G = C10.572+ 1+4 order 320 = 26·5
57th non-split extension by C10 of 2+ 1+4 acting via 2+ 1+4/C2×D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C10.572+ 1+4, C10.782- 1+4, C22⋊Q8⋊22D5, C4⋊C4.100D10, (C2×Q8).78D10, D10⋊3Q8⋊24C2, C20⋊7D4.19C2, (C2×C20).65C23, C22⋊C4.65D10, C4.Dic10⋊26C2, Dic5⋊4D4⋊16C2, (C2×C10).189C24, (C2×D20).33C22, (C22×C4).251D10, D10.13D4⋊23C2, C2.59(D4⋊6D10), C22.D20⋊17C2, C4⋊Dic5.221C22, (Q8×C10).118C22, C22.5(Q8⋊2D5), (C2×Dic5).95C23, (C22×D5).80C23, C22.210(C23×D5), C23.197(C22×D5), D10⋊C4.72C22, (C22×C10).217C23, (C22×C20).317C22, C5⋊5(C22.33C24), (C4×Dic5).124C22, C2.38(D4.10D10), C10.D4.119C22, (C22×Dic5).125C22, C4⋊C4⋊D5⋊24C2, (C5×C22⋊Q8)⋊25C2, C10.117(C2×C4○D4), C2.21(C2×Q8⋊2D5), (C2×C4×D5).114C22, (C2×C10).29(C4○D4), (C2×C10.D4)⋊30C2, (C5×C4⋊C4).169C22, (C2×C4).186(C22×D5), (C2×C5⋊D4).41C22, (C5×C22⋊C4).44C22, SmallGroup(320,1317)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10.572+ 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=a5b-1, dbd-1=ebe-1=a5b, cd=dc, ce=ec, ede-1=a5b2d >
Subgroups: 766 in 218 conjugacy classes, 95 normal (31 characteristic)
C1, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, Dic5, C20, D10, C2×C10, C2×C10, C2×C10, C2×C4⋊C4, C4×D4, C4⋊D4, C22⋊Q8, C22⋊Q8, C22.D4, C42.C2, C42⋊2C2, C4×D5, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C2×C20, C5×Q8, C22×D5, C22×C10, C22.33C24, C4×Dic5, C10.D4, C4⋊Dic5, C4⋊Dic5, D10⋊C4, C5×C22⋊C4, C5×C4⋊C4, C5×C4⋊C4, C2×C4×D5, C2×D20, C22×Dic5, C2×C5⋊D4, C22×C20, Q8×C10, Dic5⋊4D4, C22.D20, C4.Dic10, D10.13D4, C4⋊C4⋊D5, C2×C10.D4, C20⋊7D4, D10⋊3Q8, C5×C22⋊Q8, C10.572+ 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, Q8⋊2D5, C23×D5, D4⋊6D10, C2×Q8⋊2D5, D4.10D10, C10.572+ 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)
(1 153 28 11)(2 154 29 12)(3 155 30 13)(4 156 21 14)(5 157 22 15)(6 158 23 16)(7 159 24 17)(8 160 25 18)(9 151 26 19)(10 152 27 20)(31 131 43 143)(32 132 44 144)(33 133 45 145)(34 134 46 146)(35 135 47 147)(36 136 48 148)(37 137 49 149)(38 138 50 150)(39 139 41 141)(40 140 42 142)(51 116 63 128)(52 117 64 129)(53 118 65 130)(54 119 66 121)(55 120 67 122)(56 111 68 123)(57 112 69 124)(58 113 70 125)(59 114 61 126)(60 115 62 127)(71 108 83 96)(72 109 84 97)(73 110 85 98)(74 101 86 99)(75 102 87 100)(76 103 88 91)(77 104 89 92)(78 105 90 93)(79 106 81 94)(80 107 82 95)
(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 11 23 158)(2 20 24 157)(3 19 25 156)(4 18 26 155)(5 17 27 154)(6 16 28 153)(7 15 29 152)(8 14 30 151)(9 13 21 160)(10 12 22 159)(31 135 48 142)(32 134 49 141)(33 133 50 150)(34 132 41 149)(35 131 42 148)(36 140 43 147)(37 139 44 146)(38 138 45 145)(39 137 46 144)(40 136 47 143)(51 122 68 115)(52 121 69 114)(53 130 70 113)(54 129 61 112)(55 128 62 111)(56 127 63 120)(57 126 64 119)(58 125 65 118)(59 124 66 117)(60 123 67 116)(71 102 88 95)(72 101 89 94)(73 110 90 93)(74 109 81 92)(75 108 82 91)(76 107 83 100)(77 106 84 99)(78 105 85 98)(79 104 86 97)(80 103 87 96)
