?
G = C10.502+ (1+4) order 320 = 26·5
50th non-split extension by C10 of 2+ (1+4) acting via 2+ (1+4)/C2×D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C10.502+ (1+4), C10.752- (1+4), C20⋊Q8⋊23C2, C4⋊C4.94D10, (C2×Dic5)⋊4Q8, C22⋊Q8.8D5, C22.6(Q8×D5), (C2×Q8).74D10, Dic5.4(C2×Q8), (C2×C20).49C23, C22⋊C4.53D10, Dic5⋊Q8⋊12C2, C10.33(C22×Q8), (C2×C10).167C24, (C22×C4).232D10, C4⋊Dic5.47C22, C2.52(D4⋊6D10), Dic5.Q8⋊16C2, C20.48D4.19C2, (Q8×C10).102C22, C23.185(C22×D5), C22.188(C23×D5), C23.D5.31C22, (C22×C10).195C23, (C22×C20).314C22, Dic5.14D4.3C2, C5⋊3(C23.41C23), (C2×Dic5).241C23, (C4×Dic5).109C22, C23.11D10.2C2, C2.33(D4.10D10), (C2×Dic10).164C22, C10.D4.161C22, (C22×Dic5).117C22, C2.16(C2×Q8×D5), (C2×C10).6(C2×Q8), (C5×C22⋊Q8).8C2, (C5×C4⋊C4).153C22, (C2×C4).181(C22×D5), (C2×C10.D4).25C2, (C5×C22⋊C4).22C22, SmallGroup(320,1295)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 622 in 206 conjugacy classes, 103 normal (31 characteristic)
C1, C2 [×3], C2 [×2], C4 [×16], C22, C22 [×2], C22 [×2], C5, C2×C4 [×2], C2×C4 [×4], C2×C4 [×14], Q8 [×4], C23, C10 [×3], C10 [×2], C42 [×4], C22⋊C4 [×2], C22⋊C4 [×2], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×17], C22×C4, C22×C4 [×2], C2×Q8, C2×Q8 [×3], Dic5 [×4], Dic5 [×6], C20 [×6], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×C4⋊C4, C42⋊C2 [×2], C22⋊Q8, C22⋊Q8 [×3], C42.C2 [×4], C4⋊Q8 [×4], Dic10 [×3], C2×Dic5 [×12], C2×Dic5, C2×C20 [×2], C2×C20 [×4], C2×C20, C5×Q8, C22×C10, C23.41C23, C4×Dic5 [×4], C10.D4 [×14], C4⋊Dic5, C4⋊Dic5 [×2], C23.D5 [×2], C5×C22⋊C4 [×2], C5×C4⋊C4, C5×C4⋊C4 [×2], C2×Dic10, C2×Dic10 [×2], C22×Dic5 [×2], C22×C20, Q8×C10, C23.11D10 [×2], Dic5.14D4 [×2], C20⋊Q8 [×2], Dic5.Q8 [×4], C2×C10.D4, C20.48D4, Dic5⋊Q8 [×2], C5×C22⋊Q8, C10.502+ (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), C22×D5 [×7], C23.41C23, Q8×D5 [×2], C23×D5, D4⋊6D10, C2×Q8×D5, D4.10D10, C10.502+ (1+4)
Generators and relations
G = < a,b,c,d,e | a10=b4=e2=1, c2=a5, d2=a5b2, bab-1=a-1, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, dcd-1=a5c, ce=ec, ede=b2d >
►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)
(1 105 30 98)(2 104 21 97)(3 103 22 96)(4 102 23 95)(5 101 24 94)(6 110 25 93)(7 109 26 92)(8 108 27 91)(9 107 28 100)(10 106 29 99)(11 90 159 71)(12 89 160 80)(13 88 151 79)(14 87 152 78)(15 86 153 77)(16 85 154 76)(17 84 155 75)(18 83 156 74)(19 82 157 73)(20 81 158 72)(31 126 49 119)(32 125 50 118)(33 124 41 117)(34 123 42 116)(35 122 43 115)(36 121 44 114)(37 130 45 113)(38 129 46 112)(39 128 47 111)(40 127 48 120)(51 146 69 139)(52 145 70 138)(53 144 61 137)(54 143 62 136)(55 142 63 135)(56 141 64 134)(57 150 65 133)(58 149 66 132)(59 148 67 131)(60 147 68 140)
