G = C10.72+ 1+4 order 320 = 26·5
7th non-split extension by C10 of 2+ 1+4 acting via 2+ 1+4/C2×D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C10.72+ 1+4, C10.12- 1+4, C20⋊Q8⋊8C2, (C2×C20)⋊3Q8, (C2×C4)⋊3Dic10, C20.67(C2×Q8), C4⋊C4.261D10, C4.Dic10⋊8C2, C10.9(C22×Q8), (C2×C10).44C24, C4.32(C2×Dic10), (C2×C20).135C23, C20.48D4.6C2, (C22×C4).175D10, C2.11(D4⋊6D10), C22.82(C23×D5), C4⋊Dic5.358C22, (C22×C20).74C22, (C2×Dic5).14C23, (C4×Dic5).70C22, C2.11(C22×Dic10), C22.11(C2×Dic10), C10.D4.1C22, C23.223(C22×D5), C23.D5.86C22, (C22×C10).393C23, C2.5(Q8.10D10), C5⋊1(C23.41C23), (C2×Dic10).23C22, C23.21D10.20C2, (C2×C4⋊C4).27D5, (C10×C4⋊C4).20C2, (C2×C10).52(C2×Q8), (C5×C4⋊C4).293C22, (C2×C4).570(C22×D5), SmallGroup(320,1172)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10.72+ 1+4
G = < a,b,c,d,e | a10=b4=1, c2=e2=a5, d2=b2, bab-1=dad-1=eae-1=a-1, ac=ca, cbc-1=b-1, dbd-1=a5b, be=eb, dcd-1=ece-1=a5c, ede-1=b2d >
Subgroups: 606 in 206 conjugacy classes, 111 normal (17 characteristic)
C1, C2 [×3], C2 [×2], C4 [×4], C4 [×12], C22, C22 [×2], C22 [×2], C5, C2×C4 [×10], C2×C4 [×10], Q8 [×4], C23, C10 [×3], C10 [×2], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×4], C4⋊C4 [×16], C22×C4, C22×C4 [×2], C2×Q8 [×4], Dic5 [×8], C20 [×4], C20 [×4], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×C4⋊C4, C42⋊C2 [×2], C22⋊Q8 [×4], C42.C2 [×4], C4⋊Q8 [×4], Dic10 [×4], C2×Dic5 [×8], C2×C20 [×10], C2×C20 [×2], C22×C10, C23.41C23, C4×Dic5 [×4], C10.D4 [×8], C4⋊Dic5 [×8], C23.D5 [×4], C5×C4⋊C4 [×4], C2×Dic10 [×4], C22×C20, C22×C20 [×2], C20⋊Q8 [×4], C4.Dic10 [×4], C20.48D4 [×4], C23.21D10 [×2], C10×C4⋊C4, C10.72+ 1+4
Quotients: C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], D5, C2×Q8 [×6], C24, D10 [×7], C22×Q8, 2+ 1+4, 2- 1+4, Dic10 [×4], C22×D5 [×7], C23.41C23, C2×Dic10 [×6], C23×D5, C22×Dic10, D4⋊6D10, Q8.10D10, C10.72+ 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)
(1 159 25 20)(2 158 26 19)(3 157 27 18)(4 156 28 17)(5 155 29 16)(6 154 30 15)(7 153 21 14)(8 152 22 13)(9 151 23 12)(10 160 24 11)(31 139 50 145)(32 138 41 144)(33 137 42 143)(34 136 43 142)(35 135 44 141)(36 134 45 150)(37 133 46 149)(38 132 47 148)(39 131 48 147)(40 140 49 146)(51 114 70 130)(52 113 61 129)(53 112 62 128)(54 111 63 127)(55 120 64 126)(56 119 65 125)(57 118 66 124)(58 117 67 123)(59 116 68 122)(60 115 69 121)(71 99 81 105)(72 98 82 104)(73 97 83 103)(74 96 84 102)(75 95 85 101)(76 94 86 110)(77 93 87 109)(78 92 88 108)(79 91 89 107)(80 100 90 106)
