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G = C10.222- 1+4order 320 = 26·5

22nd non-split extension by C10 of 2- 1+4 acting via 2- 1+4/C2×Q8=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C10.222- 1+4, C10.552+ 1+4, C22⋊Q817D5, C4⋊C4.195D10, (C2×Q8).77D10, D10⋊Q823C2, D10⋊D4.2C2, C22⋊C4.20D10, Dic5⋊Q817C2, Dic53Q828C2, C20.23D415C2, (C2×C10).184C24, (C2×C20).628C23, (C22×C4).246D10, D10.13D421C2, C2.57(D46D10), Dic5.13(C4○D4), Dic5.5D426C2, (C2×D20).158C22, C23.D1024C2, C4⋊Dic5.219C22, (Q8×C10).114C22, (C22×D5).75C23, C22.205(C23×D5), C23.123(C22×D5), C23.D5.35C22, D10⋊C4.26C22, C23.23D1025C2, (C22×C10).212C23, (C22×C20).383C22, C55(C22.36C24), (C2×Dic5).248C23, (C4×Dic5).120C22, C10.D4.32C22, C2.23(Q8.10D10), (C2×Dic10).168C22, (C4×C5⋊D4)⋊59C2, C2.55(D5×C4○D4), C4⋊C47D528C2, C4⋊C4⋊D519C2, (C5×C22⋊Q8)⋊20C2, C10.167(C2×C4○D4), (C2×C4×D5).262C22, (C2×C4).54(C22×D5), (C5×C4⋊C4).165C22, (C2×C5⋊D4).139C22, (C5×C22⋊C4).39C22, SmallGroup(320,1312)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C10.222- 1+4
C1C5C10C2×C10C22×D5C2×C4×D5D10⋊Q8 — C10.222- 1+4
C5C2×C10 — C10.222- 1+4
C1C22C22⋊Q8

Generators and relations for C10.222- 1+4
 G = < a,b,c,d,e | a10=b4=e2=1, c2=a5, d2=a5b2, bab-1=dad-1=a-1, ac=ca, ae=ea, cbc-1=a5b-1, bd=db, ebe=a5b, cd=dc, ce=ec, ede=a5b2d >

Subgroups: 766 in 216 conjugacy classes, 93 normal (91 characteristic)
C1, C2 [×3], C2 [×3], C4 [×13], C22, C22 [×9], C5, C2×C4 [×6], C2×C4 [×10], D4 [×4], Q8 [×4], C23, C23 [×2], D5 [×2], C10 [×3], C10, C42 [×4], C22⋊C4 [×2], C22⋊C4 [×10], C4⋊C4 [×3], C4⋊C4 [×7], C22×C4, C22×C4 [×2], C2×D4 [×3], C2×Q8, C2×Q8 [×2], Dic5 [×2], Dic5 [×5], C20 [×6], D10 [×6], C2×C10, C2×C10 [×3], C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22⋊Q8 [×2], C22.D4 [×2], C4.4D4 [×3], C422C2 [×2], C4⋊Q8, Dic10 [×3], C4×D5 [×3], D20, C2×Dic5 [×6], C5⋊D4 [×3], C2×C20 [×6], C2×C20, C5×Q8, C22×D5 [×2], C22×C10, C22.36C24, C4×Dic5 [×4], C10.D4 [×6], C4⋊Dic5, D10⋊C4 [×8], C23.D5 [×2], C5×C22⋊C4 [×2], C5×C4⋊C4 [×3], C2×Dic10 [×2], C2×C4×D5 [×2], C2×D20, C2×C5⋊D4 [×2], C22×C20, Q8×C10, C23.D10, D10⋊D4, Dic5.5D4 [×2], Dic53Q8, C4⋊C47D5, D10.13D4, D10⋊Q8 [×2], C4⋊C4⋊D5, C4×C5⋊D4, C23.23D10, Dic5⋊Q8, C20.23D4, C5×C22⋊Q8, C10.222- 1+4
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×2], C24, D10 [×7], C2×C4○D4, 2+ 1+4, 2- 1+4, C22×D5 [×7], C22.36C24, C23×D5, D46D10, Q8.10D10, D5×C4○D4, C10.222- 1+4

