?
G = C10.422- (1+4) order 320 = 26·5
42nd non-split extension by C10 of 2- (1+4) acting via 2- (1+4)/C2×Q8=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C10.422- (1+4), (Q8×C10)⋊22C4, (C2×Q8)⋊9Dic5, (Q8×Dic5)⋊25C2, Q8.8(C2×Dic5), (C22×Q8).9D5, C10.68(C23×C4), (C2×Q8).207D10, C2.9(C23×Dic5), (C2×C10).304C24, C20.155(C22×C4), (C2×C20).551C23, (C22×C4).276D10, C4.19(C22×Dic5), C22.47(C23×D5), C4⋊Dic5.389C22, (Q8×C10).233C22, C23.237(C22×D5), (C22×C10).422C23, (C22×C20).284C22, C2.4(Q8.10D10), C5⋊4(C23.32C23), (C4×Dic5).177C22, (C2×Dic5).298C23, C23.D5.145C22, C22.10(C22×Dic5), C23.21D10.25C2, (Q8×C2×C10).9C2, (C5×Q8).41(C2×C4), (C2×C20).307(C2×C4), (C2×C4).30(C2×Dic5), (C2×C4).632(C22×D5), (C2×C10).311(C22×C4), SmallGroup(320,1484)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 558 in 266 conjugacy classes, 191 normal (11 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×12], C4 [×8], C22, C22 [×2], C22 [×2], C5, C2×C4 [×18], C2×C4 [×8], Q8 [×16], C23, C10, C10 [×2], C10 [×2], C42 [×12], C22⋊C4 [×4], C4⋊C4 [×12], C22×C4 [×3], C2×Q8 [×12], Dic5 [×8], C20 [×12], C2×C10, C2×C10 [×2], C2×C10 [×2], C42⋊C2 [×6], C4×Q8 [×8], C22×Q8, C2×Dic5 [×8], C2×C20 [×18], C5×Q8 [×16], C22×C10, C23.32C23, C4×Dic5 [×12], C4⋊Dic5 [×12], C23.D5 [×4], C22×C20 [×3], Q8×C10 [×12], C23.21D10 [×6], Q8×Dic5 [×8], Q8×C2×C10, C10.422- (1+4)
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], D5, C22×C4 [×14], C24, Dic5 [×8], D10 [×7], C23×C4, 2- (1+4) [×2], C2×Dic5 [×28], C22×D5 [×7], C23.32C23, C22×Dic5 [×14], C23×D5, Q8.10D10 [×2], C23×Dic5, C10.422- (1+4)
Generators and relations
G = < a,b,c,d,e | a10=b4=1, c2=a5, d2=a5b2, e2=b2, ab=ba, cac-1=dad-1=a-1, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=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 65 30 57)(2 66 21 58)(3 67 22 59)(4 68 23 60)(5 69 24 51)(6 70 25 52)(7 61 26 53)(8 62 27 54)(9 63 28 55)(10 64 29 56)(11 114 159 121)(12 115 160 122)(13 116 151 123)(14 117 152 124)(15 118 153 125)(16 119 154 126)(17 120 155 127)(18 111 156 128)(19 112 157 129)(20 113 158 130)(31 85 49 76)(32 86 50 77)(33 87 41 78)(34 88 42 79)(35 89 43 80)(36 90 44 71)(37 81 45 72)(38 82 46 73)(39 83 47 74)(40 84 48 75)(91 136 108 143)(92 137 109 144)(93 138 110 145)(94 139 101 146)(95 140 102 147)(96 131 103 148)(97 132 104 149)(98 133 105 150)(99 134 106 141)(100 135 107 142)
(1 150 6 145)(2 149 7 144)(3 148 8 143)(4 147 9 142)(5 146 10 141)(11 31 16 36)(12 40 17 35)(13 39 18 34)(14 38 19 33)(15 37 20 32)(21 132 26 137)(22 131 27 136)(23 140 28 135)(24 139 29 134)(25 138 30 133)(41 152 46 157)(42 151 47 156)(43 160 48 155)(44 159 49 154)(45 158 50 153)(51 94 56 99)(52 93 57 98)(53 92 58 97)(54 91 59 96)(55 100 60 95)(61 109 66 104)(62 108 67 103)(63 107 68 102)(64 106 69 101)(65 105 70 110)(71 114 76 119)(72 113 77 118)(73 112 78 117)(74 111 79 116)(75 120 80 115)(81 130 86 125)(82 129 87 124)(83 128 88 123)(84 127 89 122)(85 126 90 121)