(1 31 6 36)(2 32 7 37)(3 33 8 38)(4 34 9 39)(5 35 10 40)(11 140 16 135)(12 131 17 136)(13 132 18 137)(14 133 19 138)(15 134 20 139)(21 46 26 41)(22 47 27 42)(23 48 28 43)(24 49 29 44)(25 50 30 45)(51 81 56 86)(52 82 57 87)(53 83 58 88)(54 84 59 89)(55 85 60 90)(61 72 66 77)(62 73 67 78)(63 74 68 79)(64 75 69 80)(65 76 70 71)(91 111 96 116)(92 112 97 117)(93 113 98 118)(94 114 99 119)(95 115 100 120)(101 121 106 126)(102 122 107 127)(103 123 108 128)(104 124 109 129)(105 125 110 130)(141 160 146 155)(142 151 147 156)(143 152 148 157)(144 153 149 158)(145 154 150 159)
(1 99 25 105)(2 98 26 104)(3 97 27 103)(4 96 28 102)(5 95 29 101)(6 94 30 110)(7 93 21 109)(8 92 22 108)(9 91 23 107)(10 100 24 106)(11 75 160 85)(12 74 151 84)(13 73 152 83)(14 72 153 82)(15 71 154 81)(16 80 155 90)(17 79 156 89)(18 78 157 88)(19 77 158 87)(20 76 159 86)(31 114 50 130)(32 113 41 129)(33 112 42 128)(34 111 43 127)(35 120 44 126)(36 119 45 125)(37 118 46 124)(38 117 47 123)(39 116 48 122)(40 115 49 121)(51 134 70 150)(52 133 61 149)(53 132 62 148)(54 131 63 147)(55 140 64 146)(56 139 65 145)(57 138 66 144)(58 137 67 143)(59 136 68 142)(60 135 69 141)
(1 119 6 114)(2 118 7 113)(3 117 8 112)(4 116 9 111)(5 115 10 120)(11 55 16 60)(12 54 17 59)(13 53 18 58)(14 52 19 57)(15 51 20 56)(21 129 26 124)(22 128 27 123)(23 127 28 122)(24 126 29 121)(25 125 30 130)(31 99 36 94)(32 98 37 93)(33 97 38 92)(34 96 39 91)(35 95 40 100)(41 104 46 109)(42 103 47 108)(43 102 48 107)(44 101 49 106)(45 110 50 105)(61 158 66 153)(62 157 67 152)(63 156 68 151)(64 155 69 160)(65 154 70 159)(71 150 76 145)(72 149 77 144)(73 148 78 143)(74 147 79 142)(75 146 80 141)(81 134 86 139)(82 133 87 138)(83 132 88 137)(84 131 89 136)(85 140 90 135)
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), (1,159,25,20)(2,158,26,19)(3,157,27,18)(4,156,28,17)(5,155,29,16)(6,154,30,15)(7,153,21,14)(8,152,22,13)(9,151,23,12)(10,160,24,11)(31,139,50,145)(32,138,41,144)(33,137,42,143)(34,136,43,142)(35,135,44,141)(36,134,45,150)(37,133,46,149)(38,132,47,148)(39,131,48,147)(40,140,49,146)(51,114,70,130)(52,113,61,129)(53,112,62,128)(54,111,63,127)(55,120,64,126)(56,119,65,125)(57,118,66,124)(58,117,67,123)(59,116,68,122)(60,115,69,121)(71,99,81,105)(72,98,82,104)(73,97,83,103)(74,96,84,102)(75,95,85,101)(76,94,86,110)(77,93,87,109)(78,92,88,108)(79,91,89,107)(80,100,90,106), (1,31,6,36)(2,32,7,37)(3,33,8,38)(4,34,9,39)(5,35,10,40)(11,140,16,135)(12,131,17,136)(13,132,18,137)(14,133,19,138)(15,134,20,139)(21,46,26,41)(22,47,27,42)(23,48,28,43)(24,49,29,44)(25,50,30,45)(51,81,56,86)(52,82,57,87)(53,83,58,88)(54,84,59,89)(55,85,60,90)(61,72,66,77)(62,73,67,78)(63,74,68,79)(64,75,69,80)(65,76,70,71)(91,111,96,116)(92,112,97,117)(93,113,98,118)(94,114,99,119)(95,115,100,120)(101,121,106,126)(102,122,107,127)(103,123,108,128)(104,124,109,129)(105,125,110,130)(141,160,146,155)(142,151,147,156)(143,152,148,157)(144,153,149,158)(145,154,150,159), (1,99,25,105)(2,98,26,104)(3,97,27,103)(4,96,28,102)(5,95,29,101)(6,94,30,110)(7,93,21,109)(8,92,22,108)