(1 37 6 32)(2 38 7 33)(3 39 8 34)(4 40 9 35)(5 31 10 36)(11 134 16 139)(12 135 17 140)(13 136 18 131)(14 137 19 132)(15 138 20 133)(21 46 26 41)(22 47 27 42)(23 48 28 43)(24 49 29 44)(25 50 30 45)(51 71 56 76)(52 72 57 77)(53 73 58 78)(54 74 59 79)(55 75 60 80)(61 82 66 87)(62 83 67 88)(63 84 68 89)(64 85 69 90)(65 86 70 81)(91 123 96 128)(92 124 97 129)(93 125 98 130)(94 126 99 121)(95 127 100 122)(101 119 106 114)(102 120 107 115)(103 111 108 116)(104 112 109 117)(105 113 110 118)(141 154 146 159)(142 155 147 160)(143 156 148 151)(144 157 149 152)(145 158 150 153)
(1 65 25 52)(2 66 26 53)(3 67 27 54)(4 68 28 55)(5 69 29 56)(6 70 30 57)(7 61 21 58)(8 62 22 59)(9 63 23 60)(10 64 24 51)(11 126 154 114)(12 127 155 115)(13 128 156 116)(14 129 157 117)(15 130 158 118)(16 121 159 119)(17 122 160 120)(18 123 151 111)(19 124 152 112)(20 125 153 113)(31 85 44 71)(32 86 45 72)(33 87 46 73)(34 88 47 74)(35 89 48 75)(36 90 49 76)(37 81 50 77)(38 82 41 78)(39 83 42 79)(40 84 43 80)(91 143 103 131)(92 144 104 132)(93 145 105 133)(94 146 106 134)(95 147 107 135)(96 148 108 136)(97 149 109 137)(98 150 110 138)(99 141 101 139)(100 142 102 140)
(11 159)(12 160)(13 151)(14 152)(15 153)(16 154)(17 155)(18 156)(19 157)(20 158)(51 69)(52 70)(53 61)(54 62)(55 63)(56 64)(57 65)(58 66)(59 67)(60 68)(71 90)(72 81)(73 82)(74 83)(75 84)(76 85)(77 86)(78 87)(79 88)(80 89)(131 148)(132 149)(133 150)(134 141)(135 142)(136 143)(137 144)(138 145)(139 146)(140 147)
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), (1,105,30,98)(2,104,21,97)(3,103,22,96)(4,102,23,95)(5,101,24,94)(6,110,25,93)(7,109,26,92)(8,108,27,91)(9,107,28,100)(10,106,29,99)(11,90,159,71)(12,89,160,80)(13,88,151,79)(14,87,152,78)(15,86,153,77)(16,85,154,76)(17,84,155,75)(18,83,156,74)(19,82,157,73)(20,81,158,72)(31,126,49,119)(32,125,50,118)(33,124,41,117)(34,123,42,116)(35,122,43,115)(36,121,44,114)(37,130,45,113)(38,129,46,112)(39,128,47,111)(40,127,48,120)(51,146,69,139)(52,145,70,138)(53,144,61,137)(54,143,62,136)(55,142,63,135)(56,141,64,134)(57,150,65,133)(58,149,66,132)(59,148,67,131)(60,147,68,140), (1,37,6,32)(2,38,7,33)(3,39,8,34)(4,40,9,35)(5,31,10,36)(11,134,16,139)(12,135,17,140)(13,136,18,131)(14,137,19,132)(15,138,20,133)(21,46,26,41)(22,47,27,42)(23,48,28,43)(24,49,29,44)(25,50,30,45)(51,71,56,76)(52,72,57,77)(53,73,58,78)(54,74,59,79)(55,75,60,80)(61,82,66,87)(62,83,67,88)(63,84,68,89)(64,85,69,90)(65,86,70,81)(91,123,96,128)(92,124,97,129)(93,125,98,130)(94,126,99,121)(95,127,100,122)(101,119,106,114)(102,120,107,115)(103,111,108,116)(104,112,109,117)(105,113,110,118)(141,154,146