G = C10.102+ 1+4 order 320 = 26·5
10th non-split extension by C10 of 2+ 1+4 acting via 2+ 1+4/C2×D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C10.102+ 1+4, C5⋊D4⋊4Q8, C20⋊Q8⋊10C2, C5⋊1(D4⋊3Q8), C4⋊C4.264D10, D10⋊2Q8⋊9C2, D10⋊Q8⋊2C2, C22.8(Q8×D5), D10.19(C2×Q8), Dic5⋊3Q8⋊9C2, C4.92(C4○D20), (C2×C10).55C24, Dic5.20(C2×Q8), C20.194(C4○D4), C20.48D4⋊18C2, C10.26(C22×Q8), (C2×C20).138C23, Dic5.Q8⋊1C2, (C22×C4).180D10, C2.13(D4⋊6D10), C22.89(C23×D5), C4⋊Dic5.192C22, C23.227(C22×D5), C23.D5.88C22, D10⋊C4.94C22, (C22×C10).404C23, (C22×C20).103C22, (C4×Dic5).212C22, (C2×Dic5).201C23, (C22×D5).170C23, (C2×Dic10).147C22, C10.D4.149C22, C2.9(C2×Q8×D5), (D5×C4⋊C4)⋊10C2, (C2×C4⋊C4)⋊20D5, (C10×C4⋊C4)⋊17C2, (C4×C5⋊D4).3C2, C10.22(C2×C4○D4), C2.24(C2×C4○D20), (C2×C10).95(C2×Q8), (C2×C4×D5).242C22, (C5×C4⋊C4).297C22, (C2×C4).573(C22×D5), (C2×C5⋊D4).159C22, SmallGroup(320,1183)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10.102+ 1+4
G = < a,b,c,d,e | a10=b4=c2=1, d2=b2, e2=a5, bab-1=a-1, ac=ca, ad=da, ae=ea, cbc=a5b-1, dbd-1=a5b, be=eb, cd=dc, ce=ec, ede-1=b2d >
Subgroups: 726 in 228 conjugacy classes, 107 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×Q8, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C2×C10, C2×C4⋊C4, C2×C4⋊C4, C4×D4, C4×Q8, C22⋊Q8, C42.C2, C4⋊Q8, Dic10, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C2×C20, C22×D5, C22×C10, D4⋊3Q8, C4×Dic5, C4×Dic5, C10.D4, C10.D4, C4⋊Dic5, C4⋊Dic5, D10⋊C4, D10⋊C4, C23.D5, C23.D5, C5×C4⋊C4, C5×C4⋊C4, C2×Dic10, C2×Dic10, C2×C4×D5, C2×C4×D5, C2×C5⋊D4, C22×C20, C22×C20, Dic5⋊3Q8, C20⋊Q8, Dic5.Q8, D5×C4⋊C4, D10⋊Q8, D10⋊2Q8, C20.48D4, C20.48D4, C4×C5⋊D4, C4×C5⋊D4, C10×C4⋊C4, C10.102+ 1+4
Quotients: C1, C2, C22, Q8, C23, D5, C2×Q8, C4○D4, C24, D10, C22×Q8, C2×C4○D4, 2+ 1+4, C22×D5, D4⋊3Q8, C4○D20, Q8×D5, C23×D5, C2×C4○D20, D4⋊6D10, C2×Q8×D5, C10.102+ 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 11 30 157)(2 20 21 156)(3 19 22 155)(4 18 23 154)(5 17 24 153)(6 16 25 152)(7 15 26 151)(8 14 27 160)(9 13 28 159)(10 12 29 158)(31 133 49 141)(32 132 50 150)(33 131 41 149)(34 140 42 148)(35 139 43 147)(36 138 44 146)(37 137 45 145)(38 136 46 144)(39 135 47 143)(40 134 48 142)(51 121 69 113)(52 130 70 112)(53 129 61 111)(54 128 62 120)(55 127 63 119)(56 126 64 118)(57 125 65 117)(58 124 66 116)(59 123 67 115)(60 122 68 114)(71 98 89 106)(72 97 90 105)(73 96 81 104)(74 95 82 103)(75 94 83 102)(76 93 84 101)(77 92 85 110)(78 91 86 109)(79 100 87 108)(80 99 88 107)
