46(D4.10D10), (C2×Dic10).177C22, (C22×Dic5).135C22, C4⋊C4⋊D5⋊31C2, (C2×C4⋊Dic5)⋊27C2, C10.92(C2×C4○D4), C2.54(C2×D4⋊2D5), (C2×C4×D5).125C22, (C2×C10).48(C4○D4), (C5×C4⋊C4).182C22, (C2×C4).190(C22×D5), (C2×C5⋊D4).53C22, (C5×C22.D4)⋊17C2, (C5×C22⋊C4).57C22, SmallGroup(320,1337)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10.852- 1+4
G = < a,b,c,d,e | a10=b4=c2=1, d2=a5b2, e2=b2, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=b-1, dbd-1=ebe-1=a5b, cd=dc, ce=ec, ede-1=b2d >
Subgroups: 734 in 218 conjugacy classes, 95 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, Dic5, C20, D10, C2×C10, C2×C10, C2×C10, C2×C4⋊C4, C4×D4, C4⋊D4, C22⋊Q8, C22.D4, C22.D4, C42.C2, C42⋊2C2, Dic10, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C22×D5, C22×C10, C22.33C24, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C22×Dic5, C2×C5⋊D4, C22×C20, D4×C10, Dic5.14D4, C23.D10, Dic5⋊4D4, D10.12D4, C22.D20, Dic5.Q8, C4.Dic10, D10⋊2Q8, C4⋊C4⋊D5, C2×C4⋊Dic5, C23.23D10, D4×Dic5, Dic5⋊D4, C5×C22.D4, C10.852- 1+4
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, 2+ 1+4, 2- 1+4, C22×D5, C22.33C24, D4⋊2D5, C23×D5, C2×D4⋊2D5, D4⋊8D10, D4.10D10, C10.852- 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)
(1 158 23 11)(2 159 24 12)(3 160 25 13)(4 151 26 14)(5 152 27 15)(6 153 28 16)(7 154 29 17)(8 155 30 18)(9 156 21 19)(10 157 22 20)(31 131 48 148)(32 132 49 149)(33 133 50 150)(34 134 41 141)(35 135 42 142)(36 136 43 143)(37 137 44 144)(38 138 45 145)(39 139 46 146)(40 140 47 147)(51 111 68 128)(52 112 69 129)(53 113 70 130)(54 114 61 121)(55 115 62 122)(56 116 63 123)(57 117 64 124)(58 118 65 125)(59 119 66 126)(60 120 67 127)(71 108 88 91)(72 109 89 92)(73 110 90 93)(74 101 81 94)(75 102 82 95)(76 103 83 96)(77 104 84 97)(78 105 85 98)(79 106 86 99)(80 107 87 100)
(1 105)(2 106)(3 107)(4 108)(5 109)(6 110)(7 101)(8 102)(9 103)(10 104)(11 85)(12 86)(13 87)(14 88)(15 89)(16 90)(17 81)(18 82)(19 83)(20 84)(21 96)(22 97)(23 98)(24 99)(25 100)(26 91)(27 92)(28 93)(29 94)(30 95)(31 111)(32 112)(33 113)(34 114)(35 115)(36 116)(37 117)(38 118)(39 119)(40 120)(41 121)(42 122)(43 123)(44 124)(45 125)(46 126)(47 127)(48 128)(49 129)(50 130)(51 131)(52 132)(53 133)(54 134)(55 135)(56 136)(57 137)(58 138)(59 139)(60 140)(61 141)(62 142)(63 143)(64 144)(65 145)(66 146)(67 147)(68 148)(69 149)(70 150)(71 151)(72 152)(73 153)(74 154)(75 155)(76 156)(77 157)(78 158)(79 159)(80 160)