,159)(142,155,147,160)(143,156,148,151)(144,157,149,152)(145,158,150,153), (1,65,25,52)(2,66,26,53)(3,67,27,54)(4,68,28,55)(5,69,29,56)(6,70,30,57)(7,61,21,58)(8,62,22,59)(9,63,23,60)(10,64,24,51)(11,126,154,114)(12,127,155,115)(13,128,156,116)(14,129,157,117)(15,130,158,118)(16,121,159,119)(17,122,160,120)(18,123,151,111)(19,124,152,112)(20,125,153,113)(31,85,44,71)(32,86,45,72)(33,87,46,73)(34,88,47,74)(35,89,48,75)(36,90,49,76)(37,81,50,77)(38,82,41,78)(39,83,42,79)(40,84,43,80)(91,143,103,131)(92,144,104,132)(93,145,105,133)(94,146,106,134)(95,147,107,135)(96,148,108,136)(97,149,109,137)(98,150,110,138)(99,141,101,139)(100,142,102,140), (11,159)(12,160)(13,151)(14,152)(15,153)(16,154)(17,155)(18,156)(19,157)(20,158)(51,69)(52,70)(53,61)(54,62)(55,63)(56,64)(57,65)(58,66)(59,67)(60,68)(71,90)(72,81)(73,82)(74,83)(75,84)(76,85)(77,86)(78,87)(79,88)(80,89)(131,148)(132,149)(133,150)(134,141)(135,142)(136,143)(137,144)(138,145)(139,146)(140,147)>;
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), (1,105,30,98)(2,104,21,97)(3,103,22,96)(4,102,23,95)(5,101,24,94)(6,110,25,93)(7,109,26,92)(8,108,27,91)(9,107,28,100)(10,106,29,99)(11,90,159,71)(12,89,160,80)(13,88,151,79)(14,87,152,78)(15,86,153,77)(16,85,154,76)(17,84,155,75)(18,83,156,74)(19,82,157,73)(20,81,158,72)(31,126,49,119)(32,125,50,118)(33,124,41,117)(34,123,42,116)(35,122,43,115)(36,121,44,114)(37,130,45,113)(38,129,46,112)(39,128,47,111)(40,127,48,120)(51,146,69,139)(52,145,70,138)(53,144,61,137)(54,143,62,136)(55,142,63,135)(56,141,64,134)(57,150,65,133)(58,149,66,132)(59,148,67,131)(60,147,68,140), (1,37,6,32)(2,38,7,33)(3,39,8,34)(4,40,9,35)(5,31,10,36)(11,134,16,139)(12,135,17,140)(13,136,18,131)(14,137,19,132)(15,138,20,133)(21,46,26,41)(22,47,27,42)(23,48,28,43)(24,49,29,44)(25,50,30,45)(51,71,56,76)(52,72,57,77)(53,73,58,78)(54,74,59,79)(55,75,60,80)(61,82,66,87)(62,83,67,88)(63,84,68,89)(64,85,69,90)(65,86,70,81)(91,123,96,128)(92,124,97,129)(93,125,98,130)(94,126,99,121)(95,127,100,122)(101,119,106,114)(102,120,107,115)(103,111,108,116)(104,112,109,117)(105,113,110,118)(141,154,146,159)(142,155,147,160)(143,156,148,151)(144,157,149,152)(145,158,150,153), (1,65,25,52)(2,66,26,53)(3,67,27,54)(4,68,28,55)(5,69,29,56)(6,70,30,57)(7,61,21,58)(8,62,22,59)(9,63,23,60)(10,64,24,51)(11,126,154,114)(12,127,155,115)(13,128,156,116)(14,129,157,117)(15,130,158,118)(16,121,159,119)(17,122,160,120)(18,123,151,111)(19,124,152,112)(20,125,153,113)(31,85,44,71)(32,86,45,72)(33,87,46,73)(34,88,47,74)(35,89,48,75)(36,90,49,76)(37,81,50,77)(38,82,41,78)(39,83,42,79)(40,84,43,80)(91,143,103,131)(92,144,104,132)(93,145,105,133)(94,146,106,