(1 32)(2 33)(3 34)(4 35)(5 36)(6 37)(7 38)(8 39)(9 40)(10 31)(11 145)(12 146)(13 147)(14 148)(15 149)(16 150)(17 141)(18 142)(19 143)(20 144)(21 41)(22 42)(23 43)(24 44)(25 45)(26 46)(27 47)(28 48)(29 49)(30 50)(51 71)(52 72)(53 73)(54 74)(55 75)(56 76)(57 77)(58 78)(59 79)(60 80)(61 81)(62 82)(63 83)(64 84)(65 85)(66 86)(67 87)(68 88)(69 89)(70 90)(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)(131 151)(132 152)(133 153)(134 154)(135 155)(136 156)(137 157)(138 158)(139 159)(140 160)
(1 130 30 112)(2 121 21 113)(3 122 22 114)(4 123 23 115)(5 124 24 116)(6 125 25 117)(7 126 26 118)(8 127 27 119)(9 128 28 120)(10 129 29 111)(11 65 157 57)(12 66 158 58)(13 67 159 59)(14 68 160 60)(15 69 151 51)(16 70 152 52)(17 61 153 53)(18 62 154 54)(19 63 155 55)(20 64 156 56)(31 109 49 91)(32 110 50 92)(33 101 41 93)(34 102 42 94)(35 103 43 95)(36 104 44 96)(37 105 45 97)(38 106 46 98)(39 107 47 99)(40 108 48 100)(71 149 89 131)(72 150 90 132)(73 141 81 133)(74 142 82 134)(75 143 83 135)(76 144 84 136)(77 145 85 137)(78 146 86 138)(79 147 87 139)(80 148 88 140)
(1 145 6 150)(2 146 7 141)(3 147 8 142)(4 148 9 143)(5 149 10 144)(11 37 16 32)(12 38 17 33)(13 39 18 34)(14 40 19 35)(15 31 20 36)(21 138 26 133)(22 139 27 134)(23 140 28 135)(24 131 29 136)(25 132 30 137)(41 158 46 153)(42 159 47 154)(43 160 48 155)(44 151 49 156)(45 152 50 157)(51 109 56 104)(52 110 57 105)(53 101 58 106)(54 102 59 107)(55 103 60 108)(61 93 66 98)(62 94 67 99)(63 95 68 100)(64 96 69 91)(65 97 70 92)(71 129 76 124)(72 130 77 125)(73 121 78 126)(74 122 79 127)(75 123 80 128)(81 113 86 118)(82 114 87 119)(83 115 88 120)(84 116 89 111)(85 117 90 112)
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), (1,11,30,157)(2,20,21,156)(3,19,22,155)(4,18,23,154)(5,17,24,153)(6,16,25,152)(7,15,26,151)(8,14,27,160)(9,13,28,159)(10,12,29,158)(31,133,49,141)(32,132,50,150)(33,131,41,149)(34,140,42,148)(35,139,43,147)(36,138,44,146)(37,137,45,145)(38,136,46,144)(39,135,47,143)(40,134,48,142)(51,121,69,113)(52,130,70,112)(53,129,61,111)(54,128,62,120)(55,127,63,119)(56,126,64,118)(57,125,65,117)(58,124,66,116)(59,123,67,115)(60,122,68,114)(71,98,89,106)(72,97,90,105)(73,96,81,104)(74,95,82,103)(75,94,83,102)(76,93,84,101)(77,92,85,110)(78,91,86,109)(79,100,87,108)(80,99,88,107), (1,32)(2,33)(3,34)(4,35)(5,36)(6,37)(7,38)(8,39)(9,40)(10,31)(11,145)(12,146)(13,147)(14,148)(15,149)(16,150)(17,141)(18,142)(19,143)(20,144)(21,41)(22,42)(23,43)(24,44)(25,45)(26,46)(27,47)(28,48)(29,49)(30,50)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80)(61,81)(62,82)(63,83)(64,84)(65,85)(66,86)(67,87)(68,88)(69,89)(70,90)(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)(131,151)(132,152)(133,153)(134,154)(135,155)(136,156)(137,157)(138,158)(139,159)(140,160), (1,130,30,112)(2,121,21,113)(3,122,22,114)(4,123,23,115)(5,124,24,116)(6,125,25,117)(7,126,26,118)(8,127,27,119)(9,128,28