G = C10.172- 1+4 order 320 = 26·5
17th non-split extension by C10 of 2- 1+4 acting via 2- 1+4/C2×Q8=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C10.172- 1+4, C10.1162+ 1+4, (C4×D5)⋊3D4, C22⋊Q8⋊6D5, C4⋊C4.96D10, C4.188(D4×D5), C4⋊D20⋊24C2, C20⋊7D4⋊36C2, D10.16(C2×D4), C20.233(C2×D4), D10⋊D4⋊24C2, D10⋊3Q8⋊15C2, D10⋊Q8⋊18C2, (C2×C20).53C23, (C2×Q8).125D10, C22⋊C4.15D10, Dic5.88(C2×D4), C10.75(C22×D4), (C2×C10).173C24, (C22×C4).235D10, C2.33(D4⋊8D10), (C2×D20).154C22, C4⋊Dic5.214C22, (Q8×C10).106C22, (C2×Dic5).88C23, C23.118(C22×D5), C22.194(C23×D5), D10⋊C4.22C22, (C22×C10).201C23, (C22×C20).253C22, C5⋊4(C22.31C24), C10.D4.26C22, (C22×D5).205C23, C2.18(Q8.10D10), (C2×Dic10).255C22, C2.48(C2×D4×D5), (D5×C4⋊C4)⋊25C2, (C5×C22⋊Q8)⋊9C2, (C2×C4○D20)⋊23C2, (C2×Q8⋊2D5)⋊6C2, (C2×C4×D5).103C22, (C5×C4⋊C4).157C22, (C2×C4).591(C22×D5), (C2×C5⋊D4).129C22, (C5×C22⋊C4).28C22, SmallGroup(320,1301)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10.172- 1+4
G = < a,b,c,d,e | a10=b4=c2=1, d2=a5b2, e2=b2, ab=ba, cac=dad-1=a-1, ae=ea, cbc=b-1, dbd-1=a5b, be=eb, dcd-1=a5c, ce=ec, ede-1=b2d >
Subgroups: 1198 in 294 conjugacy classes, 103 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C2×Q8, C4○D4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C2×C4⋊C4, C4⋊D4, C22⋊Q8, C22⋊Q8, C2×C4○D4, Dic10, C4×D5, C4×D5, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C2×C20, C5×Q8, C22×D5, C22×D5, C22×C10, C22.31C24, C10.D4, C4⋊Dic5, D10⋊C4, C5×C22⋊C4, C5×C4⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C2×C4×D5, C2×D20, C2×D20, C4○D20, Q8⋊2D5, C2×C5⋊D4, C2×C5⋊D4, C22×C20, Q8×C10, D10⋊D4, D5×C4⋊C4, C4⋊D20, C4⋊D20, D10⋊Q8, C20⋊7D4, D10⋊3Q8, C5×C22⋊Q8, C2×C4○D20, C2×Q8⋊2D5, C10.172- 1+4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C24, D10, C22×D4, 2+ 1+4, 2- 1+4, C22×D5, C22.31C24, D4×D5, C23×D5, C2×D4×D5, Q8.10D10, D4⋊8D10, C10.172- 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)
(1 113 13 103)(2 114 14 104)(3 115 15 105)(4 116 16 106)(5 117 17 107)(6 118 18 108)(7 119 19 109)(8 120 20 110)(9 111 11 101)(10 112 12 102)(21 91 31 81)(22 92 32 82)(23 93 33 83)(24 94 34 84)(25 95 35 85)(26 96 36 86)(27 97 37 87)(28 98 38 88)(29 99 39 89)(30 100 40 90)(41 156 51 146)(42 157 52 147)(43 158 53 148)(44 159 54 149)(45 160 55 150)(46 151 56 141)(47 152 57 142)(48 153 58 143)(49 154 59 144)(50 155 60 145)(61 136 71 126)(62 137 72 127)(63 138 73 128)(64 139 74 129)(65 140 75 130)(66 131 76 121)(67 132 77 122)(68 133 78 123)(69 134 79 124)(70 135 80 125)
(1 128)(2 127)(3 126)(4 125)(5 124)(6 123)(7 122)(8 121)(9 130)(10 129)(11 140)(12 139)(13 138)(14 137)(15 136)(16 135)(17 134)(18 133)(19 132)(20 131)(21 155)(22 154)(23 153)(24 152)(25 151)(26 160)(27 159)(28 158)(29 157)(30 156)(31 145)(32 144)(33 143)(34 142)(35 141)(36 150)(37 149)(38 148)(39 147)(40 146)(41 100)(42 99)(43 98)(44 97)(45 96)(46 95)(47 94)(48 93)(49 92)(50 91)(51 90)(52 89)(53 88)(54 87)(55 86)(56 85)(57 84)(58 83)(59 82)(60 81)(61 105)(62 104)(63 103)(64 102)(65 101)(66 110)(67 109)(68 108)(69 107)(70 106)(71 115)(72 114)(73 113)(74 112)(75 111)(76 120)(77 119)(78 118)(79 117)(80 116)