134)(95,147,107,135)(96,148,108,136)(97,149,109,137)(98,150,110,138)(99,141,101,139)(100,142,102,140), (11,159)(12,160)(13,151)(14,152)(15,153)(16,154)(17,155)(18,156)(19,157)(20,158)(51,69)(52,70)(53,61)(54,62)(55,63)(56,64)(57,65)(58,66)(59,67)(60,68)(71,90)(72,81)(73,82)(74,83)(75,84)(76,85)(77,86)(78,87)(79,88)(80,89)(131,148)(132,149)(133,150)(134,141)(135,142)(136,143)(137,144)(138,145)(139,146)(140,147) );
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)], [(1,105,30,98),(2,104,21,97),(3,103,22,96),(4,102,23,95),(5,101,24,94),(6,110,25,93),(7,109,26,92),(8,108,27,91),(9,107,28,100),(10,106,29,99),(11,90,159,71),(12,89,160,80),(13,88,151,79),(14,87,152,78),(15,86,153,77),(16,85,154,76),(17,84,155,75),(18,83,156,74),(19,82,157,73),(20,81,158,72),(31,126,49,119),(32,125,50,118),(33,124,41,117),(34,123,42,116),(35,122,43,115),(36,121,44,114),(37,130,45,113),(38,129,46,112),(39,128,47,111),(40,127,48,120),(51,146,69,139),(52,145,70,138),(53,144,61,137),(54,143,62,136),(55,142,63,135),(56,141,64,134),(57,150,65,133),(58,149,66,132),(59,148,67,131),(60,147,68,140)], [(1,37,6,32),(2,38,7,33),(3,39,8,34),(4,40,9,35),(5,31,10,36),(11,134,16,139),(12,135,17,140),(13,136,18,131),(14,137,19,132),(15,138,20,133),(21,46,26,41),(22,47,27,42),(23,48,28,43),(24,49,29,44),(25,50,30,45),(51,71,56,76),(52,72,57,77),(53,73,58,78),(54,74,59,79),(55,75,60,80),(61,82,66,87),(62,83,67,88),(63,84,68,89),(64,85,69,90),(65,86,70,81),(91,123,96,128),(92,124,97,129),(93,125,98,130),(94,126,99,121),(95,127,100,122),(101,119,106,114),(102,120,107,115),(103,111,108,116),(104,112,109,117),(105,113,110,118),(141,154,146,159),(142,155,147,160),(143,156,148,151),(144,157,149,152),(145,158,150,153)], [(1,65,25,52),(2,66,26,53),(3,67,27,54),(4,68,28,55),(5,69,29,56),(6,70,30,57),(7,61,21,58),(8,62,22,59),(9,63,23,60),(10,64,24,51),(11,126,154,114),(12,127,155,115),(13,128,156,116),(14,129,157,117),(15,130,158,118),(16,121,159,119),(17,122,160,120),(18,123,151,111),(19,124,152,112),(20,125,153,113),(31,85,44,71),(32,86,45,72),(33,87,46,73),(34,88,47,74),(35,89,48,75),(36,90,49,76),(37,81,50,77),(38,82,41,78),(39,83,42,79),(40,84,43,80),(91,143,103,131),(92,144,104,132),(93,145,105,133),(94,146,106,134),(95,147,107,135),(96,148,108,136),(97,149,109,137),(98,150,110,138),(99,141,101,139),(100,142,102,140)], [(11,159),(12,160),(13,151),(14,152),(15,153),(16,154),(17,155),(18,156),(19,157),(20,158),(51,69),(52,70),(53,61),(54,62),(55,63),(56,64),(57,65),(58,66),(59,67),(60,68),(71,90),(72,81),(73,82),(74,83),(75,84),(76,85),(77,86),(78,87),(79,88),(80,89),(131,148),(132,149),(133,150),(134,141),(135,142),(136,143),(137,144),(138,145),(139,146),(140,147)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 34 | 0 | 0 |