,120)(10,129,29,111)(11,65,157,57)(12,66,158,58)(13,67,159,59)(14,68,160,60)(15,69,151,51)(16,70,152,52)(17,61,153,53)(18,62,154,54)(19,63,155,55)(20,64,156,56)(31,109,49,91)(32,110,50,92)(33,101,41,93)(34,102,42,94)(35,103,43,95)(36,104,44,96)(37,105,45,97)(38,106,46,98)(39,107,47,99)(40,108,48,100)(71,149,89,131)(72,150,90,132)(73,141,81,133)(74,142,82,134)(75,143,83,135)(76,144,84,136)(77,145,85,137)(78,146,86,138)(79,147,87,139)(80,148,88,140), (1,145,6,150)(2,146,7,141)(3,147,8,142)(4,148,9,143)(5,149,10,144)(11,37,16,32)(12,38,17,33)(13,39,18,34)(14,40,19,35)(15,31,20,36)(21,138,26,133)(22,139,27,134)(23,140,28,135)(24,131,29,136)(25,132,30,137)(41,158,46,153)(42,159,47,154)(43,160,48,155)(44,151,49,156)(45,152,50,157)(51,109,56,104)(52,110,57,105)(53,101,58,106)(54,102,59,107)(55,103,60,108)(61,93,66,98)(62,94,67,99)(63,95,68,100)(64,96,69,91)(65,97,70,92)(71,129,76,124)(72,130,77,125)(73,121,78,126)(74,122,79,127)(75,123,80,128)(81,113,86,118)(82,114,87,119)(83,115,88,120)(84,116,89,111)(85,117,90,112)>;
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), (1,11,30,157)(2,20,21,156)(3,19,22,155)(4,18,23,154)(5,17,24,153)(6,16,25,152)(7,15,26,151)(8,14,27,160)(9,13,28,159)(10,12,29,158)(31,133,49,141)(32,132,50,150)(33,131,41,149)(34,140,42,148)(35,139,43,147)(36,138,44,146)(37,137,45,145)(38,136,46,144)(39,135,47,143)(40,134,48,142)(51,121,69,113)(52,130,70,112)(53,129,61,111)(54,128,62,120)(55,127,63,119)(56,126,64,118)(57,125,65,117)(58,124,66,116)(59,123,67,115)(60,122,68,114)(71,98,89,106)(72,97,90,105)(73,96,81,104)(74,95,82,103)(75,94,83,102)(76,93,84,101)(77,92,85,110)(78,91,86,109)(79,100,87,108)(80,99,88,107), (1,32)(2,33)(3,34)(4,35)(5,36)(6,37)(7,38)(8,39)(9,40)(10,31)(11,145)(12,146)(13,147)(14,148)(15,149)(16,150)(17,141)(18,142)(19,143)(20,144)(21,41)(22,42)(23,43)(24,44)(25,45)(26,46)(27,47)(28,48)(29,49)(30,50)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80)(61,81)(62,82)(63,83)(64,84)(65,85)(66,86)(67,87)(68,88)(69,89)(70,90)(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)(131,151)(132,152)(133,153)(134,154)(135,155)(136,156)(137,157)(138,158)(139,159)(140,160), (1,130,30,112)(2,121,21,113)(3,122,22,114)(4,123,23,115)(5,124,24,116)(6,125,25,117)(7,126,26,118)(8,127,27,119)(9,128,28,120)(10,129,29,111)(11,65,157,57)(12,66,158,58)(13,67,159,59)(14,68,160,60)(15,69,151,51)(16,70,152,52)(17,61,153,53)(18,62,154,54)(19,63,155,55)(20,64,156,56)(31,109,49,91)(32,110,50,92)(33,101,41,93)(34,102,42,94)(35,103,43,95)(36,104,44,96)(37,105,45,97)(38,106,46,98)(39,107,47,99)(40,108,48,100)(71,149,89,131)(72,150,90,132)(73,141,81,133)(74,142,82,134)(75,143,83,135)(76,144,84,136)(77,145,85,137)(78,