Smallest permutation representation of C10.222- 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 76 30 90)(2 75 21 89)(3 74 22 88)(4 73 23 87)(5 72 24 86)(6 71 25 85)(7 80 26 84)(8 79 27 83)(9 78 28 82)(10 77 29 81)(11 110 151 96)(12 109 152 95)(13 108 153 94)(14 107 154 93)(15 106 155 92)(16 105 156 91)(17 104 157 100)(18 103 158 99)(19 102 159 98)(20 101 160 97)(31 70 45 56)(32 69 46 55)(33 68 47 54)(34 67 48 53)(35 66 49 52)(36 65 50 51)(37 64 41 60)(38 63 42 59)(39 62 43 58)(40 61 44 57)(111 131 125 145)(112 140 126 144)(113 139 127 143)(114 138 128 142)(115 137 129 141)(116 136 130 150)(117 135 121 149)(118 134 122 148)(119 133 123 147)(120 132 124 146)
(1 91 6 96)(2 92 7 97)(3 93 8 98)(4 94 9 99)(5 95 10 100)(11 85 16 90)(12 86 17 81)(13 87 18 82)(14 88 19 83)(15 89 20 84)(21 106 26 101)(22 107 27 102)(23 108 28 103)(24 109 29 104)(25 110 30 105)(31 116 36 111)(32 117 37 112)(33 118 38 113)(34 119 39 114)(35 120 40 115)(41 126 46 121)(42 127 47 122)(43 128 48 123)(44 129 49 124)(45 130 50 125)(51 136 56 131)(52 137 57 132)(53 138 58 133)(54 139 59 134)(55 140 60 135)(61 146 66 141)(62 147 67 142)(63 148 68 143)(64 149 69 144)(65 150 70 145)(71 156 76 151)(72 157 77 152)(73 158 78 153)(74 159 79 154)(75 160 80 155)
(1 45 25 36)(2 44 26 35)(3 43 27 34)(4 42 28 33)(5 41 29 32)(6 50 30 31)(7 49 21 40)(8 48 22 39)(9 47 23 38)(10 46 24 37)(11 145 156 136)(12 144 157 135)(13 143 158 134)(14 142 159 133)(15 141 160 132)(16 150 151 131)(17 149 152 140)(18 148 153 139)(19 147 154 138)(20 146 155 137)(51 90 70 71)(52 89 61 80)(53 88 62 79)(54 87 63 78)(55 86 64 77)(56 85 65 76)(57 84 66 75)(58 83 67 74)(59 82 68 73)(60 81 69 72)(91 130 110 111)(92 129 101 120)(93 128 102 119)(94 127 103 118)(95 126 104 117)(96 125 105 116)(97 124 106 115)(98 123 107 114)(99 122 108 113)(100 121 109 112)
(1 36)(2 37)(3 38)(4 39)(5 40)(6 31)(7 32)(8 33)(9 34)(10 35)(11 131)(12 132)(13 133)(14 134)(15 135)(16 136)(17 137)(18 138)(19 139)(20 140)(21 41)(22 42)(23 43)(24 44)(25 45)(26 46)(27 47)(28 48)(29 49)(30 50)(51 85)(52 86)(53 87)(54 88)(55 89)(56 90)(57 81)(58 82)(59 83)(60 84)(61 77)(62 78)(63 79)(64 80)(65 71)(66 72)(67 73)(68 74)(69 75)(70 76)(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)(141 157)(142 158)(143 159)(144 160)(145 151)(146 152)(147 153)(148 154)(149 155)(150 156)