(9,91,23,107)(10,100,24,106)(11,75,160,85)(12,74,151,84)(13,73,152,83)(14,72,153,82)(15,71,154,81)(16,80,155,90)(17,79,156,89)(18,78,157,88)(19,77,158,87)(20,76,159,86)(31,114,50,130)(32,113,41,129)(33,112,42,128)(34,111,43,127)(35,120,44,126)(36,119,45,125)(37,118,46,124)(38,117,47,123)(39,116,48,122)(40,115,49,121)(51,134,70,150)(52,133,61,149)(53,132,62,148)(54,131,63,147)(55,140,64,146)(56,139,65,145)(57,138,66,144)(58,137,67,143)(59,136,68,142)(60,135,69,141), (1,119,6,114)(2,118,7,113)(3,117,8,112)(4,116,9,111)(5,115,10,120)(11,55,16,60)(12,54,17,59)(13,53,18,58)(14,52,19,57)(15,51,20,56)(21,129,26,124)(22,128,27,123)(23,127,28,122)(24,126,29,121)(25,125,30,130)(31,99,36,94)(32,98,37,93)(33,97,38,92)(34,96,39,91)(35,95,40,100)(41,104,46,109)(42,103,47,108)(43,102,48,107)(44,101,49,106)(45,110,50,105)(61,158,66,153)(62,157,67,152)(63,156,68,151)(64,155,69,160)(65,154,70,159)(71,150,76,145)(72,149,77,144)(73,148,78,143)(74,147,79,142)(75,146,80,141)(81,134,86,139)(82,133,87,138)(83,132,88,137)(84,131,89,136)(85,140,90,135)>;
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), (1,159,25,20)(2,158,26,19)(3,157,27,18)(4,156,28,17)(5,155,29,16)(6,154,30,15)(7,153,21,14)(8,152,22,13)(9,151,23,12)(10,160,24,11)(31,139,50,145)(32,138,41,144)(33,137,42,143)(34,136,43,142)(35,135,44,141)(36,134,45,150)(37,133,46,149)(38,132,47,148)(39,131,48,147)(40,140,49,146)(51,114,70,130)(52,113,61,129)(53,112,62,128)(54,111,63,127)(55,120,64,126)(56,119,65,125)(57,118,66,124)(58,117,67,123)(59,116,68,122)(60,115,69,121)(71,99,81,105)(72,98,82,104)(73,97,83,103)(74,96,84,102)(75,95,85,101)(76,94,86,110)(77,93,87,109)(78,92,88,108)(79,91,89,107)(80,100,90,106), (1,31,6,36)(2,32,7,37)(3,33,8,38)(4,34,9,39)(5,35,10,40)(11,140,16,135)(12,131,17,136)(13,132,18,137)(14,133,19,138)(15,134,20,139)(21,46,26,41)(22,47,27,42)(23,48,28,43)(24,49,29,44)(25,50,30,45)(51,81,56,86)(52,82,57,87)(53,83,58,88)(54,84,59,89)(55,85,60,90)(61,72,66,77)(62,73,67,78)(63,74,68,79)(64,75,69,80)(65,76,70,71)(91,111,96,116)(92,112,97,117)(93,113,98,118)(94,114,99,119)(95,115,100,120)(101,121,106,126)(102,122,107,127)(103,123,108,128)(104,124,109,129)(105,125,110,130)(141,160,146,155)(142,151,147,156)(143,152,148,157)(144,153,149,158)(145,154,150,159), (1,99,25,105)(2,98,26,104)(3,97,27,103)(4,96,28,102)(5,95,29,101)(6,94,30,110)(7,93,21,109)(8,92,22,108)(9,91,23,107)(10,100,24,106)(11,75,160,85)(12,74,151,84)(13,73,152,83)(14,72,153,82)(15,71,154,81)(16,80,155,90)(17,79,156,89)(18,78,157,88)(19,77,158,87)(20,76,159,86)(31,114,50,130)(32,113,41,129)(33,112,42,128)(34,111,43,127)(35,120,44,126)(36,119,45,125)(37,118,46,124)(38,117,47,123)(39,116,48,122)(40,115,49,121)(51,134,70,150)(52,133,61,149)(53,132,62,148)(54,131,63,147)(55,140,64,146)(56,139,65,145