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G = C22×C32⋊C12order 432 = 24·33

Direct product of C22 and C32⋊C12

direct product, metabelian, supersoluble, monomial

Aliases: C22×C32⋊C12, C625C12, C627Dic3, C62.41D6, (C2×C62).7C6, He37(C22×C4), C62.13(C2×C6), (C2×C62).10S3, (C22×He3)⋊6C4, C6.22(C6×Dic3), C322(C22×C12), (C23×He3).4C2, C23.4(C32⋊C6), (C2×He3).27C23, C322(C22×Dic3), (C22×He3).30C22, C6.45(S3×C2×C6), (C3×C6)⋊2(C2×C12), C3.2(Dic3×C2×C6), (C2×C6).63(S3×C6), (C2×He3)⋊6(C2×C4), C3⋊Dic35(C2×C6), (C2×C3⋊Dic3)⋊6C6, (C3×C6)⋊2(C2×Dic3), (C3×C6).9(C22×C6), (C22×C6).29(C3×S3), (C3×C6).35(C22×S3), (C22×C3⋊Dic3)⋊2C3, (C2×C6).22(C3×Dic3), C2.2(C22×C32⋊C6), C22.11(C2×C32⋊C6), SmallGroup(432,376)

Series: Derived Chief Lower central Upper central

C1C32 — C22×C32⋊C12
C1C3C32C3×C6C2×He3C32⋊C12C2×C32⋊C12 — C22×C32⋊C12
C32 — C22×C32⋊C12
C1C23

Generators and relations for C22×C32⋊C12
 G = < a,b,c,d,e | a2=b2=c3=d3=e12=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1d, ede-1=d-1 >

Subgroups: 689 in 221 conjugacy classes, 102 normal (17 characteristic)
C1, C2, C2 [×6], C3, C3 [×3], C4 [×4], C22 [×7], C6, C6 [×6], C6 [×21], C2×C4 [×6], C23, C32 [×2], C32, Dic3 [×8], C12 [×4], C2×C6 [×7], C2×C6 [×21], C22×C4, C3×C6 [×2], C3×C6 [×12], C3×C6 [×7], C2×Dic3 [×12], C2×C12 [×6], C22×C6, C22×C6 [×3], He3, C3×Dic3 [×4], C3⋊Dic3 [×4], C62 [×14], C62 [×7], C22×Dic3 [×2], C22×C12, C2×He3, C2×He3 [×6], C6×Dic3 [×6], C2×C3⋊Dic3 [×6], C2×C62 [×2], C2×C62, C32⋊C12 [×4], C22×He3 [×7], Dic3×C2×C6, C22×C3⋊Dic3, C2×C32⋊C12 [×6], C23×He3, C22×C32⋊C12
Quotients: C1, C2 [×7], C3, C4 [×4], C22 [×7], S3, C6 [×7], C2×C4 [×6], C23, Dic3 [×4], C12 [×4], D6 [×3], C2×C6 [×7], C22×C4, C3×S3, C2×Dic3 [×6], C2×C12 [×6], C22×S3, C22×C6, C3×Dic3 [×4], S3×C6 [×3], C22×Dic3, C22×C12, C32⋊C6, C6×Dic3 [×6], S3×C2×C6, C32⋊C12 [×4], C2×C32⋊C6 [×3], Dic3×C2×C6, C2×C32⋊C12 [×6], C22×C32⋊C6, C22×C32⋊C12