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), (1,76,30,90)(2,75,21,89)(3,74,22,88)(4,73,23,87)(5,72,24,86)(6,71,25,85)(7,80,26,84)(8,79,27,83)(9,78,28,82)(10,77,29,81)(11,110,151,96)(12,109,152,95)(13,108,153,94)(14,107,154,93)(15,106,155,92)(16,105,156,91)(17,104,157,100)(18,103,158,99)(19,102,159,98)(20,101,160,97)(31,70,45,56)(32,69,46,55)(33,68,47,54)(34,67,48,53)(35,66,49,52)(36,65,50,51)(37,64,41,60)(38,63,42,59)(39,62,43,58)(40,61,44,57)(111,131,125,145)(112,140,126,144)(113,139,127,143)(114,138,128,142)(115,137,129,141)(116,136,130,150)(117,135,121,149)(118,134,122,148)(119,133,123,147)(120,132,124,146), (1,91,6,96)(2,92,7,97)(3,93,8,98)(4,94,9,99)(5,95,10,100)(11,85,16,90)(12,86,17,81)(13,87,18,82)(14,88,19,83)(15,89,20,84)(21,106,26,101)(22,107,27,102)(23,108,28,103)(24,109,29,104)(25,110,30,105)(31,116,36,111)(32,117,37,112)(33,118,38,113)(34,119,39,114)(35,120,40,115)(41,126,46,121)(42,127,47,122)(43,128,48,123)(44,129,49,124)(45,130,50,125)(51,136,56,131)(52,137,57,132)(53,138,58,133)(54,139,59,134)(55,140,60,135)(61,146,66,141)(62,147,67,142)(63,148,68,143)(64,149,69,144)(65,150,70,145)(71,156,76,151)(72,157,77,152)(73,158,78,153)(74,159,79,154)(75,160,80,155), (1,45,25,36)(2,44,26,35)(3,43,27,34)(4,42,28,33)(5,41,29,32)(6,50,30,31)(7,49,21,40)(8,48,22,39)(9,47,23,38)(10,46,24,37)(11,145,156,136)(12,144,157,135)(13,143,158,134)(14,142,159,133)(15,141,160,132)(16,150,151,131)(17,149,152,140)(18,148,153,139)(19,147,154,138)(20,146,155,137)(51,90,70,71)(52,89,61,80)(53,88,62,79)(54,87,63,78)(55,86,64,77)(56,85,65,76)(57,84,66,75)(58,83,67,74)(59,82,68,73)(60,81,69,72)(91,130,110,111)(92,129,101,120)(93,128,102,119)(94,127,103,118)(95,126,104,117)(96,125,105,116)(97,124,106,115)(98,123,107,114)(99,122,108,113)(100,121,109,112), (1,36)(2,37)(3,38)(4,39)(5,40)(6,31)(7,32)(8,33)(9,34)(10,35)(11,131)(12,132)(13,133)(14,134)(15,135)(16,136)(17,137)(18,138)(19,139)(20,140)(21,41)(22,42)(23,43)(24,44)(25,45)(26,46)(27,47)(28,48)(29,49)(30,50)(51,85)(52,86)(53,87)(54,88)(55,89)(56,90)(57,81)(58,82)(59,83)(60,84)(61,77)(62,78)(63,79)(64,80)(65,71)(66,72)(67,73)(68,74)(69,75)(70,76)(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)(141,157)(142,158)(143,159)(144,160)(145,151)(146,152)(147,153)(148,154)(149,155)(150,156)>;