(1 16 28 158)(2 15 29 157)(3 14 30 156)(4 13 21 155)(5 12 22 154)(6 11 23 153)(7 20 24 152)(8 19 25 151)(9 18 26 160)(10 17 27 159)(31 135 43 147)(32 134 44 146)(33 133 45 145)(34 132 46 144)(35 131 47 143)(36 140 48 142)(37 139 49 141)(38 138 50 150)(39 137 41 149)(40 136 42 148)(51 127 63 115)(52 126 64 114)(53 125 65 113)(54 124 66 112)(55 123 67 111)(56 122 68 120)(57 121 69 119)(58 130 70 118)(59 129 61 117)(60 128 62 116)(71 102 83 100)(72 101 84 99)(73 110 85 98)(74 109 86 97)(75 108 87 96)(76 107 88 95)(77 106 89 94)(78 105 90 93)(79 104 81 92)(80 103 82 91)
(1 33 23 50)(2 34 24 41)(3 35 25 42)(4 36 26 43)(5 37 27 44)(6 38 28 45)(7 39 29 46)(8 40 30 47)(9 31 21 48)(10 32 22 49)(11 145 158 138)(12 146 159 139)(13 147 160 140)(14 148 151 131)(15 149 152 132)(16 150 153 133)(17 141 154 134)(18 142 155 135)(19 143 156 136)(20 144 157 137)(51 88 68 71)(52 89 69 72)(53 90 70 73)(54 81 61 74)(55 82 62 75)(56 83 63 76)(57 84 64 77)(58 85 65 78)(59 86 66 79)(60 87 67 80)(91 123 108 116)(92 124 109 117)(93 125 110 118)(94 126 101 119)(95 127 102 120)(96 128 103 111)(97 129 104 112)(98 130 105 113)(99 121 106 114)(100 122 107 115)
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), (1,158,23,11)(2,159,24,12)(3,160,25,13)(4,151,26,14)(5,152,27,15)(6,153,28,16)(7,154,29,17)(8,155,30,18)(9,156,21,19)(10,157,22,20)(31,131,48,148)(32,132,49,149)(33,133,50,150)(34,134,41,141)(35,135,42,142)(36,136,43,143)(37,137,44,144)(38,138,45,145)(39,139,46,146)(40,140,47,147)(51,111,68,128)(52,112,69,129)(53,113,70,130)(54,114,61,121)(55,115,62,122)(56,116,63,123)(57,117,64,124)(58,118,65,125)(59,119,66,126)(60,120,67,127)(71,108,88,91)(72,109,89,92)(73,110,90,93)(74,101,81,94)(75,102,82,95)(76,103,83,96)(77,104,84,97)(78,105,85,98)(79,106,86,99)(80,107,87,100), (1,105)(2,106)(3,107)(4,108)(5,109)(6,110)(7,101)(8,102)(9,103)(10,104)(11,85)(12,86)(13,87)(14,88)(15,89)(16,90)(17,81)(18,82)(19,83)(20,84)(21,96)(22,97)(23,98)(24,99)(25,100)(26,91)(27,92)(28,93)(29,94)(30,95)(31,111)(32,112)(33,113)(34,114)(35,115)(36,116)(37,117)(38,118)(39,119)(40,120)(41,121)(42,122)(43,123)(44,124)(45,125)(46,126)(47,127)(48,128)(49,129)(50,130)(51,131)(52,132)(53,133)(54,134)(55,135)(56,136)(57,137)(58,138)(59,139)(60,140)(61,141)(62,142)(63,143)(64,144)(65,145)(66,146)(67,147)(68,148)(69,149)(70,150)(71,151)(72,152)(73,153)(74,154)(75,155)(76,156)(77,157)(78,158)(79,159)(80,160), (1,16,28,158)(2,15,29,157)(3,14,30,156)(4,13,21,155)(5,12,22,154)(6,11,23,153)(7,20,24,152)(8,19,25,151)(9,18,26,160)(10,17,27,159)(31,135,43,147)(32,134,44,146)(33,133,45,145)(34,132,46,144)(35,131,47,143)(36,140,48,142)(37,139,49,141)(38,138,50,150)(39,137,41,149)(40,136,42,148)(51,127,63,115)(52,126,64,114)(53,125,65,113)(54,124,66,112)(55,123,67,111)(56,122,68,120)(57,121,69,119)(58,130,70,118)(59,129,61,117)(60,128,62,116)(71,102,83,100)(72,101,84,99)(73,110,85,98)(74,109,86,97)(75,108,87,96)(76,107,88,95)(77,106,89,94)(78,105,90,93)(79,104,81,92)(80,103