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G = C22×C4.F5order 320 = 26·5

Direct product of C22 and C4.F5

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C22×C4.F5, Dic5.15C24, C5⋊C81C23, C2.4(C23×F5), (C2×C10)⋊4M4(2), C101(C2×M4(2)), C10.2(C23×C4), C51(C22×M4(2)), (C22×C4).24F5, C4.43(C22×F5), C23.65(C2×F5), (C22×C20).26C4, C20.83(C22×C4), (C4×D5).83C23, (C23×D5).18C4, D10.44(C22×C4), C22.55(C22×F5), Dic5.44(C22×C4), (C2×Dic5).362C23, (C22×Dic5).282C22, (C22×C5⋊C8)⋊8C2, (C2×C5⋊C8)⋊9C22, (C2×C4×D5).38C4, (C4×D5).90(C2×C4), (C2×C4).146(C2×F5), (D5×C22×C4).30C2, (C2×C20).133(C2×C4), (C2×C4×D5).399C22, (C22×C10).78(C2×C4), (C2×C10).96(C22×C4), (C2×Dic5).198(C2×C4), (C22×D5).132(C2×C4), SmallGroup(320,1588)

Series: Derived Chief Lower central Upper central

C1C10 — C22×C4.F5
C1C5C10Dic5C5⋊C8C2×C5⋊C8C22×C5⋊C8 — C22×C4.F5
C5C10 — C22×C4.F5
C1C23C22×C4

Generators and relations for C22×C4.F5
 G = < a,b,c,d,e | a2=b2=c4=d5=1, e4=c2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=d3 >

Subgroups: 906 in 298 conjugacy classes, 156 normal (13 characteristic)
C1, C2, C2 [×6], C2 [×4], C4 [×4], C4 [×4], C22 [×7], C22 [×16], C5, C8 [×8], C2×C4 [×6], C2×C4 [×22], C23, C23 [×10], D5 [×4], C10, C10 [×6], C2×C8 [×12], M4(2) [×16], C22×C4, C22×C4 [×13], C24, Dic5, Dic5 [×3], C20 [×4], D10 [×4], D10 [×12], C2×C10 [×7], C22×C8 [×2], C2×M4(2) [×12], C23×C4, C5⋊C8 [×8], C4×D5 [×16], C2×Dic5 [×6], C2×C20 [×6], C22×D5 [×6], C22×D5 [×4], C22×C10, C22×M4(2), C4.F5 [×16], C2×C5⋊C8 [×12], C2×C4×D5 [×12], C22×Dic5, C22×C20, C23×D5, C2×C4.F5 [×12], C22×C5⋊C8 [×2], D5×C22×C4, C22×C4.F5
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], M4(2) [×4], C22×C4 [×14], C24, F5, C2×M4(2) [×6], C23×C4, C2×F5 [×7], C22×M4(2), C4.F5 [×4], C22×F5 [×7], C2×C4.F5 [×6], C23×F5, C22×C4.F5