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), (1,76,30,90)(2,75,21,89)(3,74,22,88)(4,73,23,87)(5,72,24,86)(6,71,25,85)(7,80,26,84)(8,79,27,83)(9,78,28,82)(10,77,29,81)(11,110,151,96)(12,109,152,95)(13,108,153,94)(14,107,154,93)(15,106,155,92)(16,105,156,91)(17,104,157,100)(18,103,158,99)(19,102,159,98)(20,101,160,97)(31,70,45,56)(32,69,46,55)(33,68,47,54)(34,67,48,53)(35,66,49,52)(36,65,50,51)(37,64,41,60)(38,63,42,59)(39,62,43,58)(40,61,44,57)(111,131,125,145)(112,140,126,144)(113,139,127,143)(114,138,128,142)(115,137,129,141)(116,136,130,150)(117,135,121,149)(118,134,122,148)(119,133,123,147)(120,132,124,146), (1,91,6,96)(2,92,7,97)(3,93,8,98)(4,94,9,99)(5,95,10,100)(11,85,16,90)(12,86,17,81)(13,87,18,82)(14,88,19,83)(15,89,20,84)(21,106,26,101)(22,107,27,102)(23,108,28,103)(24,109,29,104)(25,110,30,105)(31,116,36,111)(32,117,37,112)(33,118,38,113)(34,119,39,114)(35,120,40,115)(41,126,46,121)(42,127,47,122)(43,128,48,123)(44,129,49,124)(45,130,50,125)(51,136,56,131)(52,137,57,132)(53,138,58,133)(54,139,59,134)(55,140,60,135)(61,146,66,141)(62,147,67,142)(63,148,68,143)(64,149,69,144)(65,150,70,145)(71,156,76,151)(72,157,77,152)(73,158,78,153)(74,159,79,154)(75,160,80,155), (1,45,25,36)(2,44,26,35)(3,43,27,34)(4,42,28,33)(5,41,29,32)(6,50,30,31)(7,49,21,40)(8,48,22,39)(9,47,23,38)(10,46,24,37)(11,145,156,136)(12,144,157,135)(13,143,158,134)(14,142,159,133)(15,141,160,132)(16,150,151,131)(17,149,152,140)(18,148,153,139)(19,147,154,138)(20,146,155,137)(51,90,70,71)(52,89,61,80)(53,88,62,79)(54,87,63,78)(55,86,64,77)(56,85,65,76)(57,84,66,75)(58,83,67,74)(59,82,68,73)(60,81,69,72)(91,130,110,111)(92,129,101,120)(93,128,102,119)(94,127,103,118)(95,126,104,117)(96,125,105,116)(97,124,106,115)(98,123,107,114)(99,122,108,113)(100,121,109,112), (1,36)(2,37)(3,38)(4,39)(5,40)(6,31)(7,32)(8,33)(9,34)(10,35)(11,131)(12,132)(13,133)(14,134)(15,135)(16,136)(17,137)(18,138)(19,139)(20,140)(21,41)(22,42)(23,43)(24,44)(25,45)(26,46)(27,47)(28,48)(29,49)(30,50)(51,85)(52,86)(53,87)(54,88)(55,89)(56,90)(57,81)(58,82)(59,83)(60,84)(61,77)(62,78)(63,79)(64,80)(65,71)(66,72)(67,73)(68,74)(69,75)(70,76)(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)(141,157)(142,158)(143,159)(144,160)(145,151)(146,152)(147,153)(148,154)(149,155)(150,156) );

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)], [(1,76,30,90),(2,75,21,89),(3,74,22,88),(4,73,23,87),(5,72,24,86),(6,71,25,85),(7,80,26,84),(8,79,27,83),(9,78,28,82),(10,77,29,81),(11,110,151,96),(12,109,152,95),(13,108,153,94),(14,107,154,93),(15,106,155,92),(16,105,156,91),(17,104,157,100),(18,103,158,99),(19,102,159,98),(20,101,160,97),(31,70,45,56),(32,69,46,55),(33,68,47,54),(34,67,48,53),(35,66,49,52),(36,65,50,51),(37,64,41,60),(38,63,42,59),(39,62,43,58),(40,61,44,57),(111,131,125,145),(112,140,126,144),(113,139,127,143),(114,138,128,142),(115,137,129,141),(116,136,130,150),(117,135,121,149),(118,134,122,148),(119,133,123,147),(120,132,124,146)], [(1,91,6,96),(2,92,7,97),(3,93,8,98),(4,94,9,99),(5,95,10,100),(11,85,16,90),(12,86,17,81),(13,87,18,82),(14,88,19,83),(15,89,20,84),(21,106,26,101),(22,107,27,102),(23,108,28,103),(24,109,29,104),(25,110,30,105),(31,116,36,111),(32,117,37,112),(33,118,38,113),(34,119,39,114),(35,120,40,115),(41,126,46,121),(42,127,47,122),(43,128,48,123),(44,129,49,124),(45,130,50,125),(51,136,56,131),(52,137,57,132),(53,138,58,133),(54,139,59,134),(55,140,60,135),(61,146,66,141),(62,147,67,142),(63,148,68,143),(64,149,69,144),(65,150,70,145),(71,156,76,151),(72,157,77,152),(73,158,78,153),(74,159,79,154),(75,160,80,155)], [(1,45,25,36),(2,44,26,35),(3,43,27,34),(4,42,28,33),(5,41,29,32),(6,50,30,31),(7,49,21,40),(8,48,22,39),(9,47,23,38),(10,46,24,37),(11,145,156,136),(12,144,157,135),(13,143,158,134),(14,142,159,133),(15,141,160,132),(16,150,151,131),(17,149,152,140),(18,148,153,139),(19,147,154,138),(20,146,155,137),(51,90,70,71),(52,89,61,80),(53,88,62,79),(54,87,63,78),(55,86,64,77),(56,85,65,76),(57,84,66,75),(58,83,67,74),(59,82,68,73),(60,81,69,72),(91,130,110,111),(92,129,101,120),(93,128,102,119),(94,127,103,118),(95,126,104,117),(96,125,105,116),(97,124,106,115),(98,123,107,114),(99,122,108,113),(100,121,109,112)], [(1,36),(2,37),(3,38),(4,39),(5,40),(6,31),(7,32),(8,33),(9,34),(10,35),(11,131),(12,132),(13,133),(14,134),(15,135),(16,136),(17,137),(18,138),(19,139),(20,140),(21,41),(22,42),(23,43),(24,44),(25,45),(26,46),(27,47),(28,48),(29,49),(30,50),(51,85),(52,86),(53,87),(54,88),(55,89),(56,90),(57,81),(58,82),(59,83),(60,84),(61,77),(62,78),(63,79),(64,80),(65,71),(66,72),(67,73),(68,74),(69,75),(70,76),(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),(141,157),(142,158),(143,159),(144,160),(145,151),(146,152),(147,153),(148,154),(149,155),(150,156)])

