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## G = He3⋊Q16order 432 = 24·33

### The semidirect product of He3 and Q16 acting via Q16/C2=D4

Aliases: He3⋊Q16, C6.3S3≀C2, He32Q8.C2, C3.(C32⋊Q16), C2.5(He3⋊D4), (C2×He3).3D4, He32C8.1C2, He33C4.3C22, SmallGroup(432,236)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — He3 — He3⋊3C4 — He3⋊Q16
 Chief series C1 — C3 — He3 — C2×He3 — He3⋊3C4 — He3⋊2Q8 — He3⋊Q16
 Lower central He3 — C2×He3 — He3⋊3C4 — He3⋊Q16
 Upper central C1 — C2

Generators and relations for He3⋊Q16
G = < a,b,c,d,e | a3=b3=c3=d8=1, e2=d4, ab=ba, cac-1=dcd-1=ece-1=ab-1, dad-1=bc-1, eae-1=b-1c, bc=cb, bd=db, ebe-1=b-1, ede-1=d-1 >

Character table of He3⋊Q16

 class 1 2 3A 3B 3C 4A 4B 4C 6A 6B 6C 8A 8B 12A 12B 12C 12D 12E 12F 24A 24B 24C 24D size 1 1 2 12 12 18 36 36 2 12 12 18 18 18 18 36 36 36 36 18 18 18 18 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 1 1 -1 1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 -1 -1 linear of order 2 ρ3 1 1 1 1 1 1 -1 1 1 1 1 -1 -1 1 1 1 1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ4 1 1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 linear of order 2 ρ5 2 2 2 2 2 -2 0 0 2 2 2 0 0 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ6 2 -2 2 2 2 0 0 0 -2 -2 -2 √2 -√2 0 0 0 0 0 0 -√2 -√2 √2 √2 symplectic lifted from Q16, Schur index 2 ρ7 2 -2 2 2 2 0 0 0 -2 -2 -2 -√2 √2 0 0 0 0 0 0 √2 √2 -√2 -√2 symplectic lifted from Q16, Schur index 2 ρ8 4 4 4 1 -2 0 2 0 4 -2 1 0 0 0 0 0 0 -1 -1 0 0 0 0 orthogonal lifted from S3≀C2 ρ9 4 4 4 -2 1 0 0 2 4 1 -2 0 0 0 0 -1 -1 0 0 0 0 0 0 orthogonal lifted from S3≀C2 ρ10 4 4 4 -2 1 0 0 -2 4 1 -2 0 0 0 0 1 1 0 0 0 0 0 0 orthogonal lifted from S3≀C2 ρ11 4 4 4 1 -2 0 -2 0 4 -2 1 0 0 0 0 0 0 1 1 0 0 0 0 orthogonal lifted from S3≀C2 ρ12 4 -4 4 1 -2 0 0 0 -4 2 -1 0 0 0 0 0 0 -√3 √3 0 0 0 0 symplectic lifted from C32⋊Q16, Schur index 2 ρ13 4 -4 4 -2 1 0 0 0 -4 -1 2 0 0 0 0 -√3 √3 0 0 0 0 0 0 symplectic lifted from C32⋊Q16, Schur index 2 ρ14 4 -4 4 -2 1 0 0 0 -4 -1 2 0 0 0 0 √3 -√3 0 0 0 0 0 0 symplectic lifted from C32⋊Q16, Schur index 2 ρ15 4 -4 4 1 -2 0 0 0 -4 2 -1 0 0 0 0 0 0 √3 -√3 0 0 0 0 symplectic lifted from C32⋊Q16, Schur index 2 ρ16 6 6 -3 0 0 -2 0 0 -3 0 0 2 2 1 1 0 0 0 0 -1 -1 -1 -1 orthogonal lifted from He3⋊D4 ρ17 6 6 -3 0 0 -2 0 0 -3 0 0 -2 -2 1 1 0 0 0 0 1 1 1 1 orthogonal lifted from He3⋊D4 ρ18 6 6 -3 0 0 2 0 0 -3 0 0 0 0 -1 -1 0 0 0 0 √3 -√3 -√3 √3 orthogonal lifted from He3⋊D4 ρ19 6 6 -3 0 0 2 0 0 -3 0 0 0 0 -1 -1 0 0 0 0 -√3 √3 √3 -√3 orthogonal lifted from He3⋊D4 ρ20 6 -6 -3 0 0 0 0 0 3 0 0 -√2 √2 √3 -√3 0 0 0 0 ζ87ζ3+ζ85ζ3+ζ85 ζ83ζ3+ζ83+ζ8ζ3 ζ83ζ32+ζ8ζ32+ζ8 ζ87ζ32+ζ87+ζ85ζ32 symplectic faithful, Schur index 2 ρ21 6 -6 -3 0 0 0 0 0 3 0 0 √2 -√2 √3 -√3 0 0 0 0 ζ87ζ32+ζ87+ζ85ζ32 ζ83ζ32+ζ8ζ32+ζ8 ζ83ζ3+ζ83+ζ8ζ3 ζ87ζ3+ζ85ζ3+ζ85 symplectic faithful, Schur index 2 ρ22 6 -6 -3 0 0 0 0 0 3 0 0 √2 -√2 -√3 √3 0 0 0 0 ζ83ζ32+ζ8ζ32+ζ8 ζ87ζ32+ζ87+ζ85ζ32 ζ87ζ3+ζ85ζ3+ζ85 ζ83ζ3+ζ83+ζ8ζ3 symplectic faithful, Schur index 2 ρ23 6 -6 -3 0 0 0 0 0 3 0 0 -√2 √2 -√3 √3 0 0 0 0 ζ83ζ3+ζ83+ζ8ζ3 ζ87ζ3+ζ85ζ3+ζ85 ζ87ζ32+ζ87+ζ85ζ32 ζ83ζ32+ζ8ζ32+ζ8 symplectic faithful, Schur index 2