(1 113 25 125)(2 112 26 124)(3 111 27 123)(4 120 28 122)(5 119 29 121)(6 118 30 130)(7 117 21 129)(8 116 22 128)(9 115 23 127)(10 114 24 126)(11 69 154 56)(12 68 155 55)(13 67 156 54)(14 66 157 53)(15 65 158 52)(16 64 159 51)(17 63 160 60)(18 62 151 59)(19 61 152 58)(20 70 153 57)(31 101 44 99)(32 110 45 98)(33 109 46 97)(34 108 47 96)(35 107 48 95)(36 106 49 94)(37 105 50 93)(38 104 41 92)(39 103 42 91)(40 102 43 100)(71 134 85 146)(72 133 86 145)(73 132 87 144)(74 131 88 143)(75 140 89 142)(76 139 90 141)(77 138 81 150)(78 137 82 149)(79 136 83 148)(80 135 84 147)
(1 70 30 52)(2 61 21 53)(3 62 22 54)(4 63 23 55)(5 64 24 56)(6 65 25 57)(7 66 26 58)(8 67 27 59)(9 68 28 60)(10 69 29 51)(11 126 159 119)(12 127 160 120)(13 128 151 111)(14 129 152 112)(15 130 153 113)(16 121 154 114)(17 122 155 115)(18 123 156 116)(19 124 157 117)(20 125 158 118)(31 90 49 71)(32 81 50 72)(33 82 41 73)(34 83 42 74)(35 84 43 75)(36 85 44 76)(37 86 45 77)(38 87 46 78)(39 88 47 79)(40 89 48 80)(91 148 108 131)(92 149 109 132)(93 150 110 133)(94 141 101 134)(95 142 102 135)(96 143 103 136)(97 144 104 137)(98 145 105 138)(99 146 106 139)(100 147 107 140)
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), (1,65,30,57)(2,66,21,58)(3,67,22,59)(4,68,23,60)(5,69,24,51)(6,70,25,52)(7,61,26,53)(8,62,27,54)(9,63,28,55)(10,64,29,56)(11,114,159,121)(12,115,160,122)(13,116,151,123)(14,117,152,124)(15,118,153,125)(16,119,154,126)(17,120,155,127)(18,111,156,128)(19,112,157,129)(20,113,158,130)(31,85,49,76)(32,86,50,77)(33,87,41,78)(34,88,42,79)(35,89,43,80)(36,90,44,71)(37,81,45,72)(38,82,46,73)(39,83,47,74)(40,84,48,75)(91,136,108,143)(92,137,109,144)(93,138,110,145)(94,139,101,146)(95,140,102,147)(96,131,103,148)(97,132,104,149)(98,133,105,150)(99,134,106,141)(100,135,107,142), (1,150,6,145)(2,149,7,144)(3,148,8,143)(4,147,9,142)(5,146,10,141)(11,31,16,36)(12,40,17,35)(13,39,18,34)(14,38,19,33)(15,37,20,32)(21,132,26,137)(22,131,27,136)(23,140,28,135)(24,139,29,134)(25,138,30,133)(41,152,46,157)(42,151,47,156)(43,160,48,155)(44,159,49,154)(45,158,50,153)(51,94,56,99)(52,93,57,98)(53,92,58,97)(54,91,59,96)(55,100,60,95)(61,109,66,104)(62,108,67,103)(63,107,68,102)(64,106,69,101)(65,105,70,110)(71,114,76,119)(72,113,77,118)(73,112,78,117)(74,111,79,116)(75,120,80,115)(81,130,86,125)(82,129,87,124)(83,128,88,123)(84,127,89,122)(85,126,90,121), (1,113,25,125)(2,112,26,124)(3,111,27,123)(4,120,28,122)(5,119,29,121)(6,118,30,130)(7,117,21,129)(8,116,22,128)(9,115,23,127)(10,114,24,126)(11,69,154,56)(12,68,155,55)(13,67,156,54)(14,66