(1 33 18 28)(2 32 19 27)(3 31 20 26)(4 40 11 25)(5 39 12 24)(6 38 13 23)(7 37 14 22)(8 36 15 21)(9 35 16 30)(10 34 17 29)(41 75 56 70)(42 74 57 69)(43 73 58 68)(44 72 59 67)(45 71 60 66)(46 80 51 65)(47 79 52 64)(48 78 53 63)(49 77 54 62)(50 76 55 61)(81 105 96 120)(82 104 97 119)(83 103 98 118)(84 102 99 117)(85 101 100 116)(86 110 91 115)(87 109 92 114)(88 108 93 113)(89 107 94 112)(90 106 95 111)(121 145 136 160)(122 144 137 159)(123 143 138 158)(124 142 139 157)(125 141 140 156)(126 150 131 155)(127 149 132 154)(128 148 133 153)(129 147 134 152)(130 146 135 151)
(1 93 13 83)(2 94 14 84)(3 95 15 85)(4 96 16 86)(5 97 17 87)(6 98 18 88)(7 99 19 89)(8 100 20 90)(9 91 11 81)(10 92 12 82)(21 111 31 101)(22 112 32 102)(23 113 33 103)(24 114 34 104)(25 115 35 105)(26 116 36 106)(27 117 37 107)(28 118 38 108)(29 119 39 109)(30 120 40 110)(41 131 51 121)(42 132 52 122)(43 133 53 123)(44 134 54 124)(45 135 55 125)(46 136 56 126)(47 137 57 127)(48 138 58 128)(49 139 59 129)(50 140 60 130)(61 151 71 141)(62 152 72 142)(63 153 73 143)(64 154 74 144)(65 155 75 145)(66 156 76 146)(67 157 77 147)(68 158 78 148)(69 159 79 149)(70 160 80 150)
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), (1,113,13,103)(2,114,14,104)(3,115,15,105)(4,116,16,106)(5,117,17,107)(6,118,18,108)(7,119,19,109)(8,120,20,110)(9,111,11,101)(10,112,12,102)(21,91,31,81)(22,92,32,82)(23,93,33,83)(24,94,34,84)(25,95,35,85)(26,96,36,86)(27,97,37,87)(28,98,38,88)(29,99,39,89)(30,100,40,90)(41,156,51,146)(42,157,52,147)(43,158,53,148)(44,159,54,149)(45,160,55,150)(46,151,56,141)(47,152,57,142)(48,153,58,143)(49,154,59,144)(50,155,60,145)(61,136,71,126)(62,137,72,127)(63,138,73,128)(64,139,74,129)(65,140,75,130)(66,131,76,121)(67,132,77,122)(68,133,78,123)(69,134,79,124)(70,135,80,125), (1,128)(2,127)(3,126)(4,125)(5,124)(6,123)(7,122)(8,121)(9,130)(10,129)(11,140)(12,139)(13,138)(14,137)(15,136)(16,135)(17,134)(18,133)(19,132)(20,131)(21,155)(22,154)(23,153)(24,152)(25,151)(26,160)(27,159)(28,158)(29,157)(30,156)(31,145)(32,144)(33,143)(34,142)(35,141)(36,150)(37,149)(38,148)(39,147)(40,146)(41,100)(42,99)(43,98)(44,97)(45,96)(46,95)(47,94)(48,93)(49,92)(50,91)(51,90)(52,89)(53,88)(54,87)(55,86)(56,85)(57,84)(58,83)(59,82)(60,81)(61,105)(62,104)(63,103)(64,102)(65,101)(66,110)(67,109)(68,108)(69,107)(70,106)(71,115)(72,114)(73,113)(74,112)(75,111)(76,120)(77,119)(78,118)(79,117)(80,116), (1,33,18,28)(2,32,19,27)(3,31,20,26)(4,40,11,25)(5,39,12,24)(6,38,13,23)(7,37,14,22)(8,36,15,21)(9,35,16