Smallest permutation representation of He3⋊Q16
On 144 points
Generators in S144
(1 122 43)(2 49 132)(3 50 92)(4 93 46)(5 126 47)(6 53 136)(7 54 96)(8 89 42)(9 57 137)(10 138 71)(11 104 72)(12 60 36)(13 61 141)(14 142 67)(15 100 68)(16 64 40)(17 121 55)(18 131 56)(19 120 91)(20 113 124)(21 125 51)(22 135 52)(23 116 95)(24 117 128)(25 90 119)(26 44 123)(27 45 133)(28 134 114)(29 94 115)(30 48 127)(31 41 129)(32 130 118)(33 77 70)(34 83 78)(35 59 106)(37 73 66)(38 87 74)(39 63 110)(58 105 103)(62 109 99)(65 97 80)(69 101 76)(75 143 88)(79 139 84)(81 144 111)(82 102 112)(85 140 107)(86 98 108)
(1 25 18)(2 26 19)(3 27 20)(4 28 21)(5 29 22)(6 30 23)(7 31 24)(8 32 17)(9 77 112)(10 78 105)(11 79 106)(12 80 107)(13 73 108)(14 74 109)(15 75 110)(16 76 111)(33 102 137)(34 103 138)(35 104 139)(36 97 140)(37 98 141)(38 99 142)(39 100 143)(40 101 144)(41 117 54)(42 118 55)(43 119 56)(44 120 49)(45 113 50)(46 114 51)(47 115 52)(48 116 53)(57 70 82)(58 71 83)(59 72 84)(60 65 85)(61 66 86)(62 67 87)(63 68 88)(64 69 81)(89 130 121)(90 131 122)(91 132 123)(92 133 124)(93 134 125)(94 135 126)(95 136 127)(96 129 128)
(1 122 119)(2 132 120)(3 113 124)(4 114 134)(5 126 115)(6 136 116)(7 117 128)(8 118 130)(9 82 102)(10 83 34)(11 104 84)(12 36 85)(13 86 98)(14 87 38)(15 100 88)(16 40 81)(17 42 89)(18 131 43)(19 91 44)(20 45 133)(21 46 93)(22 135 47)(23 95 48)(24 41 129)(25 90 56)(26 123 49)(27 50 92)(28 51 125)(29 94 52)(30 127 53)(31 54 96)(32 55 121)(33 112 70)(35 72 106)(37 108 66)(39 68 110)(57 137 77)(58 103 78)(59 79 139)(60 80 97)(61 141 73)(62 99 74)(63 75 143)(64 76 101)(65 107 140)(67 142 109)(69 111 144)(71 138 105)
(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)
(1 61 5 57)(2 60 6 64)(3 59 7 63)(4 58 8 62)(9 56 13 52)(10 55 14 51)(11 54 15 50)(12 53 16 49)(17 67 21 71)(18 66 22 70)(19 65 23 69)(20 72 24 68)(25 86 29 82)(26 85 30 81)(27 84 31 88)(28 83 32 87)(33 90 37 94)(34 89 38 93)(35 96 39 92)(36 95 40 91)(41 110 45 106)(42 109 46 105)(43 108 47 112)(44 107 48 111)(73 115 77 119)(74 114 78 118)(75 113 79 117)(76 120 80 116)(97 127 101 123)(98 126 102 122)(99 125 103 121)(100 124 104 128)(129 143 133 139)(130 142 134 138)(131 141 135 137)(132 140 136 144)