,157,53)(15,65,158,52)(16,64,159,51)(17,63,160,60)(18,62,151,59)(19,61,152,58)(20,70,153,57)(31,101,44,99)(32,110,45,98)(33,109,46,97)(34,108,47,96)(35,107,48,95)(36,106,49,94)(37,105,50,93)(38,104,41,92)(39,103,42,91)(40,102,43,100)(71,134,85,146)(72,133,86,145)(73,132,87,144)(74,131,88,143)(75,140,89,142)(76,139,90,141)(77,138,81,150)(78,137,82,149)(79,136,83,148)(80,135,84,147), (1,70,30,52)(2,61,21,53)(3,62,22,54)(4,63,23,55)(5,64,24,56)(6,65,25,57)(7,66,26,58)(8,67,27,59)(9,68,28,60)(10,69,29,51)(11,126,159,119)(12,127,160,120)(13,128,151,111)(14,129,152,112)(15,130,153,113)(16,121,154,114)(17,122,155,115)(18,123,156,116)(19,124,157,117)(20,125,158,118)(31,90,49,71)(32,81,50,72)(33,82,41,73)(34,83,42,74)(35,84,43,75)(36,85,44,76)(37,86,45,77)(38,87,46,78)(39,88,47,79)(40,89,48,80)(91,148,108,131)(92,149,109,132)(93,150,110,133)(94,141,101,134)(95,142,102,135)(96,143,103,136)(97,144,104,137)(98,145,105,138)(99,146,106,139)(100,147,107,140)>;
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), (1,65,30,57)(2,66,21,58)(3,67,22,59)(4,68,23,60)(5,69,24,51)(6,70,25,52)(7,61,26,53)(8,62,27,54)(9,63,28,55)(10,64,29,56)(11,114,159,121)(12,115,160,122)(13,116,151,123)(14,117,152,124)(15,118,153,125)(16,119,154,126)(17,120,155,127)(18,111,156,128)(19,112,157,129)(20,113,158,130)(31,85,49,76)(32,86,50,77)(33,87,41,78)(34,88,42,79)(35,89,43,80)(36,90,44,71)(37,81,45,72)(38,82,46,73)(39,83,47,74)(40,84,48,75)(91,136,108,143)(92,137,109,144)(93,138,110,145)(94,139,101,146)(95,140,102,147)(96,131,103,148)(97,132,104,149)(98,133,105,150)(99,134,106,141)(100,135,107,142), (1,150,6,145)(2,149,7,144)(3,148,8,143)(4,147,9,142)(5,146,10,141)(11,31,16,36)(12,40,17,35)(13,39,18,34)(14,38,19,33)(15,37,20,32)(21,132,26,137)(22,131,27,136)(23,140,28,135)(24,139,29,134)(25,138,30,133)(41,152,46,157)(42,151,47,156)(43,160,48,155)(44,159,49,154)(45,158,50,153)(51,94,56,99)(52,93,57,98)(53,92,58,97)(54,91,59,96)(55,100,60,95)(61,109,66,104)(62,108,67,103)(63,107,68,102)(64,106,69,101)(65,105,70,110)(71,114,76,119)(72,113,77,118)(73,112,78,117)(74,111,79,116)(75,120,80,115)(81,130,86,125)(82,129,87,124)(83,128,88,123)(84,127,89,122)(85,126,90,121), (1,113,25,125)(2,112,26,124)(3,111,27,123)(4,120,28,122)(5,119,29,121)(6,118,30,130)(7,117,21,129)(8,116,22,128)(9,115,23,127)(10,114,24,126)(11,69,154,56)(12,68,155,55)(13,67,156,54)(14,66,157,53)(15,65,158,52)(16,64,159,51)(17,63,160,60)(18,62,151,59)(19,61,152,58)(20,70,153,57)(31,101,44,99)(32,110,45,98)(33,109,46,97)(34,108,47,96)(35,107,48,95)(36,106,49,94)(37,105,50,93)(38,104,41,92)(39,103,42,91)(40,102,43,100)(71,134,85,146)(72,133,86,145)(73,132,87,144)(74,131,88,143)(75,140,89,142)(76,139,90,141)(77,138,81,150)(78,137,82,149)(79,136,83,148)(80,135,84,