Smallest permutation representation of C22×C4.F5
On 160 points
Generators in S160
(1 139)(2 140)(3 141)(4 142)(5 143)(6 144)(7 137)(8 138)(9 34)(10 35)(11 36)(12 37)(13 38)(14 39)(15 40)(16 33)(17 119)(18 120)(19 113)(20 114)(21 115)(22 116)(23 117)(24 118)(25 123)(26 124)(27 125)(28 126)(29 127)(30 128)(31 121)(32 122)(41 68)(42 69)(43 70)(44 71)(45 72)(46 65)(47 66)(48 67)(49 100)(50 101)(51 102)(52 103)(53 104)(54 97)(55 98)(56 99)(57 95)(58 96)(59 89)(60 90)(61 91)(62 92)(63 93)(64 94)(73 83)(74 84)(75 85)(76 86)(77 87)(78 88)(79 81)(80 82)(105 160)(106 153)(107 154)(108 155)(109 156)(110 157)(111 158)(112 159)(129 147)(130 148)(131 149)(132 150)(133 151)(134 152)(135 145)(136 146)
(1 95)(2 96)(3 89)(4 90)(5 91)(6 92)(7 93)(8 94)(9 42)(10 43)(11 44)(12 45)(13 46)(14 47)(15 48)(16 41)(17 153)(18 154)(19 155)(20 156)(21 157)(22 158)(23 159)(24 160)(25 78)(26 79)(27 80)(28 73)(29 74)(30 75)(31 76)(32 77)(33 68)(34 69)(35 70)(36 71)(37 72)(38 65)(39 66)(40 67)(49 136)(50 129)(51 130)(52 131)(53 132)(54 133)(55 134)(56 135)(57 139)(58 140)(59 141)(60 142)(61 143)(62 144)(63 137)(64 138)(81 124)(82 125)(83 126)(84 127)(85 128)(86 121)(87 122)(88 123)(97 151)(98 152)(99 145)(100 146)(101 147)(102 148)(103 149)(104 150)(105 118)(106 119)(107 120)(108 113)(109 114)(110 115)(111 116)(112 117)
(1 89 5 93)(2 94 6 90)(3 91 7 95)(4 96 8 92)(9 17 13 21)(10 22 14 18)(11 19 15 23)(12 24 16 20)(25 103 29 99)(26 100 30 104)(27 97 31 101)(28 102 32 98)(33 114 37 118)(34 119 38 115)(35 116 39 120)(36 113 40 117)(41 156 45 160)(42 153 46 157)(43 158 47 154)(44 155 48 159)(49 128 53 124)(50 125 54 121)(51 122 55 126)(52 127 56 123)(57 141 61 137)(58 138 62 142)(59 143 63 139)(60 140 64 144)(65 110 69 106)(66 107 70 111)(67 112 71 108)(68 109 72 105)(73 148 77 152)(74 145 78 149)(75 150 79 146)(76 147 80 151)(81 136 85 132)(82 133 86 129)(83 130 87 134)(84 135 88 131)
(1 50 21 48 88)(2 41 51 81 22)(3 82 42 23 52)(4 24 83 53 43)(5 54 17 44 84)(6 45 55 85 18)(7 86 46 19 56)(8 20 87 49 47)(9 159 131 89 125)(10 90 160 126 132)(11 127 91 133 153)(12 134 128 154 92)(13 155 135 93 121)(14 94 156 122 136)(15 123 95 129 157)(16 130 124 158 96)(25 57 147 110 40)(26 111 58 33 148)(27 34 112 149 59)(28 150 35 60 105)(29 61 151 106 36)(30 107 62 37 152)(31 38 108 145 63)(32 146 39 64 109)(65 113 99 137 76)(66 138 114 77 100)(67 78 139 101 115)(68 102 79 116 140)(69 117 103 141 80)(70 142 118 73 104)(71 74 143 97 119)(72 98 75 120 144)
(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)

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

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

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

56 conjugacy classes

class 1 2A···2G2H2I2J2K4A4B4C4D4E···4L 5 8A···8P10A···10G20A···20H
order12···2222244444···458···810···1020···20
size11···11010101022225···5410···104···44···4

56 irreducible representations

dim111111124444
type+++++++
imageC1C2C2C2C4C4C4M4(2)F5C2×F5C2×F5C4.F5
kernelC22×C4.F5C2×C4.F5C22×C5⋊C8D5×C22×C4C2×C4×D5C22×C20C23×D5C2×C10C22×C4C2×C4C23C22
# reps11221122281618

Matrix representation of C22×C4.F5 in GL8(𝔽41)

400000000
040000000
00100000
00010000
00001000
00000100
00000010
00000001
,
400000000
040000000
00100000
00010000
000040000
000004000
000000400
000000040
,
90000000
032000000
003290000
00090000
0000727014
00000342714
00001427340
0000140277
,
10000000
01000000
00100000
00010000
000040100
000040010
000040001
000040000
,
09000000
400000000
004050000
003910000
00002525152
000040273027
00001411141
000039261616

G:=sub<GL(8,GF(41))| [40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[9,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,9,9,0,0,0,0,0,0,0,0,7,0,14,14,0,0,0,0,27,34,27,0,0,0,0,0,0,27,34,27,0,0,0,0,14,14,0,7],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,40,40,40,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0],[0,40,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,40,39,0,0,0,0,0,0,5,1,0,0,0,0,0,0,0,0,25,40,14,39,0,0,0,0,25,27,11,26,0,0,0,0,15,30,14,16,0,0,0,0,2,27,1,16] >;

C22×C4.F5 in GAP, Magma, Sage, TeX

C_2^2\times C_4.F_5
% in TeX

G:=Group("C2^2xC4.F5");
// GroupNames label

G:=SmallGroup(320,1588);
// by ID

G=gap.SmallGroup(320,1588);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,1123,136,102,6278,818]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^4=d^5=1,e^4=c^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=d^3>;
// generators/relations

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