Smallest permutation representation of C22×C32⋊C12
On 144 points
Generators in S144
(1 32)(2 29)(3 30)(4 31)(5 25)(6 26)(7 27)(8 28)(9 34)(10 35)(11 36)(12 33)(13 23)(14 24)(15 21)(16 22)(17 46)(18 47)(19 48)(20 45)(37 43)(38 44)(39 41)(40 42)(49 126)(50 127)(51 128)(52 129)(53 130)(54 131)(55 132)(56 121)(57 122)(58 123)(59 124)(60 125)(61 90)(62 91)(63 92)(64 93)(65 94)(66 95)(67 96)(68 85)(69 86)(70 87)(71 88)(72 89)(73 113)(74 114)(75 115)(76 116)(77 117)(78 118)(79 119)(80 120)(81 109)(82 110)(83 111)(84 112)(97 143)(98 144)(99 133)(100 134)(101 135)(102 136)(103 137)(104 138)(105 139)(106 140)(107 141)(108 142)
(1 48)(2 45)(3 46)(4 47)(5 37)(6 38)(7 39)(8 40)(9 14)(10 15)(11 16)(12 13)(17 30)(18 31)(19 32)(20 29)(21 35)(22 36)(23 33)(24 34)(25 43)(26 44)(27 41)(28 42)(49 138)(50 139)(51 140)(52 141)(53 142)(54 143)(55 144)(56 133)(57 134)(58 135)(59 136)(60 137)(61 117)(62 118)(63 119)(64 120)(65 109)(66 110)(67 111)(68 112)(69 113)(70 114)(71 115)(72 116)(73 86)(74 87)(75 88)(76 89)(77 90)(78 91)(79 92)(80 93)(81 94)(82 95)(83 96)(84 85)(97 131)(98 132)(99 121)(100 122)(101 123)(102 124)(103 125)(104 126)(105 127)(106 128)(107 129)(108 130)
(1 99 94)(2 91 108)(3 105 88)(4 85 102)(5 70 134)(6 143 67)(7 64 140)(8 137 61)(9 66 138)(10 135 63)(11 72 144)(12 141 69)(13 52 113)(14 110 49)(15 58 119)(16 116 55)(17 50 115)(18 112 59)(19 56 109)(20 118 53)(21 123 79)(22 76 132)(23 129 73)(24 82 126)(25 87 100)(26 97 96)(27 93 106)(28 103 90)(29 62 142)(30 139 71)(31 68 136)(32 133 65)(33 107 86)(34 95 104)(35 101 92)(36 89 98)(37 114 57)(38 54 111)(39 120 51)(40 60 117)(41 80 128)(42 125 77)(43 74 122)(44 131 83)(45 78 130)(46 127 75)(47 84 124)(48 121 81)
(1 28 33)(2 34 25)(3 26 35)(4 36 27)(5 29 9)(6 10 30)(7 31 11)(8 12 32)(13 19 40)(14 37 20)(15 17 38)(16 39 18)(21 46 44)(22 41 47)(23 48 42)(24 43 45)(49 57 53)(50 54 58)(51 59 55)(52 56 60)(61 69 65)(62 66 70)(63 71 67)(64 68 72)(73 81 77)(74 78 82)(75 83 79)(76 80 84)(85 89 93)(86 94 90)(87 91 95)(88 96 92)(97 101 105)(98 106 102)(99 103 107)(100 108 104)(109 117 113)(110 114 118)(111 119 115)(112 116 120)(121 125 129)(122 130 126)(123 127 131)(124 132 128)(133 137 141)(134 142 138)(135 139 143)(136 144 140)
(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)

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

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

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

80 conjugacy classes

class 1 2A···2G3A3B3C3D3E3F4A···4H6A···6G6H···6U6V···6AP12A···12P
order12···23333334···46···66···66···612···12
size11···12336669···92···23···36···69···9

80 irreducible representations

dim11111111222222666
type++++-++-+
imageC1C2C2C3C4C6C6C12S3Dic3D6C3×S3C3×Dic3S3×C6C32⋊C6C32⋊C12C2×C32⋊C6
kernelC22×C32⋊C12C2×C32⋊C12C23×He3C22×C3⋊Dic3C22×He3C2×C3⋊Dic3C2×C62C62C2×C62C62C62C22×C6C2×C6C2×C6C23C22C22
# reps1612812216143286143

Matrix representation of C22×C32⋊C12 in GL10(𝔽13)

1000000000
0100000000
00120000000
00012000000
0000100000
0000010000
0000001000
0000000100
0000000010
0000000001
,
12000000000
01200000000
0010000000
0001000000
0000100000
0000010000
0000001000
0000000100
0000000010
0000000001
,
01200000000
11200000000
001212000000
0010000000
00000000012
00000000112
00000120000
00001120000
00000001200
00000011200
,
1000000000
0100000000
0010000000
0001000000
00001210000
00001200000
00000012100
00000012000
00000000121
00000000120
,
8500000000
0500000000
0070000000
0066000000
0000850000
0000050000
0000000050
0000000058
0000000800
0000008000

G:=sub<GL(10,GF(13))| [1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,1,0],[8,0,0,0,0,0,0,0,0,0,5,5,0,0,0,0,0,0,0,0,0,0,7,6,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,5,5,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,5,5,0,0,0,0,0,0,0,0,0,8,0,0] >;

C22×C32⋊C12 in GAP, Magma, Sage, TeX

C_2^2\times C_3^2\rtimes C_{12}
% in TeX

G:=Group("C2^2xC3^2:C12");
// GroupNames label

G:=SmallGroup(432,376);
// by ID

G=gap.SmallGroup(432,376);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,168,4037,1034,14118]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^3=d^3=e^12=1,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*d,e*d*e^-1=d^-1>;
// generators/relations

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