,30)(10,34,17,29)(41,75,56,70)(42,74,57,69)(43,73,58,68)(44,72,59,67)(45,71,60,66)(46,80,51,65)(47,79,52,64)(48,78,53,63)(49,77,54,62)(50,76,55,61)(81,105,96,120)(82,104,97,119)(83,103,98,118)(84,102,99,117)(85,101,100,116)(86,110,91,115)(87,109,92,114)(88,108,93,113)(89,107,94,112)(90,106,95,111)(121,145,136,160)(122,144,137,159)(123,143,138,158)(124,142,139,157)(125,141,140,156)(126,150,131,155)(127,149,132,154)(128,148,133,153)(129,147,134,152)(130,146,135,151), (1,93,13,83)(2,94,14,84)(3,95,15,85)(4,96,16,86)(5,97,17,87)(6,98,18,88)(7,99,19,89)(8,100,20,90)(9,91,11,81)(10,92,12,82)(21,111,31,101)(22,112,32,102)(23,113,33,103)(24,114,34,104)(25,115,35,105)(26,116,36,106)(27,117,37,107)(28,118,38,108)(29,119,39,109)(30,120,40,110)(41,131,51,121)(42,132,52,122)(43,133,53,123)(44,134,54,124)(45,135,55,125)(46,136,56,126)(47,137,57,127)(48,138,58,128)(49,139,59,129)(50,140,60,130)(61,151,71,141)(62,152,72,142)(63,153,73,143)(64,154,74,144)(65,155,75,145)(66,156,76,146)(67,157,77,147)(68,158,78,148)(69,159,79,149)(70,160,80,150)>;
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), (1,113,13,103)(2,114,14,104)(3,115,15,105)(4,116,16,106)(5,117,17,107)(6,118,18,108)(7,119,19,109)(8,120,20,110)(9,111,11,101)(10,112,12,102)(21,91,31,81)(22,92,32,82)(23,93,33,83)(24,94,34,84)(25,95,35,85)(26,96,36,86)(27,97,37,87)(28,98,38,88)(29,99,39,89)(30,100,40,90)(41,156,51,146)(42,157,52,147)(43,158,53,148)(44,159,54,149)(45,160,55,150)(46,151,56,141)(47,152,57,142)(48,153,58,143)(49,154,59,144)(50,155,60,145)(61,136,71,126)(62,137,72,127)(63,138,73,128)(64,139,74,129)(65,140,75,130)(66,131,76,121)(67,132,77,122)(68,133,78,123)(69,134,79,124)(70,135,80,125), (1,128)(2,127)(3,126)(4,125)(5,124)(6,123)(7,122)(8,121)(9,130)(10,129)(11,140)(12,139)(13,138)(14,137)(15,136)(16,135)(17,134)(18,133)(19,132)(20,131)(21,155)(22,154)(23,153)(24,152)(25,151)(26,160)(27,159)(28,158)(29,157)(30,156)(31,145)(32,144)(33,143)(34,142)(35,141)(36,150)(37,149)(38,148)(39,147)(40,146)(41,100)(42,99)(43,98)(44,97)(45,96)(46,95)(47,94)(48,93)(49,92)(50,91)(51,90)(52,89)(53,88)(54,87)(55,86)(56,85)(57,84)(58,83)(59,82)(60,81)(61,105)(62,104)(63,103)(64,102)(65,101)(66,110)(67,109)(68,108)(69,107)(70,106)(71,115)(72,114)(73,113)(74,112)(75,111)(76,120)(77,119)(78,118)(79,117)(80,116), (1,33,18,28)(2,32,19,27)(3,31,20,26)(4,40,11,25)(5,39,12,24)(6,38,13,23)(7,37,14,22)(8,36,15,21)(9,35,16,30)(10,34,17,29)(41,75,56,70)(42,74,57,69)(43,73,58,68)(44,72,59,67)(45,71,60,66)(46,80,51,65)(47,79,52,64)(48,78,53,63)(49,77,54,62)(50,76,55,61)(81,105,96,120)(82,104,97,119)(83,103,98,118)(84,102,99,117)(85,101,100,116)(86,110,91,115)(87,109,92,114)(88,108,93,113)(89,107,94,112)(90,106,95,111)(121,145,136,160)(122,144,137,159)(123,143,138,158)(124,142,139,157)(125,141,140,156)(126,150,131,155)(127,149