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## G = C32×Q8⋊3S3order 432 = 24·33

### Direct product of C32 and Q8⋊3S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C32×Q8⋊3S3
 Chief series C1 — C3 — C6 — C3×C6 — C32×C6 — S3×C3×C6 — S3×C3×C12 — C32×Q8⋊3S3
 Lower central C3 — C6 — C32×Q8⋊3S3
 Upper central C1 — C3×C6 — Q8×C32

Generators and relations for C32×Q83S3
G = < a,b,c,d,e,f | a3=b3=c4=e3=f2=1, d2=c2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd-1=fcf=c-1, ce=ec, de=ed, df=fd, fef=e-1 >

Subgroups: 576 in 288 conjugacy classes, 138 normal (16 characteristic)
C1, C2, C2, C3, C3, C3, C4, C4, C22, S3, C6, C6, C6, C2×C4, D4, Q8, C32, C32, C32, Dic3, C12, C12, D6, C2×C6, C4○D4, C3×S3, C3×C6, C3×C6, C3×C6, C4×S3, D12, C2×C12, C3×D4, C3×Q8, C3×Q8, C3×Q8, C33, C3×Dic3, C3×C12, C3×C12, S3×C6, C62, Q83S3, C3×C4○D4, S3×C32, C32×C6, S3×C12, C3×D12, C6×C12, D4×C32, Q8×C32, Q8×C32, Q8×C32, C32×Dic3, C32×C12, S3×C3×C6, C3×Q83S3, C32×C4○D4, S3×C3×C12, C32×D12, Q8×C33, C32×Q83S3
Quotients: C1, C2, C3, C22, S3, C6, C23, C32, D6, C2×C6, C4○D4, C3×S3, C3×C6, C22×S3, C22×C6, S3×C6, C62, Q83S3, C3×C4○D4, S3×C32, S3×C2×C6, C2×C62, S3×C3×C6, C3×Q83S3, C32×C4○D4, S3×C62, C32×Q83S3

Smallest permutation representation of C32×Q83S3
On 144 points
Generators in S144
(1 59 7)(2 60 8)(3 57 5)(4 58 6)(9 30 27)(10 31 28)(11 32 25)(12 29 26)(13 39 34)(14 40 35)(15 37 36)(16 38 33)(17 144 126)(18 141 127)(19 142 128)(20 143 125)(21 118 113)(22 119 114)(23 120 115)(24 117 116)(41 46 95)(42 47 96)(43 48 93)(44 45 94)(49 75 70)(50 76 71)(51 73 72)(52 74 69)(53 66 62)(54 67 63)(55 68 64)(56 65 61)(77 82 131)(78 83 132)(79 84 129)(80 81 130)(85 111 106)(86 112 107)(87 109 108)(88 110 105)(89 102 98)(90 103 99)(91 104 100)(92 101 97)(121 139 135)(122 140 136)(123 137 133)(124 138 134)
(1 29 14)(2 30 15)(3 31 16)(4 32 13)(5 10 33)(6 11 34)(7 12 35)(8 9 36)(17 22 139)(18 23 140)(19 24 137)(20 21 138)(25 39 58)(26 40 59)(27 37 60)(28 38 57)(41 64 71)(42 61 72)(43 62 69)(44 63 70)(45 54 49)(46 55 50)(47 56 51)(48 53 52)(65 73 96)(66 74 93)(67 75 94)(68 76 95)(77 100 107)(78 97 108)(79 98 105)(80 99 106)(81 90 85)(82 91 86)(83 92 87)(84 89 88)(101 109 132)(102 110 129)(103 111 130)(104 112 131)(113 124 125)(114 121 126)(115 122 127)(116 123 128)(117 133 142)(118 134 143)(119 135 144)(120 136 141)
(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 46 3 48)(2 45 4 47)(5 43 7 41)(6 42 8 44)(9 63 11 61)(10 62 12 64)(13 51 15 49)(14 50 16 52)(17 109 19 111)(18 112 20 110)(21 129 23 131)(22 132 24 130)(25 65 27 67)(26 68 28 66)(29 55 31 53)(30 54 32 56)(33 69 35 71)(34 72 36 70)(37 75 39 73)(38 74 40 76)(57 93 59 95)(58 96 60 94)(77 118 79 120)(78 117 80 119)(81 114 83 116)(82 113 84 115)(85 126 87 128)(86 125 88 127)(89 122 91 124)(90 121 92 123)(97 133 99 135)(98 136 100 134)(101 137 103 139)(102 140 104 138)(105 141 107 143)(106 144 108 142)
(1 14 29)(2 15 30)(3 16 31)(4 13 32)(5 33 10)(6 34 11)(7 35 12)(8 36 9)(17 22 139)(18 23 140)(19 24 137)(20 21 138)(25 58 39)(26 59 40)(27 60 37)(28 57 38)(41 71 64)(42 72 61)(43 69 62)(44 70 63)(45 49 54)(46 50 55)(47 51 56)(48 52 53)(65 96 73)(66 93 74)(67 94 75)(68 95 76)(77 100 107)(78 97 108)(79 98 105)(80 99 106)(81 90 85)(82 91 86)(83 92 87)(84 89 88)(101 109 132)(102 110 129)(103 111 130)(104 112 131)(113 124 125)(114 121 126)(115 122 127)(116 123 128)(117 133 142)(118 134 143)(119 135 144)(120 136 141)
(1 84)(2 83)(3 82)(4 81)(5 77)(6 80)(7 79)(8 78)(9 97)(10 100)(11 99)(12 98)(13 85)(14 88)(15 87)(16 86)(17 73)(18 76)(19 75)(20 74)(21 93)(22 96)(23 95)(24 94)(25 103)(26 102)(27 101)(28 104)(29 89)(30 92)(31 91)(32 90)(33 107)(34 106)(35 105)(36 108)(37 109)(38 112)(39 111)(40 110)(41 120)(42 119)(43 118)(44 117)(45 116)(46 115)(47 114)(48 113)(49 128)(50 127)(51 126)(52 125)(53 124)(54 123)(55 122)(56 121)(57 131)(58 130)(59 129)(60 132)(61 135)(62 134)(63 133)(64 136)(65 139)(66 138)(67 137)(68 140)(69 143)(70 142)(71 141)(72 144)