)(57,138,66,144)(58,137,67,143)(59,136,68,142)(60,135,69,141), (1,119,6,114)(2,118,7,113)(3,117,8,112)(4,116,9,111)(5,115,10,120)(11,55,16,60)(12,54,17,59)(13,53,18,58)(14,52,19,57)(15,51,20,56)(21,129,26,124)(22,128,27,123)(23,127,28,122)(24,126,29,121)(25,125,30,130)(31,99,36,94)(32,98,37,93)(33,97,38,92)(34,96,39,91)(35,95,40,100)(41,104,46,109)(42,103,47,108)(43,102,48,107)(44,101,49,106)(45,110,50,105)(61,158,66,153)(62,157,67,152)(63,156,68,151)(64,155,69,160)(65,154,70,159)(71,150,76,145)(72,149,77,144)(73,148,78,143)(74,147,79,142)(75,146,80,141)(81,134,86,139)(82,133,87,138)(83,132,88,137)(84,131,89,136)(85,140,90,135) );
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)], [(1,159,25,20),(2,158,26,19),(3,157,27,18),(4,156,28,17),(5,155,29,16),(6,154,30,15),(7,153,21,14),(8,152,22,13),(9,151,23,12),(10,160,24,11),(31,139,50,145),(32,138,41,144),(33,137,42,143),(34,136,43,142),(35,135,44,141),(36,134,45,150),(37,133,46,149),(38,132,47,148),(39,131,48,147),(40,140,49,146),(51,114,70,130),(52,113,61,129),(53,112,62,128),(54,111,63,127),(55,120,64,126),(56,119,65,125),(57,118,66,124),(58,117,67,123),(59,116,68,122),(60,115,69,121),(71,99,81,105),(72,98,82,104),(73,97,83,103),(74,96,84,102),(75,95,85,101),(76,94,86,110),(77,93,87,109),(78,92,88,108),(79,91,89,107),(80,100,90,106)], [(1,31,6,36),(2,32,7,37),(3,33,8,38),(4,34,9,39),(5,35,10,40),(11,140,16,135),(12,131,17,136),(13,132,18,137),(14,133,19,138),(15,134,20,139),(21,46,26,41),(22,47,27,42),(23,48,28,43),(24,49,29,44),(25,50,30,45),(51,81,56,86),(52,82,57,87),(53,83,58,88),(54,84,59,89),(55,85,60,90),(61,72,66,77),(62,73,67,78),(63,74,68,79),(64,75,69,80),(65,76,70,71),(91,111,96,116),(92,112,97,117),(93,113,98,118),(94,114,99,119),(95,115,100,120),(101,121,106,126),(102,122,107,127),(103,123,108,128),(104,124,109,129),(105,125,110,130),(141,160,146,155),(142,151,147,156),(143,152,148,157),(144,153,149,158),(145,154,150,159)], [(1,99,25,105),(2,98,26,104),(3,97,27,103),(4,96,28,102),(5,95,29,101),(6,94,30,110),(7,93,21,109),(8,92,22,108),(9,91,23,107),(10,100,24,106),(11,75,160,85),(12,74,151,84),(13,73,152,83),(14,72,153,82),(15,71,154,81),(16,80,155,90),(17,79,156,89),(18,78,157,88),(19,77,158,87),(20,76,159,86),(31,114,50,130),(32,113,41,129),(33,112,42,128),(34,111,43,127),(35,120,44,126),(36,119,45,125),(37,118,46,124),(38,117,47,123),(39,116,48,122),(40,115,49,121),(51,134,70,150),(52,133,61,149),(53,132,62,148),(54,131,63,147),(55,140,64,146),(56,139,65,145),(57,138,66,144),(58,137,67,143),(59,136,68,142),(60,135,69,141)], [(1,119,6,114),(2,118,7,113),(3,117,8,112),(4,116,9,111),(5,115,10,120),(11,55,16,60),(12,54,17,59),(13,53,18,58),(14,52,19,57),(15,51,20,56),(21,129,26,124),(22,128,27,123),(23,127,28,122),(24,126,29,121),(25,125,30,130),(31,99,36,94),(32,98,37,93),(33,97,38,92),(34,96,39,91),(35,95,40,100),(41,104,46,109),(42,103,47,108),(43,102,48,107),(44,101,49,106),(45,110,50,105),(61,158,66,153),(62,157,67,152),(63,156,68,151),(64,155,69,160),(65,154,70,159),(71,150,76,145),(72,149,77,144),(73,148,78,143),(74,147,79,142),(75,146,80,141),(81,134,86,139),(82,133,87,138),(83,132,88,137),(84,131,89,136),(85,140,90,135)])