(1 38 28 50)(2 39 29 41)(3 40 30 42)(4 31 21 43)(5 32 22 44)(6 33 23 45)(7 34 24 46)(8 35 25 47)(9 36 26 48)(10 37 27 49)(11 145 153 133)(12 146 154 134)(13 147 155 135)(14 148 156 136)(15 149 157 137)(16 150 158 138)(17 141 159 139)(18 142 160 140)(19 143 151 131)(20 144 152 132)(51 83 63 71)(52 84 64 72)(53 85 65 73)(54 86 66 74)(55 87 67 75)(56 88 68 76)(57 89 69 77)(58 90 70 78)(59 81 61 79)(60 82 62 80)(91 128 103 116)(92 129 104 117)(93 130 105 118)(94 121 106 119)(95 122 107 120)(96 123 108 111)(97 124 109 112)(98 125 110 113)(99 126 101 114)(100 127 102 115)
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), (1,153,28,11)(2,154,29,12)(3,155,30,13)(4,156,21,14)(5,157,22,15)(6,158,23,16)(7,159,24,17)(8,160,25,18)(9,151,26,19)(10,152,27,20)(31,131,43,143)(32,132,44,144)(33,133,45,145)(34,134,46,146)(35,135,47,147)(36,136,48,148)(37,137,49,149)(38,138,50,150)(39,139,41,141)(40,140,42,142)(51,116,63,128)(52,117,64,129)(53,118,65,130)(54,119,66,121)(55,120,67,122)(56,111,68,123)(57,112,69,124)(58,113,70,125)(59,114,61,126)(60,115,62,127)(71,108,83,96)(72,109,84,97)(73,110,85,98)(74,101,86,99)(75,102,87,100)(76,103,88,91)(77,104,89,92)(78,105,90,93)(79,106,81,94)(80,107,82,95), (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), (1,11,23,158)(2,20,24,157)(3,19,25,156)(4,18,26,155)(5,17,27,154)(6,16,28,153)(7,15,29,152)(8,14,30,151)(9,13,21,160)(10,12,22,159)(31,135,48,142)(32,134,49,141)(33,133,50,150)(34,132,41,149)(35,131,42,148)(36,140,43,147)(37,139,44,146)(38,138,45,145)(39,137,46,144)(40,136,47,143)(51,122,68,115)(52,121,69,114)(53,130,70,113)(54,129,61,112)(55,128,62,111)(56,127,63,120)(57,126,64,119)(58,125,65,118)(59,124,66,117)(60,123,67,116)(71,102,88,95)(72,101,89,94)(73,110,90,93)(74,109,81,92)(75,108,82,91)(76,107,83,100)(77,106,84,99)(78,105,85,98)(79,104,86,97)(80,103,87,96), (1,38,28,50)(2,39,29,41)(3,40,30,42)(4,31,21,43)(5,32,22,44)(6,33,23,45)(7,34,24,46)(8,35,25,47)(9,36,26,48)(10,37,27,49)(11,145,153,133)(12,146,154,134)(13,147,155,135)(14,148,156,136)(15,149,157,137)(16,150,158,138)(17,141,159,139)(18,142,160,140)(19,143,151,131)(20,144,152,132)(51,83,63,71)(52,84,64,72)(53,85,65,73)(54,86,66,74)(55,87,67,75)(56,88,68,76)(57,89,69,77)(58,90,70,78)(59,81,61,79)(60,82,62,80)(91,128,103,116)(92,129,104,117)(93,130,105,118)(94,121,106,119)(95,122,107,120)(96,123,108,111)(97,124,109,112)(98,125,110,113)(99,126,101,114)(100,127,102,115)>;
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), (1,153,28,11)(2,154,29,12)(3,155,30,13)(4,156,21,14)(5,157,22,15)(6,158,23,16)(7,159,24,17)(8,160,25,18)(9,151,26,19)(10,152,27,20)(31,131,43,143)(32,132,44,144)(33,133,45,145)(34,134,46,146)(35,135,47,147)(36,136,48,148)(37,137,49,149)(38,138,50,150)(39,139,41,141)(40,140,42,142)(51,116,63,128)(52,117,64,129)(53,118,65,130)(54,119,66,121)(55,120,67,122)(56,111,68,123)(57,112,69,124)(58,113,70,125)(59,114,61,126)(60,115,62,127)(71,108,83,96)(72,109,84,97)(73,110,85,98)(74,101,86,99)(75,102,87,100)(76,103,88,91)(77,104,89,92)(78,105,90,93)(79,106,81,94)(80,107,82,95), (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), (1,11,23,158)(2,20,24,157)(3,19,25,156)(4,18,26,155)(5,17,27,154)(6,16,28,153)(7,15,29,152)(8,14,30,151)