,132,154)(128,148,133,153)(129,147,134,152)(130,146,135,151), (1,93,13,83)(2,94,14,84)(3,95,15,85)(4,96,16,86)(5,97,17,87)(6,98,18,88)(7,99,19,89)(8,100,20,90)(9,91,11,81)(10,92,12,82)(21,111,31,101)(22,112,32,102)(23,113,33,103)(24,114,34,104)(25,115,35,105)(26,116,36,106)(27,117,37,107)(28,118,38,108)(29,119,39,109)(30,120,40,110)(41,131,51,121)(42,132,52,122)(43,133,53,123)(44,134,54,124)(45,135,55,125)(46,136,56,126)(47,137,57,127)(48,138,58,128)(49,139,59,129)(50,140,60,130)(61,151,71,141)(62,152,72,142)(63,153,73,143)(64,154,74,144)(65,155,75,145)(66,156,76,146)(67,157,77,147)(68,158,78,148)(69,159,79,149)(70,160,80,150) );
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)], [(1,113,13,103),(2,114,14,104),(3,115,15,105),(4,116,16,106),(5,117,17,107),(6,118,18,108),(7,119,19,109),(8,120,20,110),(9,111,11,101),(10,112,12,102),(21,91,31,81),(22,92,32,82),(23,93,33,83),(24,94,34,84),(25,95,35,85),(26,96,36,86),(27,97,37,87),(28,98,38,88),(29,99,39,89),(30,100,40,90),(41,156,51,146),(42,157,52,147),(43,158,53,148),(44,159,54,149),(45,160,55,150),(46,151,56,141),(47,152,57,142),(48,153,58,143),(49,154,59,144),(50,155,60,145),(61,136,71,126),(62,137,72,127),(63,138,73,128),(64,139,74,129),(65,140,75,130),(66,131,76,121),(67,132,77,122),(68,133,78,123),(69,134,79,124),(70,135,80,125)], [(1,128),(2,127),(3,126),(4,125),(5,124),(6,123),(7,122),(8,121),(9,130),(10,129),(11,140),(12,139),(13,138),(14,137),(15,136),(16,135),(17,134),(18,133),(19,132),(20,131),(21,155),(22,154),(23,153),(24,152),(25,151),(26,160),(27,159),(28,158),(29,157),(30,156),(31,145),(32,144),(33,143),(34,142),(35,141),(36,150),(37,149),(38,148),(39,147),(40,146),(41,100),(42,99),(43,98),(44,97),(45,96),(46,95),(47,94),(48,93),(49,92),(50,91),(51,90),(52,89),(53,88),(54,87),(55,86),(56,85),(57,84),(58,83),(59,82),(60,81),(61,105),(62,104),(63,103),(64,102),(65,101),(66,110),(67,109),(68,108),(69,107),(70,106),(71,115),(72,114),(73,113),(74,112),(75,111),(76,120),(77,119),(78,118),(79,117),(80,116)], [(1,33,18,28),(2,32,19,27),(3,31,20,26),(4,40,11,25),(5,39,12,24),(6,38,13,23),(7,37,14,22),(8,36,15,21),(9,35,16,30),(10,34,17,29),(41,75,56,70),(42,74,57,69),(43,73,58,68),(44,72,59,67),(45,71,60,66),(46,80,51,65),(47,79,52,64),(48,78,53,63),(49,77,54,62),(50,76,55,61),(81,105,96,120),(82,104,97,119),(83,103,98,118),(84,102,99,117),(85,101,100,116),(86,110,91,115),(87,109,92,114),(88,108,93,113),(89,107,94,112),(90,106,95,111),(121,145,136,160),(122,144,137,159),(123,143,138,158),(124,142,139,157),(125,141,140,156),(126,150,131,155),(127,149,132,154),(128,148,133,153),(129,147,134,152),(130,146,135,151)], [(1,93,13,83),(2,94,14,84),(3,95,15,85),(4,96,16,86),(5,97,17,87),(6,98,18,88),(7,99,19,89),(8,100,20,90),(9,91,11,81),(10,92,12,82),(21,111,31,101),(22,112,32,102),(23,113,33,103),