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

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

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

Matrix representation of He3⋊Q16 in GL6(𝔽73)

 0 0 0 1 0 0 0 0 72 72 0 0 0 0 0 0 0 1 0 0 0 0 72 72 0 1 0 0 0 0 72 72 0 0 0 0
,
 0 1 0 0 0 0 72 72 0 0 0 0 0 0 0 1 0 0 0 0 72 72 0 0 0 0 0 0 0 1 0 0 0 0 72 72
,
 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 72 0 0 0 0 1 0 0 1 0 0 0 0 72 72 0 0 0 0
,
 15 53 38 58 38 58 20 35 15 53 15 53 38 58 15 53 38 58 15 53 20 35 15 53 15 53 15 53 20 35 20 35 20 35 38 58
,
 61 51 0 0 0 0 63 12 0 0 0 0 0 0 61 51 0 0 0 0 63 12 0 0 0 0 0 0 63 12 0 0 0 0 22 10

G:=sub<GL(6,GF(73))| [0,0,0,0,0,72,0,0,0,0,1,72,0,72,0,0,0,0,1,72,0,0,0,0,0,0,0,72,0,0,0,0,1,72,0,0],[0,72,0,0,0,0,1,72,0,0,0,0,0,0,0,72,0,0,0,0,1,72,0,0,0,0,0,0,0,72,0,0,0,0,1,72],[0,0,0,0,0,72,0,0,0,0,1,72,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0],[15,20,38,15,15,20,53,35,58,53,53,35,38,15,15,20,15,20,58,53,53,35,53,35,38,15,38,15,20,38,58,53,58,53,35,58],[61,63,0,0,0,0,51,12,0,0,0,0,0,0,61,63,0,0,0,0,51,12,0,0,0,0,0,0,63,22,0,0,0,0,12,10] >;

He3⋊Q16 in GAP, Magma, Sage, TeX

{\rm He}_3\rtimes Q_{16}
% in TeX

G:=Group("He3:Q16");
// GroupNames label

G:=SmallGroup(432,236);
// by ID

G=gap.SmallGroup(432,236);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,56,85,64,254,135,58,1124,851,298,348,1027,537,14118,7069]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^8=1,e^2=d^4,a*b=b*a,c*a*c^-1=d*c*d^-1=e*c*e^-1=a*b^-1,d*a*d^-1=b*c^-1,e*a*e^-1=b^-1*c,b*c=c*b,b*d=d*b,e*b*e^-1=b^-1,e*d*e^-1=d^-1>;
// generators/relations

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