,82,91), (1,33,23,50)(2,34,24,41)(3,35,25,42)(4,36,26,43)(5,37,27,44)(6,38,28,45)(7,39,29,46)(8,40,30,47)(9,31,21,48)(10,32,22,49)(11,145,158,138)(12,146,159,139)(13,147,160,140)(14,148,151,131)(15,149,152,132)(16,150,153,133)(17,141,154,134)(18,142,155,135)(19,143,156,136)(20,144,157,137)(51,88,68,71)(52,89,69,72)(53,90,70,73)(54,81,61,74)(55,82,62,75)(56,83,63,76)(57,84,64,77)(58,85,65,78)(59,86,66,79)(60,87,67,80)(91,123,108,116)(92,124,109,117)(93,125,110,118)(94,126,101,119)(95,127,102,120)(96,128,103,111)(97,129,104,112)(98,130,105,113)(99,121,106,114)(100,122,107,115)>;
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), (1,158,23,11)(2,159,24,12)(3,160,25,13)(4,151,26,14)(5,152,27,15)(6,153,28,16)(7,154,29,17)(8,155,30,18)(9,156,21,19)(10,157,22,20)(31,131,48,148)(32,132,49,149)(33,133,50,150)(34,134,41,141)(35,135,42,142)(36,136,43,143)(37,137,44,144)(38,138,45,145)(39,139,46,146)(40,140,47,147)(51,111,68,128)(52,112,69,129)(53,113,70,130)(54,114,61,121)(55,115,62,122)(56,116,63,123)(57,117,64,124)(58,118,65,125)(59,119,66,126)(60,120,67,127)(71,108,88,91)(72,109,89,92)(73,110,90,93)(74,101,81,94)(75,102,82,95)(76,103,83,96)(77,104,84,97)(78,105,85,98)(79,106,86,99)(80,107,87,100), (1,105)(2,106)(3,107)(4,108)(5,109)(6,110)(7,101)(8,102)(9,103)(10,104)(11,85)(12,86)(13,87)(14,88)(15,89)(16,90)(17,81)(18,82)(19,83)(20,84)(21,96)(22,97)(23,98)(24,99)(25,100)(26,91)(27,92)(28,93)(29,94)(30,95)(31,111)(32,112)(33,113)(34,114)(35,115)(36,116)(37,117)(38,118)(39,119)(40,120)(41,121)(42,122)(43,123)(44,124)(45,125)(46,126)(47,127)(48,128)(49,129)(50,130)(51,131)(52,132)(53,133)(54,134)(55,135)(56,136)(57,137)(58,138)(59,139)(60,140)(61,141)(62,142)(63,143)(64,144)(65,145)(66,146)(67,147)(68,148)(69,149)(70,150)(71,151)(72,152)(73,153)(74,154)(75,155)(76,156)(77,157)(78,158)(79,159)(80,160), (1,16,28,158)(2,15,29,157)(3,14,30,156)(4,13,21,155)(5,12,22,154)(6,11,23,153)(7,20,24,152)(8,19,25,151)(9,18,26,160)(10,17,27,159)(31,135,43,147)(32,134,44,146)(33,133,45,145)(34,132,46,144)(35,131,47,143)(36,140,48,142)(37,139,49,141)(38,138,50,150)(39,137,41,149)(40,136,42,148)(51,127,63,115)(52,126,64,114)(53,125,65,113)(54,124,66,112)(55,123,67,111)(56,122,68,120)(57,121,69,119)(58,130,70,118)(59,129,61,117)(60,128,62,116)(71,102,83,100)(72,101,84,99)(73,110,85,98)(74,109,86,97)(75,108,87,96)(76,107,88,95)(77,106,89,94)(78,105,90,93)(79,104,81,92)(80,103,82,91), (1,33,23,50)(2,34,24,41)(3,35,25,42)(4,36,26,43)(5,37,27,44)(6,38,28,45)(7,39,29,46)(8,40,30,47)(9,31,21,48)(10,32,22,49)(11,145,158,138)(12,146,159,139)(13,147,160,140)(14,148,151,131)(15,149,152,132)(16,150,153,133)(17,141,154,134)(18,142,155,135)(19,143,156,136)(20,144,157,137)(51,88,68,71)(52,89,69,72)(53,90,70,73)(54,81,61,74)(55,82,62,75)(56,83,63,76)(57,84,64,77)(58,85,65,78)(59,86,66,79)(60,87,67,80)(91,123,108,116)(92,124,109,117)(93,125,110,118)(94,126,101,119)(95,127,102,120)(96,128,103,111)(97,129,104,112)(98,130,105,113)(99,121,106,114)(100,122,107,115) );