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

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

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

135 conjugacy classes

 class 1 2A 2B 2C 2D 3A ··· 3H 3I ··· 3Q 4A 4B 4C 4D 4E 6A ··· 6H 6I ··· 6Q 6R ··· 6AO 12A ··· 12X 12Y ··· 12AN 12AO ··· 12BO order 1 2 2 2 2 3 ··· 3 3 ··· 3 4 4 4 4 4 6 ··· 6 6 ··· 6 6 ··· 6 12 ··· 12 12 ··· 12 12 ··· 12 size 1 1 6 6 6 1 ··· 1 2 ··· 2 2 2 2 3 3 1 ··· 1 2 ··· 2 6 ··· 6 2 ··· 2 3 ··· 3 4 ··· 4

135 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 S3 D6 C4○D4 C3×S3 S3×C6 C3×C4○D4 Q8⋊3S3 C3×Q8⋊3S3 kernel C32×Q8⋊3S3 S3×C3×C12 C32×D12 Q8×C33 C3×Q8⋊3S3 S3×C12 C3×D12 Q8×C32 Q8×C32 C3×C12 C33 C3×Q8 C12 C32 C32 C3 # reps 1 3 3 1 8 24 24 8 1 3 2 8 24 16 1 8

Matrix representation of C32×Q83S3 in GL4(𝔽13) generated by

 1 0 0 0 0 1 0 0 0 0 9 0 0 0 0 9
,
 3 0 0 0 0 3 0 0 0 0 1 0 0 0 0 1
,
 12 0 0 0 0 12 0 0 0 0 12 5 0 0 10 1
,
 1 0 0 0 0 1 0 0 0 0 1 6 0 0 4 12
,
 9 0 0 0 0 3 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 1 0 0 0 0 0 5 4 0 0 7 8
G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,9,0,0,0,0,9],[3,0,0,0,0,3,0,0,0,0,1,0,0,0,0,1],[12,0,0,0,0,12,0,0,0,0,12,10,0,0,5,1],[1,0,0,0,0,1,0,0,0,0,1,4,0,0,6,12],[9,0,0,0,0,3,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,5,7,0,0,4,8] >;

C32×Q83S3 in GAP, Magma, Sage, TeX

C_3^2\times Q_8\rtimes_3S_3
% in TeX

G:=Group("C3^2xQ8:3S3");
// GroupNames label

G:=SmallGroup(432,707);
// by ID

G=gap.SmallGroup(432,707);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-3,-2,-3,512,1598,807,394,14118]);
// Polycyclic

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

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