147), (1,70,30,52)(2,61,21,53)(3,62,22,54)(4,63,23,55)(5,64,24,56)(6,65,25,57)(7,66,26,58)(8,67,27,59)(9,68,28,60)(10,69,29,51)(11,126,159,119)(12,127,160,120)(13,128,151,111)(14,129,152,112)(15,130,153,113)(16,121,154,114)(17,122,155,115)(18,123,156,116)(19,124,157,117)(20,125,158,118)(31,90,49,71)(32,81,50,72)(33,82,41,73)(34,83,42,74)(35,84,43,75)(36,85,44,76)(37,86,45,77)(38,87,46,78)(39,88,47,79)(40,89,48,80)(91,148,108,131)(92,149,109,132)(93,150,110,133)(94,141,101,134)(95,142,102,135)(96,143,103,136)(97,144,104,137)(98,145,105,138)(99,146,106,139)(100,147,107,140) );
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)], [(1,65,30,57),(2,66,21,58),(3,67,22,59),(4,68,23,60),(5,69,24,51),(6,70,25,52),(7,61,26,53),(8,62,27,54),(9,63,28,55),(10,64,29,56),(11,114,159,121),(12,115,160,122),(13,116,151,123),(14,117,152,124),(15,118,153,125),(16,119,154,126),(17,120,155,127),(18,111,156,128),(19,112,157,129),(20,113,158,130),(31,85,49,76),(32,86,50,77),(33,87,41,78),(34,88,42,79),(35,89,43,80),(36,90,44,71),(37,81,45,72),(38,82,46,73),(39,83,47,74),(40,84,48,75),(91,136,108,143),(92,137,109,144),(93,138,110,145),(94,139,101,146),(95,140,102,147),(96,131,103,148),(97,132,104,149),(98,133,105,150),(99,134,106,141),(100,135,107,142)], [(1,150,6,145),(2,149,7,144),(3,148,8,143),(4,147,9,142),(5,146,10,141),(11,31,16,36),(12,40,17,35),(13,39,18,34),(14,38,19,33),(15,37,20,32),(21,132,26,137),(22,131,27,136),(23,140,28,135),(24,139,29,134),(25,138,30,133),(41,152,46,157),(42,151,47,156),(43,160,48,155),(44,159,49,154),(45,158,50,153),(51,94,56,99),(52,93,57,98),(53,92,58,97),(54,91,59,96),(55,100,60,95),(61,109,66,104),(62,108,67,103),(63,107,68,102),(64,106,69,101),(65,105,70,110),(71,114,76,119),(72,113,77,118),(73,112,78,117),(74,111,79,116),(75,120,80,115),(81,130,86,125),(82,129,87,124),(83,128,88,123),(84,127,89,122),(85,126,90,121)], [(1,113,25,125),(2,112,26,124),(3,111,27,123),(4,120,28,122),(5,119,29,121),(6,118,30,130),(7,117,21,129),(8,116,22,128),(9,115,23,127),(10,114,24,126),(11,69,154,56),(12,68,155,55),(13,67,156,54),(14,66,157,53),(15,65,158,52),(16,64,159,51),(17,63,160,60),(18,62,151,59),(19,61,152,58),(20,70,153,57),(31,101,44,99),(32,110,45,98),(33,109,46,97),(34,108,47,96),(35,107,48,95),(36,106,49,94),(37,105,50,93),(38,104,41,92),(39,103,42,91),(40,102,43,100),(71,134,85,146),(72,133,86,145),(73,132,87,144),(74,131,88,143),(75,140,89,142),(76,139,90,141),(77,138,81,150),(78,137,82,149),(79,136,83,148),(80,135,84,147)], [(1,70,30,52),(2,61,21,53),(3,62,22,54),(4,63,23,55),(5,64,24,56),(6,65,25,57),(7,66,26,58),(8,67,27,59),(9,68,28,60),(10,69,29,51),(11,126,159,119),(12,127,160,120),(13,128,151,111),(14,129,152,112),(15,130,153,113),(16,121,154,114),(17,122,155,115