50 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C···4G4H4I4J4K4L4M4N4O5A5B10A···10F10G10H10I10J20A···20H20I···20P
order1222222444···4444444445510···101010101020···2020···20
size111142020224···41010101020202020222···244444···48···8

50 irreducible representations

dim1111111111111122222244444
type++++++++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2D5C4○D4D10D10D10D102+ 1+42- 1+4D46D10Q8.10D10D5×C4○D4
kernelC10.222- 1+4C23.D10D10⋊D4Dic5.5D4Dic53Q8C4⋊C47D5D10.13D4D10⋊Q8C4⋊C4⋊D5C4×C5⋊D4C23.23D10Dic5⋊Q8C20.23D4C5×C22⋊Q8C22⋊Q8Dic5C22⋊C4C4⋊C4C22×C4C2×Q8C10C10C2C2C2
# reps1112111211111124462211444

Matrix representation of C10.222- 1+4 in GL6(𝔽41)

4000000
0400000
00343400
007100
00003434
000071
,
3370000
23380000
00001740
0000324
0024100
00381700
,
900000
090000
00270340
00027034
00340140
00034014
,
4000000
0400000
00174000
0032400
00001740
0000324
,
4000000
1910000
00174000
0012400
00001740
0000124

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,34,7,0,0,0,0,34,1,0,0,0,0,0,0,34,7,0,0,0,0,34,1],[3,23,0,0,0,0,37,38,0,0,0,0,0,0,0,0,24,38,0,0,0,0,1,17,0,0,17,3,0,0,0,0,40,24,0,0],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,27,0,34,0,0,0,0,27,0,34,0,0,34,0,14,0,0,0,0,34,0,14],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,17,3,0,0,0,0,40,24,0,0,0,0,0,0,17,3,0,0,0,0,40,24],[40,19,0,0,0,0,0,1,0,0,0,0,0,0,17,1,0,0,0,0,40,24,0,0,0,0,0,0,17,1,0,0,0,0,40,24] >;

C10.222- 1+4 in GAP, Magma, Sage, TeX

C_{10}._{22}2_-^{1+4}
% in TeX

G:=Group("C10.22ES-(2,2)");
// GroupNames label

G:=SmallGroup(320,1312);
// by ID

G=gap.SmallGroup(320,1312);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,758,555,100,1571,297,12550]);
// 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=d*a*d^-1=a^-1,a*c=c*a,a*e=e*a,c*b*c^-1=a^5*b^-1,b*d=d*b,e*b*e=a^5*b,c*d=d*c,c*e=e*c,e*d*e=a^5*b^2*d>;
// generators/relations

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