(24,114,34,104),(25,115,35,105),(26,116,36,106),(27,117,37,107),(28,118,38,108),(29,119,39,109),(30,120,40,110),(41,131,51,121),(42,132,52,122),(43,133,53,123),(44,134,54,124),(45,135,55,125),(46,136,56,126),(47,137,57,127),(48,138,58,128),(49,139,59,129),(50,140,60,130),(61,151,71,141),(62,152,72,142),(63,153,73,143),(64,154,74,144),(65,155,75,145),(66,156,76,146),(67,157,77,147),(68,158,78,148),(69,159,79,149),(70,160,80,150)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | ··· | 4G | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | ··· | 20P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 10 | 10 | 20 | 20 | 20 | 2 | 2 | 4 | ··· | 4 | 10 | 10 | 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 | D4 | D5 | D10 | D10 | D10 | D10 | 2+ 1+4 | 2- 1+4 | D4×D5 | Q8.10D10 | D4⋊8D10 |
| kernel | C10.172- 1+4 | D10⋊D4 | D5×C4⋊C4 | C4⋊D20 | D10⋊Q8 | C20⋊7D4 | D10⋊3Q8 | C5×C22⋊Q8 | C2×C4○D20 | C2×Q8⋊2D5 | C4×D5 | C22⋊Q8 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×Q8 | C10 | C10 | C4 | C2 | C2 |
| # reps | 1 | 4 | 1 | 3 | 2 | 1 | 1 | 1 | 1 | 1 | 4 | 2 | 4 | 6 | 2 | 2 | 1 | 1 | 4 | 4 | 4 |
Matrix representation of C10.172- 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 | 33 | 39 | 35 | 35 |
| 0 | 0 | 32 | 8 | 6 | 40 |
| 1 | 39 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 | 0 |
| 0 | 0 | 2 | 13 | 9 | 0 |
| 0 | 0 | 11 | 4 | 0 | 9 |
| 1 | 39 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 30 | 36 | 4 | 26 |
| 0 | 0 | 35 | 17 | 9 | 37 |
| 0 | 0 | 2 | 13 | 9 | 0 |
| 0 | 0 | 16 | 37 | 32 | 26 |
| 40 | 2 | 0 | 0 | 0 | 0 |
| 40 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 35 | 0 | 0 |
| 0 | 0 | 20 | 23 | 0 | 0 |
| 0 | 0 | 32 | 39 | 18 | 35 |
| 0 | 0 | 35 | 9 | 20 | 23 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 13 | 0 | 0 |
| 0 | 0 | 28 | 39 | 0 | 0 |
| 0 | 0 | 1 | 23 | 39 | 28 |
| 0 | 0 | 18 | 16 | 13 | 2 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,35,6,33,32,0,0,35,40,39,8,0,0,0,0,35,6,0,0,0,0,35,40],[1,0,0,0,0,0,39,40,0,0,0,0,0,0,32,0,2,11,0,0,0,32,13,4,0,0,0,0,9,0,0,0,0,0,0,9],[1,0,0,0,0,0,39,40,0,0,0,0,0,0,30,35,2,16,0,0,36,17,13,37,0,0,4,9,9,32,0,0,26,37,0,26],[40,40,0,0,0,0,2,1,0,0,0,0,0,0,18,20,32,35,0,0,35,23,39,9,0,0,0,0,18,20,0,0,0,0,35,23],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,2,28,1,18,0,0,13,39,23,16,0,0,0,0,39,13,0,0,0,0,28,2] >;
C10.172- 1+4 in GAP, Magma, Sage, TeX
C_{10}._{17}2_-^{1+4} % in TeX
G:=Group("C10.17ES-(2,2)"); // GroupNames label
G:=SmallGroup(320,1301);
// by ID
G=gap.SmallGroup(320,1301);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,219,268,675,297,12550]);
// 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,c*a*c=d*a*d^-1=a^-1,a*e=e*a,c*b*c=b^-1,d*b*d^-1=a^5*b,b*e=e*b,d*c*d^-1=a^5*c,c*e=e*c,e*d*e^-1=b^2*d>;
// generators/relations