),(18,123,156,116),(19,124,157,117),(20,125,158,118),(31,90,49,71),(32,81,50,72),(33,82,41,73),(34,83,42,74),(35,84,43,75),(36,85,44,76),(37,86,45,77),(38,87,46,78),(39,88,47,79),(40,89,48,80),(91,148,108,131),(92,149,109,132),(93,150,110,133),(94,141,101,134),(95,142,102,135),(96,143,103,136),(97,144,104,137),(98,145,105,138),(99,146,106,139),(100,147,107,140)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 23 | 0 | 0 | 0 | 0 | 0 |
| 17 | 25 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 18 | 0 |
| 0 | 0 | 3 | 40 | 0 | 18 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 21 | 3 | 0 | 0 |
| 0 | 0 | 3 | 20 | 0 | 0 |
| 0 | 0 | 9 | 38 | 23 | 6 |
| 0 | 0 | 38 | 17 | 21 | 18 |
| 36 | 38 | 0 | 0 | 0 | 0 |
| 36 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 38 | 23 | 6 |
| 0 | 0 | 19 | 21 | 17 | 40 |
| 0 | 0 | 21 | 3 | 0 | 0 |
| 0 | 0 | 25 | 24 | 15 | 11 |
| 5 | 3 | 0 | 0 | 0 | 0 |
| 5 | 36 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 40 | 1 | 2 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 36 | 2 | 40 | 38 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 21 | 3 | 0 | 0 |
| 0 | 0 | 3 | 20 | 0 | 0 |
| 0 | 0 | 32 | 3 | 18 | 35 |
| 0 | 0 | 25 | 35 | 20 | 23 |
G:=sub<GL(6,GF(41))| [23,17,0,0,0,0,0,25,0,0,0,0,0,0,16,0,0,3,0,0,0,16,0,40,0,0,0,0,18,0,0,0,0,0,0,18],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,21,3,9,38,0,0,3,20,38,17,0,0,0,0,23,21,0,0,0,0,6,18],[36,36,0,0,0,0,38,5,0,0,0,0,0,0,9,19,21,25,0,0,38,21,3,24,0,0,23,17,0,15,0,0,6,40,0,11],[5,5,0,0,0,0,3,36,0,0,0,0,0,0,3,0,0,36,0,0,40,0,40,2,0,0,1,1,0,40,0,0,2,0,0,38],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,21,3,32,25,0,0,3,20,3,35,0,0,0,0,18,20,0,0,0,0,35,23] >;
74 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4L | 4M | ··· | 4AB | 5A | 5B | 10A | ··· | 10N | 20A | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | ··· | 2 | 10 | ··· | 10 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
74 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | - | ||
| image | C1 | C2 | C2 | C2 | C4 | D5 | D10 | Dic5 | D10 | 2- (1+4) | Q8.10D10 |
| kernel | C10.422- (1+4) | C23.21D10 | Q8×Dic5 | Q8×C2×C10 | Q8×C10 | C22×Q8 | C22×C4 | C2×Q8 | C2×Q8 | C10 | C2 |
| # reps | 1 | 6 | 8 | 1 | 16 | 2 | 6 | 16 | 8 | 2 | 8 |
In GAP, Magma, Sage, TeX
C_{10}._{42}2_-^{(1+4)} % in TeX
G:=Group("C10.42ES-(2,2)"); // GroupNames label
G:=SmallGroup(320,1484);
// by ID
G=gap.SmallGroup(320,1484);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,232,387,184,1123,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^10=b^4=1,c^2=a^5,d^2=a^5*b^2,e^2=b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b^2*d>;
// generators/relations
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