(9,13,21,160)(10,12,22,159)(31,135,48,142)(32,134,49,141)(33,133,50,150)(34,132,41,149)(35,131,42,148)(36,140,43,147)(37,139,44,146)(38,138,45,145)(39,137,46,144)(40,136,47,143)(51,122,68,115)(52,121,69,114)(53,130,70,113)(54,129,61,112)(55,128,62,111)(56,127,63,120)(57,126,64,119)(58,125,65,118)(59,124,66,117)(60,123,67,116)(71,102,88,95)(72,101,89,94)(73,110,90,93)(74,109,81,92)(75,108,82,91)(76,107,83,100)(77,106,84,99)(78,105,85,98)(79,104,86,97)(80,103,87,96), (1,38,28,50)(2,39,29,41)(3,40,30,42)(4,31,21,43)(5,32,22,44)(6,33,23,45)(7,34,24,46)(8,35,25,47)(9,36,26,48)(10,37,27,49)(11,145,153,133)(12,146,154,134)(13,147,155,135)(14,148,156,136)(15,149,157,137)(16,150,158,138)(17,141,159,139)(18,142,160,140)(19,143,151,131)(20,144,152,132)(51,83,63,71)(52,84,64,72)(53,85,65,73)(54,86,66,74)(55,87,67,75)(56,88,68,76)(57,89,69,77)(58,90,70,78)(59,81,61,79)(60,82,62,80)(91,128,103,116)(92,129,104,117)(93,130,105,118)(94,121,106,119)(95,122,107,120)(96,123,108,111)(97,124,109,112)(98,125,110,113)(99,126,101,114)(100,127,102,115) );
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)], [(1,153,28,11),(2,154,29,12),(3,155,30,13),(4,156,21,14),(5,157,22,15),(6,158,23,16),(7,159,24,17),(8,160,25,18),(9,151,26,19),(10,152,27,20),(31,131,43,143),(32,132,44,144),(33,133,45,145),(34,134,46,146),(35,135,47,147),(36,136,48,148),(37,137,49,149),(38,138,50,150),(39,139,41,141),(40,140,42,142),(51,116,63,128),(52,117,64,129),(53,118,65,130),(54,119,66,121),(55,120,67,122),(56,111,68,123),(57,112,69,124),(58,113,70,125),(59,114,61,126),(60,115,62,127),(71,108,83,96),(72,109,84,97),(73,110,85,98),(74,101,86,99),(75,102,87,100),(76,103,88,91),(77,104,89,92),(78,105,90,93),(79,106,81,94),(80,107,82,95)], [(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)], [(1,11,23,158),(2,20,24,157),(3,19,25,156),(4,18,26,155),(5,17,27,154),(6,16,28,153),(7,15,29,152),(8,14,30,151),(9,13,21,160),(10,12,22,159),(31,135,48,142),(32,134,49,141),(33,133,50,150),(34,132,41,149),(35,131,42,148),(36,140,43,147),(37,139,44,146),(38,138,45,145),(39,137,46,144),(40,136,47,143),(51,122,68,115),(52,121,69,114),(53,130,70,113),(54,129,61,112),(55,128,62,111),(56,127,63,120),(57,126,64,119),(58,125,65,118),(59,124,66,117),(60,123,67,116),(71,102,88,95),(72,101,89,94),(73,110,90,93),(74,109,81,92),(75,108,82,91),(76,107,83,100),(77,106,84,99),(78,105,85,98),(79,104,86,97),(80,103,87,96)], [(1,38,28,50),(2,39,29,41),(3,40,30,42),(4,31,21,43),(5,32,22,44),(6,33,23,45),(7,34,24,46),(8,35,25,47),(9,36,26,48),(10,37,27,49),(11,145,153,133),(12,146,154,134),(13,147,155,135),(14,148,156,136),(15,149,157,137),(16,150,158,138),(17,141,159,139),(18,142,160,140),(19,143,151,131),(20,144,152,132),(51,83,63,71),(52,84,64,72),(53,85,65,73),(54,86,66,74),(55,87,67,75),(56,88,68,76),(57,89,69,77),(58,90,70,78),(59,81,61,79),(60,82,62,80),(91,128,103,116),(92,129,104,117),(93,130,105,118),(94,121,106,119),(95,122,107,120),(96,123,108,111),(97,124,109,112),(98,125,110,113),(99,126,101,114),(100,127,102,115)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | ··· | 20P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 2 | 2 | 20 | 20 | 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 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | D10 | D10 | D10 | D10 | 2+ 1+4 | 2- 1+4 | Q8⋊2D5 | D4⋊6D10 | D4.10D10 |