146,86,138)(79,147,87,139)(80,148,88,140), (1,145,6,150)(2,146,7,141)(3,147,8,142)(4,148,9,143)(5,149,10,144)(11,37,16,32)(12,38,17,33)(13,39,18,34)(14,40,19,35)(15,31,20,36)(21,138,26,133)(22,139,27,134)(23,140,28,135)(24,131,29,136)(25,132,30,137)(41,158,46,153)(42,159,47,154)(43,160,48,155)(44,151,49,156)(45,152,50,157)(51,109,56,104)(52,110,57,105)(53,101,58,106)(54,102,59,107)(55,103,60,108)(61,93,66,98)(62,94,67,99)(63,95,68,100)(64,96,69,91)(65,97,70,92)(71,129,76,124)(72,130,77,125)(73,121,78,126)(74,122,79,127)(75,123,80,128)(81,113,86,118)(82,114,87,119)(83,115,88,120)(84,116,89,111)(85,117,90,112) );
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)], [(1,11,30,157),(2,20,21,156),(3,19,22,155),(4,18,23,154),(5,17,24,153),(6,16,25,152),(7,15,26,151),(8,14,27,160),(9,13,28,159),(10,12,29,158),(31,133,49,141),(32,132,50,150),(33,131,41,149),(34,140,42,148),(35,139,43,147),(36,138,44,146),(37,137,45,145),(38,136,46,144),(39,135,47,143),(40,134,48,142),(51,121,69,113),(52,130,70,112),(53,129,61,111),(54,128,62,120),(55,127,63,119),(56,126,64,118),(57,125,65,117),(58,124,66,116),(59,123,67,115),(60,122,68,114),(71,98,89,106),(72,97,90,105),(73,96,81,104),(74,95,82,103),(75,94,83,102),(76,93,84,101),(77,92,85,110),(78,91,86,109),(79,100,87,108),(80,99,88,107)], [(1,32),(2,33),(3,34),(4,35),(5,36),(6,37),(7,38),(8,39),(9,40),(10,31),(11,145),(12,146),(13,147),(14,148),(15,149),(16,150),(17,141),(18,142),(19,143),(20,144),(21,41),(22,42),(23,43),(24,44),(25,45),(26,46),(27,47),(28,48),(29,49),(30,50),(51,71),(52,72),(53,73),(54,74),(55,75),(56,76),(57,77),(58,78),(59,79),(60,80),(61,81),(62,82),(63,83),(64,84),(65,85),(66,86),(67,87),(68,88),(69,89),(70,90),(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),(131,151),(132,152),(133,153),(134,154),(135,155),(136,156),(137,157),(138,158),(139,159),(140,160)], [(1,130,30,112),(2,121,21,113),(3,122,22,114),(4,123,23,115),(5,124,24,116),(6,125,25,117),(7,126,26,118),(8,127,27,119),(9,128,28,120),(10,129,29,111),(11,65,157,57),(12,66,158,58),(13,67,159,59),(14,68,160,60),(15,69,151,51),(16,70,152,52),(17,61,153,53),(18,62,154,54),(19,63,155,55),(20,64,156,56),(31,109,49,91),(32,110,50,92),(33,101,41,93),(34,102,42,94),(35,103,43,95),(36,104,44,96),(37,105,45,97),(38,106,46,98),(39,107,47,99),(40,108,48,100),(71,149,89,131),(72,150,90,132),(73,141,81,133),(74,142,82,134),(75,143,83,135),(76,144,84,136),(77,145,85,137),(78,146,86,138),(79,147,87,139),(80,148,88,140)], [(1,145,6,150),(2,146,7,141),(3,147,8,142),(4,148,9,143),(5,149,10,144),(11,37,16,32),(12,38,17,33),(13,39,18,34),(14,40,19,35),(15,31,20,36),(21,138,26,133),(22,139,27,134),(23,140,28,135),(24,131,29,136),(