Copied to
clipboard

?

G = C22×C4.F5  order 320 = 26·5

Direct product of C22 and C4.F5

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

Aliases: C22×C4.F5, Dic5.15C24, C5⋊C8⋊1C23, C2.4(C23×F5), C10⋊1(C2×M4(2)), (C2×C10)⋊4M4(2), C10.2(C23×C4), C5⋊1(C22×M4(2)), (C22×C4).24F5, C4.43(C22×F5), C23.65(C2×F5), C20.83(C22×C4), (C22×C20).26C4, (C4×D5).83C23, (C23×D5).18C4, D10.44(C22×C4), C22.55(C22×F5), Dic5.44(C22×C4), (C2×Dic5).362C23, (C22×Dic5).282C22, (C2×C5⋊C8)⋊9C22, (C22×C5⋊C8)⋊8C2, (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, (C2×C10).96(C22×C4), (C22×C10).78(C2×C4), (C2×Dic5).198(C2×C4), (C22×D5).132(C2×C4), SmallGroup(320,1588)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C10 — C22×C4.F5
Chief series
C1 — C5 — C10 — Dic5 — C5⋊C8 — C2×C5⋊C8 — C22×C5⋊C8 — C22×C4.F5
Lower central
C5 — C10 — C22×C4.F5

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

Generators and relations
 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 >

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

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

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

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

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

Matrix representation ►G ⊆ 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] >;
 

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

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
 

׿
×
⋊
ℤ
𝔽
○
≀