| 0 | 0 | 7 | 34 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 34 |
| 0 | 0 | 0 | 0 | 7 | 34 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 24 | 40 | 0 | 0 |
| 0 | 0 | 3 | 17 | 0 | 0 |
| 0 | 0 | 2 | 0 | 24 | 40 |
| 0 | 0 | 14 | 39 | 3 | 17 |
| 1 | 37 | 0 | 0 | 0 | 0 |
| 21 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 30 | 9 | 0 | 0 |
| 0 | 0 | 32 | 11 | 0 | 0 |
| 0 | 0 | 27 | 15 | 11 | 32 |
| 0 | 0 | 26 | 9 | 9 | 30 |
| 32 | 36 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 22 | 2 | 39 | 0 |
| 0 | 0 | 39 | 36 | 0 | 39 |
| 0 | 0 | 14 | 17 | 19 | 39 |
| 0 | 0 | 24 | 10 | 2 | 5 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 22 | 2 | 40 | 0 |
| 0 | 0 | 39 | 36 | 0 | 40 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,7,0,0,0,0,34,34,0,0,0,0,0,0,1,7,0,0,0,0,34,34],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,24,3,2,14,0,0,40,17,0,39,0,0,0,0,24,3,0,0,0,0,40,17],[1,21,0,0,0,0,37,40,0,0,0,0,0,0,30,32,27,26,0,0,9,11,15,9,0,0,0,0,11,9,0,0,0,0,32,30],[32,0,0,0,0,0,36,9,0,0,0,0,0,0,22,39,14,24,0,0,2,36,17,10,0,0,39,0,19,2,0,0,0,39,39,5],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,22,39,0,0,0,1,2,36,0,0,0,0,40,0,0,0,0,0,0,40] >;
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | ··· | 4P | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | ··· | 20P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 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 | 4 | ··· | 4 | 10 | 10 | 10 | 10 | 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 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | + | + | + | + | - | - | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | Q8 | D5 | D10 | D10 | D10 | D10 | 2+ (1+4) | 2- (1+4) | Q8×D5 | D4⋊6D10 | D4.10D10 |
| kernel | C10.502+ (1+4) | C23.11D10 | Dic5.14D4 | C20⋊Q8 | Dic5.Q8 | C2×C10.D4 | C20.48D4 | Dic5⋊Q8 | C5×C22⋊Q8 | C2×Dic5 | C22⋊Q8 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×Q8 | C10 | C10 | C22 | C2 | C2 |
| # reps | 1 | 2 | 2 | 2 | 4 | 1 | 1 | 2 | 1 | 4 | 2 | 4 | 6 | 2 | 2 | 1 | 1 | 4 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_{10}._{50}2_+^{(1+4)} % in TeX
G:=Group("C10.50ES+(2,2)"); // GroupNames label
G:=SmallGroup(320,1295);
// by ID
G=gap.SmallGroup(320,1295);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,387,100,1123,185,80,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^10=b^4=e^2=1,c^2=a^5,d^2=a^5*b^2,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,d*c*d^-1=a^5*c,c*e=e*c,e*d*e=b^2*d>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