25,132,30,137),(41,158,46,153),(42,159,47,154),(43,160,48,155),(44,151,49,156),(45,152,50,157),(51,109,56,104),(52,110,57,105),(53,101,58,106),(54,102,59,107),(55,103,60,108),(61,93,66,98),(62,94,67,99),(63,95,68,100),(64,96,69,91),(65,97,70,92),(71,129,76,124),(72,130,77,125),(73,121,78,126),(74,122,79,127),(75,123,80,128),(81,113,86,118),(82,114,87,119),(83,115,88,120),(84,116,89,111),(85,117,90,112)]])
65 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | ··· | 4Q | 5A | 5B | 10A | ··· | 10N | 20A | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 10 | 10 | 2 | ··· | 2 | 4 | 4 | 4 | 10 | 10 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
65 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | Q8 | D5 | C4○D4 | D10 | D10 | C4○D20 | 2+ 1+4 | Q8×D5 | D4⋊6D10 |
| kernel | C10.102+ 1+4 | Dic5⋊3Q8 | C20⋊Q8 | Dic5.Q8 | D5×C4⋊C4 | D10⋊Q8 | D10⋊2Q8 | C20.48D4 | C4×C5⋊D4 | C10×C4⋊C4 | C5⋊D4 | C2×C4⋊C4 | C20 | C4⋊C4 | C22×C4 | C4 | C10 | C22 | C2 |
| # reps | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 3 | 3 | 1 | 4 | 2 | 4 | 8 | 6 | 16 | 1 | 4 | 4 |
Matrix representation of C10.102+ 1+4 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 35 | 0 | 0 |
| 0 | 0 | 6 | 35 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 9 | 1 | 0 | 0 | 0 | 0 |
| 2 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 6 | 0 | 0 |
| 0 | 0 | 1 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 21 |
| 0 | 0 | 0 | 0 | 21 | 38 |
| 40 | 9 | 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 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 1 | 32 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 21 | 38 |
| 0 | 0 | 0 | 0 | 38 | 20 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 38 | 20 |
| 0 | 0 | 0 | 0 | 20 | 3 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,6,0,0,0,0,35,35,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[9,2,0,0,0,0,1,32,0,0,0,0,0,0,35,1,0,0,0,0,6,6,0,0,0,0,0,0,3,21,0,0,0,0,21,38],[40,0,0,0,0,0,9,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,32,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,21,38,0,0,0,0,38,20],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,38,20,0,0,0,0,20,3] >;
C10.102+ 1+4 in GAP, Magma, Sage, TeX
C_{10}._{10}2_+^{1+4} % in TeX
G:=Group("C10.10ES+(2,2)"); // GroupNames label
G:=SmallGroup(320,1183);
// by ID
G=gap.SmallGroup(320,1183);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,387,100,1571,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^10=b^4=c^2=1,d^2=b^2,e^2=a^5,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=a^5*b^-1,d*b*d^-1=a^5*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b^2*d>;
// generators/relations