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## G = C33⋊12SD16order 432 = 24·33

### 4th semidirect product of C33 and SD16 acting via SD16/C4=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32×C12 — C33⋊12SD16
 Chief series C1 — C3 — C32 — C33 — C32×C6 — C32×C12 — C32×D12 — C33⋊12SD16
 Lower central C33 — C32×C6 — C32×C12 — C33⋊12SD16
 Upper central C1 — C2 — C4

Generators and relations for C3312SD16
G = < a,b,c,d,e | a3=b3=c3=d8=e2=1, ab=ba, ac=ca, dad-1=a-1, ae=ea, bc=cb, dbd-1=b-1, be=eb, dcd-1=ece=c-1, ede=d3 >

Subgroups: 656 in 152 conjugacy classes, 46 normal (18 characteristic)
C1, C2, C2, C3, C3, C3, C4, C4, C22, S3, C6, C6, C6, C8, D4, Q8, C32, C32, C32, Dic3, C12, C12, C12, D6, C2×C6, SD16, C3×S3, C3×C6, C3×C6, C3×C6, C3⋊C8, Dic6, D12, C3×D4, C3×Q8, C33, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, C3×C12, S3×C6, C62, D4.S3, Q82S3, S3×C32, C32×C6, C324C8, C3×Dic6, C3×D12, C324Q8, D4×C32, C3×C3⋊Dic3, C32×C12, S3×C3×C6, Dic6⋊S3, C329SD16, C337C8, C32×D12, C3×C324Q8, C3312SD16
Quotients: C1, C2, C22, S3, D4, D6, SD16, C3⋊S3, C3⋊D4, S32, C2×C3⋊S3, D4.S3, Q82S3, D6⋊S3, C327D4, S3×C3⋊S3, Dic6⋊S3, C329SD16, C336D4, C3312SD16

Smallest permutation representation of C3312SD16
On 144 points
Generators in S144
(1 135 66)(2 67 136)(3 129 68)(4 69 130)(5 131 70)(6 71 132)(7 133 72)(8 65 134)(9 33 54)(10 55 34)(11 35 56)(12 49 36)(13 37 50)(14 51 38)(15 39 52)(16 53 40)(17 111 120)(18 113 112)(19 105 114)(20 115 106)(21 107 116)(22 117 108)(23 109 118)(24 119 110)(25 89 59)(26 60 90)(27 91 61)(28 62 92)(29 93 63)(30 64 94)(31 95 57)(32 58 96)(41 75 100)(42 101 76)(43 77 102)(44 103 78)(45 79 104)(46 97 80)(47 73 98)(48 99 74)(81 138 123)(82 124 139)(83 140 125)(84 126 141)(85 142 127)(86 128 143)(87 144 121)(88 122 137)
(1 91 106)(2 107 92)(3 93 108)(4 109 94)(5 95 110)(6 111 96)(7 89 112)(8 105 90)(9 100 88)(10 81 101)(11 102 82)(12 83 103)(13 104 84)(14 85 97)(15 98 86)(16 87 99)(17 58 132)(18 133 59)(19 60 134)(20 135 61)(21 62 136)(22 129 63)(23 64 130)(24 131 57)(25 113 72)(26 65 114)(27 115 66)(28 67 116)(29 117 68)(30 69 118)(31 119 70)(32 71 120)(33 41 122)(34 123 42)(35 43 124)(36 125 44)(37 45 126)(38 127 46)(39 47 128)(40 121 48)(49 140 78)(50 79 141)(51 142 80)(52 73 143)(53 144 74)(54 75 137)(55 138 76)(56 77 139)
(1 27 20)(2 21 28)(3 29 22)(4 23 30)(5 31 24)(6 17 32)(7 25 18)(8 19 26)(9 122 75)(10 76 123)(11 124 77)(12 78 125)(13 126 79)(14 80 127)(15 128 73)(16 74 121)(33 137 100)(34 101 138)(35 139 102)(36 103 140)(37 141 104)(38 97 142)(39 143 98)(40 99 144)(41 54 88)(42 81 55)(43 56 82)(44 83 49)(45 50 84)(46 85 51)(47 52 86)(48 87 53)(57 110 70)(58 71 111)(59 112 72)(60 65 105)(61 106 66)(62 67 107)(63 108 68)(64 69 109)(89 113 133)(90 134 114)(91 115 135)(92 136 116)(93 117 129)(94 130 118)(95 119 131)(96 132 120)
(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 49)(2 52)(3 55)(4 50)(5 53)(6 56)(7 51)(8 54)(9 65)(10 68)(11 71)(12 66)(13 69)(14 72)(15 67)(16 70)(17 43)(18 46)(19 41)(20 44)(21 47)(22 42)(23 45)(24 48)(25 85)(26 88)(27 83)(28 86)(29 81)(30 84)(31 87)(32 82)(33 134)(34 129)(35 132)(36 135)(37 130)(38 133)(39 136)(40 131)(57 121)(58 124)(59 127)(60 122)(61 125)(62 128)(63 123)(64 126)(73 107)(74 110)(75 105)(76 108)(77 111)(78 106)(79 109)(80 112)(89 142)(90 137)(91 140)(92 143)(93 138)(94 141)(95 144)(96 139)(97 113)(98 116)(99 119)(100 114)(101 117)(102 120)(103 115)(104 118)

