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## G = C33⋊16SD16order 432 = 24·33

### 8th semidirect product of C33 and SD16 acting via SD16/C4=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32×C12 — C33⋊16SD16
 Chief series C1 — C3 — C32 — C33 — C32×C6 — C32×C12 — C32×C3⋊C8 — C33⋊16SD16
 Lower central C33 — C32×C6 — C32×C12 — C33⋊16SD16
 Upper central C1 — C2 — C4

Generators and relations for C3316SD16
G = < a,b,c,d,e | a3=b3=c3=d8=e2=1, ab=ba, ac=ca, ad=da, eae=a-1, bc=cb, bd=db, ebe=b-1, dcd-1=c-1, ce=ec, ede=d3 >

Subgroups: 1056 in 168 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⋊S3, C3×C6, C3×C6, C3×C6, C3⋊C8, C24, Dic6, D12, C3×D4, C33, C3⋊Dic3, C3×C12, C3×C12, C3×C12, S3×C6, C2×C3⋊S3, C24⋊C2, D4.S3, C3×C3⋊S3, C32×C6, C3×C3⋊C8, C3×C24, C3×D12, C324Q8, C12⋊S3, C335C4, C32×C12, C6×C3⋊S3, D12.S3, C242S3, C32×C3⋊C8, C3×C12⋊S3, C338Q8, C3316SD16
Quotients: C1, C2, C22, S3, D4, D6, SD16, C3⋊S3, D12, C3⋊D4, S32, C2×C3⋊S3, C24⋊C2, D4.S3, C3⋊D12, C12⋊S3, S3×C3⋊S3, D12.S3, C242S3, C338D4, C3316SD16

Smallest permutation representation of C3316SD16
On 144 points
Generators in S144
(1 65 76)(2 66 77)(3 67 78)(4 68 79)(5 69 80)(6 70 73)(7 71 74)(8 72 75)(9 134 44)(10 135 45)(11 136 46)(12 129 47)(13 130 48)(14 131 41)(15 132 42)(16 133 43)(17 105 140)(18 106 141)(19 107 142)(20 108 143)(21 109 144)(22 110 137)(23 111 138)(24 112 139)(25 118 37)(26 119 38)(27 120 39)(28 113 40)(29 114 33)(30 115 34)(31 116 35)(32 117 36)(49 62 125)(50 63 126)(51 64 127)(52 57 128)(53 58 121)(54 59 122)(55 60 123)(56 61 124)(81 92 103)(82 93 104)(83 94 97)(84 95 98)(85 96 99)(86 89 100)(87 90 101)(88 91 102)
(1 141 125)(2 142 126)(3 143 127)(4 144 128)(5 137 121)(6 138 122)(7 139 123)(8 140 124)(9 118 97)(10 119 98)(11 120 99)(12 113 100)(13 114 101)(14 115 102)(15 116 103)(16 117 104)(17 56 72)(18 49 65)(19 50 66)(20 51 67)(21 52 68)(22 53 69)(23 54 70)(24 55 71)(25 94 44)(26 95 45)(27 96 46)(28 89 47)(29 90 48)(30 91 41)(31 92 42)(32 93 43)(33 87 130)(34 88 131)(35 81 132)(36 82 133)(37 83 134)(38 84 135)(39 85 136)(40 86 129)(57 79 109)(58 80 110)(59 73 111)(60 74 112)(61 75 105)(62 76 106)(63 77 107)(64 78 108)
(1 106 49)(2 50 107)(3 108 51)(4 52 109)(5 110 53)(6 54 111)(7 112 55)(8 56 105)(9 83 25)(10 26 84)(11 85 27)(12 28 86)(13 87 29)(14 30 88)(15 81 31)(16 32 82)(17 75 124)(18 125 76)(19 77 126)(20 127 78)(21 79 128)(22 121 80)(23 73 122)(24 123 74)(33 48 101)(34 102 41)(35 42 103)(36 104 43)(37 44 97)(38 98 45)(39 46 99)(40 100 47)(57 144 68)(58 69 137)(59 138 70)(60 71 139)(61 140 72)(62 65 141)(63 142 66)(64 67 143)(89 129 113)(90 114 130)(91 131 115)(92 116 132)(93 133 117)(94 118 134)(95 135 119)(96 120 136)
(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 31)(2 26)(3 29)(4 32)(5 27)(6 30)(7 25)(8 28)(9 112)(10 107)(11 110)(12 105)(13 108)(14 111)(15 106)(16 109)(17 129)(18 132)(19 135)(20 130)(21 133)(22 136)(23 131)(24 134)(33 67)(34 70)(35 65)(36 68)(37 71)(38 66)(39 69)(40 72)(41 138)(42 141)(43 144)(44 139)(45 142)(46 137)(47 140)(48 143)(49 81)(50 84)(51 87)(52 82)(53 85)(54 88)(55 83)(56 86)(57 104)(58 99)(59 102)(60 97)(61 100)(62 103)(63 98)(64 101)(73 115)(74 118)(75 113)(76 116)(77 119)(78 114)(79 117)(80 120)(89 124)(90 127)(91 122)(92 125)(93 128)(94 123)(95 126)(96 121)

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

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

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

60 conjugacy classes

 class 1 2A 2B 3A ··· 3E 3F 3G 3H 3I 4A 4B 6A ··· 6E 6F 6G 6H 6I 6J 6K 8A 8B 12A ··· 12H 12I ··· 12Q 24A ··· 24P 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 24 ··· 24 size 1 1 36 2 ··· 2 4 4 4 4 2 108 2 ··· 2 4 4 4 4 36 36 6 6 2 ··· 2 4 ··· 4 6 ··· 6

60 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + - + - image C1 C2 C2 C2 S3 S3 D4 D6 SD16 D12 C3⋊D4 C24⋊C2 S32 D4.S3 C3⋊D12 D12.S3 kernel C33⋊16SD16 C32×C3⋊C8 C3×C12⋊S3 C33⋊8Q8 C3×C3⋊C8 C12⋊S3 C32×C6 C3×C12 C33 C3×C6 C3×C6 C32 C12 C32 C6 C3 # reps 1 1 1 1 4 1 1 5 2 8 2 16 4 1 4 8

Matrix representation of C3316SD16 in GL8(𝔽73)

 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 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 72 1 0 0 0 0 0 0 72 0 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 0 72 0 0 0 0 0 0 1 72
,
 14 18 0 0 0 0 0 0 65 47 0 0 0 0 0 0 0 0 66 14 0 0 0 0 0 0 59 7 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 0 1 0 0 0 0 0 0 1 0
,
 28 42 0 0 0 0 0 0 70 45 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 72 1 0 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1

G:=sub<GL(8,GF(73))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72],[14,65,0,0,0,0,0,0,18,47,0,0,0,0,0,0,0,0,66,59,0,0,0,0,0,0,14,7,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[28,70,0,0,0,0,0,0,42,45,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1] >;

C3316SD16 in GAP, Magma, Sage, TeX

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

G:=Group("C3^3:16SD16");
// GroupNames label

G:=SmallGroup(432,443);
// by ID

G=gap.SmallGroup(432,443);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,28,197,135,58,571,2028,14118]);
// 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,a*d=d*a,e*a*e=a^-1,b*c=c*b,b*d=d*b,e*b*e=b^-1,d*c*d^-1=c^-1,c*e=e*c,e*d*e=d^3>;
// generators/relations

׿
×
𝔽