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)], [(1,158,23,11),(2,159,24,12),(3,160,25,13),(4,151,26,14),(5,152,27,15),(6,153,28,16),(7,154,29,17),(8,155,30,18),(9,156,21,19),(10,157,22,20),(31,131,48,148),(32,132,49,149),(33,133,50,150),(34,134,41,141),(35,135,42,142),(36,136,43,143),(37,137,44,144),(38,138,45,145),(39,139,46,146),(40,140,47,147),(51,111,68,128),(52,112,69,129),(53,113,70,130),(54,114,61,121),(55,115,62,122),(56,116,63,123),(57,117,64,124),(58,118,65,125),(59,119,66,126),(60,120,67,127),(71,108,88,91),(72,109,89,92),(73,110,90,93),(74,101,81,94),(75,102,82,95),(76,103,83,96),(77,104,84,97),(78,105,85,98),(79,106,86,99),(80,107,87,100)], [(1,105),(2,106),(3,107),(4,108),(5,109),(6,110),(7,101),(8,102),(9,103),(10,104),(11,85),(12,86),(13,87),(14,88),(15,89),(16,90),(17,81),(18,82),(19,83),(20,84),(21,96),(22,97),(23,98),(24,99),(25,100),(26,91),(27,92),(28,93),(29,94),(30,95),(31,111),(32,112),(33,113),(34,114),(35,115),(36,116),(37,117),(38,118),(39,119),(40,120),(41,121),(42,122),(43,123),(44,124),(45,125),(46,126),(47,127),(48,128),(49,129),(50,130),(51,131),(52,132),(53,133),(54,134),(55,135),(56,136),(57,137),(58,138),(59,139),(60,140),(61,141),(62,142),(63,143),(64,144),(65,145),(66,146),(67,147),(68,148),(69,149),(70,150),(71,151),(72,152),(73,153),(74,154),(75,155),(76,156),(77,157),(78,158),(79,159),(80,160)], [(1,16,28,158),(2,15,29,157),(3,14,30,156),(4,13,21,155),(5,12,22,154),(6,11,23,153),(7,20,24,152),(8,19,25,151),(9,18,26,160),(10,17,27,159),(31,135,43,147),(32,134,44,146),(33,133,45,145),(34,132,46,144),(35,131,47,143),(36,140,48,142),(37,139,49,141),(38,138,50,150),(39,137,41,149),(40,136,42,148),(51,127,63,115),(52,126,64,114),(53,125,65,113),(54,124,66,112),(55,123,67,111),(56,122,68,120),(57,121,69,119),(58,130,70,118),(59,129,61,117),(60,128,62,116),(71,102,83,100),(72,101,84,99),(73,110,85,98),(74,109,86,97),(75,108,87,96),(76,107,88,95),(77,106,89,94),(78,105,90,93),(79,104,81,92),(80,103,82,91)], [(1,33,23,50),(2,34,24,41),(3,35,25,42),(4,36,26,43),(5,37,27,44),(6,38,28,45),(7,39,29,46),(8,40,30,47),(9,31,21,48),(10,32,22,49),(11,145,158,138),(12,146,159,139),(13,147,160,140),(14,148,151,131),(15,149,152,132),(16,150,153,133),(17,141,154,134),(18,142,155,135),(19,143,156,136),(20,144,157,137),(51,88,68,71),(52,89,69,72),(53,90,70,73),(54,81,61,74),(55,82,62,75),(56,83,63,76),(57,84,64,77),(58,85,65,78),(59,86,66,79),(60,87,67,80),(91,123,108,116),(92,124,109,117),(93,125,110,118),(94,126,101,119),(95,127,102,120),(96,128,103,111),(97,129,104,112),(98,130,105,113),(99,121,106,114),(100,122,107,115)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4E | 4F | 4G | 4H | 4I | 4J | ··· | 4N | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 20A | ··· | 20H | 20I | ··· | 20N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 20 | 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⋊2D5 | D4⋊8D10 | D4.10D10 |
| kernel | C10.852- 1+4 | Dic5.14D4 | C23.D10 | Dic5⋊4D4 | D10.12D4 | C22.D20 | Dic5.Q8 | C4.Dic10 | D10⋊2Q8 | C4⋊C4⋊D5 | C2×C4⋊Dic5 | C23.23D10 | D4×Dic5 | Dic5⋊D4 | C5×C22.D4 | C22.D4 | C2×C10 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C10 | C10 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 6 | 4 | 2 | 2 | 1 | 1 | 4 | 4 | 4 |
Matrix representation of C10.852- 1+4 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 7 | 0 | 0 |
| 0 | 0 | 34 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 7 |
| 0 | 0 | 0 | 0 | 34 | 40 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 21 | 0 |
| 0 | 0 | 0 | 3 | 0 | 21 |
| 0 | 0 | 21 | 0 | 38 | 0 |
| 0 | 0 | 0 | 21 | 0 | 38 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 26 | 25 | 8 | 14 |
| 0 | 0 | 16 | 15 | 27 | 33 |
| 0 | 0 | 8 | 14 | 15 | 16 |
| 0 | 0 | 27 | 33 | 25 | 26 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 38 | 0 | 20 | 0 |
| 0 | 0 | 21 | 3 | 24 | 21 |
| 0 | 0 | 20 | 0 | 3 | 0 |
| 0 | 0 | 24 | 21 | 20 | 38 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 9 | 0 | 0 |
| 0 | 0 | 32 | 30 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 9 |
| 0 | 0 | 0 | 0 | 32 | 30 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,7,34,0,0,0,0,7,40,0,0,0,0,0,0,7,34,0,0,0,0,7,40],[0,9,0,0,0,0,32,0,0,0,0,0,0,0,3,0,21,0,0,0,0,3,0,21,0,0,21,0,38,0,0,0,0,21,0,38],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,26,16,8,27,0,0,25,15,14,33,0,0,8,27,15,25,0,0,14,33,16,26],[0,32,0,0,0,0,32,0,0,0,0,0,0,0,38,21,20,24,0,0,0,3,0,21,0,0,20,24,3,20,0,0,0,21,0,38],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,11,32,0,0,0,0,9,30,0,0,0,0,0,0,11,32,0,0,0,0,9,30] >;
C10.852- 1+4 in GAP, Magma, Sage, TeX
C_{10}._{85}2_-^{1+4} % in TeX
G:=Group("C10.85ES-(2,2)"); // GroupNames label
G:=SmallGroup(320,1337);
// by ID
G=gap.SmallGroup(320,1337);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,387,100,675,409,136,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^10=b^4=c^2=1,d^2=a^5*b^2,e^2=b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,c*b*c=b^-1,d*b*d^-1=e*b*e^-1=a^5*b,c*d=d*c,c*e=e*c,e*d*e^-1=b^2*d>;
// generators/relations