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4P | 5A | 5B | 10A | ··· | 10N | 20A | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | + | - | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | Q8 | D5 | D10 | D10 | Dic10 | 2+ 1+4 | 2- 1+4 | D4⋊6D10 | Q8.10D10 |
| kernel | C10.72+ 1+4 | C20⋊Q8 | C4.Dic10 | C20.48D4 | C23.21D10 | C10×C4⋊C4 | C2×C20 | C2×C4⋊C4 | C4⋊C4 | C22×C4 | C2×C4 | C10 | C10 | C2 | C2 |
| # reps | 1 | 4 | 4 | 4 | 2 | 1 | 4 | 2 | 8 | 6 | 16 | 1 | 1 | 4 | 4 |
Matrix representation of C10.72+ 1+4 ►in GL6(𝔽41)
| 25 | 0 | 0 | 0 | 0 | 0 |
| 36 | 23 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 39 | 0 | 31 | 0 |
| 0 | 0 | 23 | 0 | 0 | 31 |
| 33 | 5 | 0 | 0 | 0 | 0 |
| 28 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 6 | 1 |
| 0 | 0 | 0 | 0 | 34 | 19 |
| 0 | 0 | 4 | 21 | 5 | 35 |
| 0 | 0 | 36 | 38 | 4 | 28 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 37 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 39 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 40 |
| 0 | 0 | 0 | 26 | 1 | 0 |
| 10 | 4 | 0 | 0 | 0 | 0 |
| 26 | 31 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 2 | 0 |
| 0 | 0 | 15 | 0 | 32 | 40 |
| 0 | 0 | 31 | 0 | 29 | 0 |
| 0 | 0 | 24 | 40 | 15 | 0 |
| 33 | 5 | 0 | 0 | 0 | 0 |
| 28 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 15 | 0 | 0 | 39 |
| 0 | 0 | 29 | 0 | 40 | 9 |
| 0 | 0 | 17 | 1 | 0 | 12 |
| 0 | 0 | 31 | 0 | 0 | 26 |
G:=sub<GL(6,GF(41))| [25,36,0,0,0,0,0,23,0,0,0,0,0,0,4,0,39,23,0,0,0,4,0,0,0,0,0,0,31,0,0,0,0,0,0,31],[33,28,0,0,0,0,5,8,0,0,0,0,0,0,8,0,4,36,0,0,0,0,21,38,0,0,6,34,5,4,0,0,1,19,35,28],[9,37,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,39,9,12,26,0,0,0,0,0,1,0,0,0,0,40,0],[10,26,0,0,0,0,4,31,0,0,0,0,0,0,12,15,31,24,0,0,0,0,0,40,0,0,2,32,29,15,0,0,0,40,0,0],[33,28,0,0,0,0,5,8,0,0,0,0,0,0,15,29,17,31,0,0,0,0,1,0,0,0,0,40,0,0,0,0,39,9,12,26] >;
C10.72+ 1+4 in GAP, Magma, Sage, TeX
C_{10}._72_+^{1+4} % in TeX
G:=Group("C10.7ES+(2,2)"); // GroupNames label
G:=SmallGroup(320,1172);
// by ID
G=gap.SmallGroup(320,1172);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,232,758,675,570,80,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^10=b^4=1,c^2=e^2=a^5,d^2=b^2,b*a*b^-1=d*a*d^-1=e*a*e^-1=a^-1,a*c=c*a,c*b*c^-1=b^-1,d*b*d^-1=a^5*b,b*e=e*b,d*c*d^-1=e*c*e^-1=a^5*c,e*d*e^-1=b^2*d>;
// generators/relations