| kernel | C10.572+ 1+4 | Dic5⋊4D4 | C22.D20 | C4.Dic10 | D10.13D4 | C4⋊C4⋊D5 | C2×C10.D4 | C20⋊7D4 | D10⋊3Q8 | C5×C22⋊Q8 | C22⋊Q8 | C2×C10 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×Q8 | C10 | C10 | C22 | C2 | C2 |
| # reps | 1 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 1 | 2 | 4 | 4 | 6 | 2 | 2 | 1 | 1 | 4 | 4 | 4 |
Matrix representation of C10.572+ 1+4 ►in GL8(𝔽41)
| 34 | 34 | 0 | 0 | 0 | 0 | 0 | 0 |
| 7 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 34 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 1 | 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 | 30 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 11 | 0 | 0 | 0 | 0 |
| 30 | 32 | 0 | 0 | 0 | 0 | 0 | 0 |
| 9 | 11 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 0 | 26 | 0 |
| 0 | 0 | 0 | 0 | 0 | 15 | 0 | 26 |
| 0 | 0 | 0 | 0 | 26 | 0 | 26 | 0 |
| 0 | 0 | 0 | 0 | 0 | 26 | 0 | 26 |
| 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 | 31 | 40 | 31 |
| 0 | 0 | 0 | 0 | 39 | 1 | 39 | 1 |
| 0 | 0 | 0 | 0 | 40 | 31 | 1 | 10 |
| 0 | 0 | 0 | 0 | 39 | 1 | 2 | 40 |
| 0 | 0 | 30 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 27 | 11 | 0 | 0 | 0 | 0 |
| 30 | 32 | 0 | 0 | 0 | 0 | 0 | 0 |
| 27 | 11 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 18 | 15 | 23 | 26 |
| 0 | 0 | 0 | 0 | 18 | 23 | 23 | 18 |
| 0 | 0 | 0 | 0 | 23 | 26 | 23 | 26 |
| 0 | 0 | 0 | 0 | 23 | 18 | 23 | 18 |
| 30 | 32 | 0 | 0 | 0 | 0 | 0 | 0 |
| 9 | 11 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 30 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 22 | 30 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 11 | 28 |
| 0 | 0 | 0 | 0 | 0 | 0 | 22 | 30 |
G:=sub<GL(8,GF(41))| [34,7,0,0,0,0,0,0,34,1,0,0,0,0,0,0,0,0,34,7,0,0,0,0,0,0,34,1,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,35,34,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,35,34],[0,0,30,9,0,0,0,0,0,0,32,11,0,0,0,0,30,9,0,0,0,0,0,0,32,11,0,0,0,0,0,0,0,0,0,0,15,0,26,0,0,0,0,0,0,15,0,26,0,0,0,0,26,0,26,0,0,0,0,0,0,26,0,26],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,39,40,39,0,0,0,0,31,1,31,1,0,0,0,0,40,39,1,2,0,0,0,0,31,1,10,40],[0,0,30,27,0,0,0,0,0,0,32,11,0,0,0,0,30,27,0,0,0,0,0,0,32,11,0,0,0,0,0,0,0,0,0,0,18,18,23,23,0,0,0,0,15,23,26,18,0,0,0,0,23,23,23,23,0,0,0,0,26,18,26,18],[30,9,0,0,0,0,0,0,32,11,0,0,0,0,0,0,0,0,11,32,0,0,0,0,0,0,9,30,0,0,0,0,0,0,0,0,11,22,0,0,0,0,0,0,28,30,0,0,0,0,0,0,0,0,11,22,0,0,0,0,0,0,28,30] >;
C10.572+ 1+4 in GAP, Magma, Sage, TeX
C_{10}._{57}2_+^{1+4} % in TeX
G:=Group("C10.57ES+(2,2)"); // GroupNames label
G:=SmallGroup(320,1317);
// by ID
G=gap.SmallGroup(320,1317);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,387,100,675,409,80,12550]);
// 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=a^5*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=a^5*b^2*d>;
// generators/relations