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

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

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

48 conjugacy classes

 class 1 2A 2B 3A ··· 3E 3F 3G 3H 3I 4A 4B 6A ··· 6E 6F 6G 6H 6I 6J ··· 6Q 8A 8B 12A ··· 12M 12N 12O order 1 2 2 3 ··· 3 3 3 3 3 4 4 6 ··· 6 6 6 6 6 6 ··· 6 8 8 12 ··· 12 12 12 size 1 1 12 2 ··· 2 4 4 4 4 2 36 2 ··· 2 4 4 4 4 12 ··· 12 54 54 4 ··· 4 36 36

48 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + - + - image C1 C2 C2 C2 S3 S3 D4 D6 SD16 C3⋊D4 S32 D4.S3 Q8⋊2S3 D6⋊S3 Dic6⋊S3 kernel C33⋊12SD16 C33⋊7C8 C32×D12 C3×C32⋊4Q8 C3×D12 C32⋊4Q8 C32×C6 C3×C12 C33 C3×C6 C12 C32 C32 C6 C3 # reps 1 1 1 1 4 1 1 5 2 10 4 4 1 4 8

Matrix representation of C3312SD16 in GL8(𝔽73)

 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 71 70 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 1 72 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 71 70 0 0 0 0 0 0 1 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
,
 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 0 0 0 0 0 0 0 0 72 72 0 0 0 0 0 0 1 0
,
 12 69 0 0 0 0 0 0 18 0 0 0 0 0 0 0 0 0 6 2 0 0 0 0 0 0 19 67 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 72 72
,
 29 58 0 0 0 0 0 0 56 44 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 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 72 72

G:=sub<GL(8,GF(73))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,71,1,0,0,0,0,0,0,70,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,71,1,0,0,0,0,0,0,70,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,72,0],[12,18,0,0,0,0,0,0,69,0,0,0,0,0,0,0,0,0,6,19,0,0,0,0,0,0,2,67,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,72],[29,56,0,0,0,0,0,0,58,44,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,72] >;

C3312SD16 in GAP, Magma, Sage, TeX

C_3^3\rtimes_{12}{\rm SD}_{16}
% in TeX

G:=Group("C3^3:12SD16");
// GroupNames label

G:=SmallGroup(432,439);
// by ID

G=gap.SmallGroup(432,439);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,85,64,254,135,58,571,2028,14118]);
// Polycyclic

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

׿
×
𝔽