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## G = C33⋊13SD16order 432 = 24·33

### 5th semidirect product of C33 and SD16 acting via SD16/C4=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32×C12 — C33⋊13SD16
 Chief series C1 — C3 — C32 — C33 — C32×C6 — C32×C12 — C32×Dic6 — C33⋊13SD16
 Lower central C33 — C32×C6 — C32×C12 — C33⋊13SD16
 Upper central C1 — C2 — C4

Generators and relations for C3313SD16
G = < a,b,c,d,e | a3=b3=c3=d8=e2=1, ab=ba, ac=ca, dad-1=eae=a-1, bc=cb, dbd-1=ebe=b-1, dcd-1=c-1, ce=ec, ede=d3 >

Subgroups: 736 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⋊S3, C3×C6, C3×C6, C3×C6, C3⋊C8, Dic6, D12, C3×D4, C3×Q8, C33, C3×Dic3, C3×C12, C3×C12, C3×C12, S3×C6, C2×C3⋊S3, D4.S3, Q82S3, C3×C3⋊S3, C32×C6, C324C8, C3×Dic6, C3×D12, C12⋊S3, Q8×C32, C32×Dic3, C32×C12, C6×C3⋊S3, Dic6⋊S3, C3211SD16, C337C8, C32×Dic6, C3×C12⋊S3, C3313SD16
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, C3211SD16, C336D4, C3313SD16

Smallest permutation representation of C3313SD16
On 144 points
Generators in S144
(1 54 20)(2 21 55)(3 56 22)(4 23 49)(5 50 24)(6 17 51)(7 52 18)(8 19 53)(9 129 63)(10 64 130)(11 131 57)(12 58 132)(13 133 59)(14 60 134)(15 135 61)(16 62 136)(25 119 86)(26 87 120)(27 113 88)(28 81 114)(29 115 82)(30 83 116)(31 117 84)(32 85 118)(33 43 126)(34 127 44)(35 45 128)(36 121 46)(37 47 122)(38 123 48)(39 41 124)(40 125 42)(65 144 90)(66 91 137)(67 138 92)(68 93 139)(69 140 94)(70 95 141)(71 142 96)(72 89 143)(73 108 98)(74 99 109)(75 110 100)(76 101 111)(77 112 102)(78 103 105)(79 106 104)(80 97 107)
(1 31 15)(2 16 32)(3 25 9)(4 10 26)(5 27 11)(6 12 28)(7 29 13)(8 14 30)(17 58 81)(18 82 59)(19 60 83)(20 84 61)(21 62 85)(22 86 63)(23 64 87)(24 88 57)(33 137 102)(34 103 138)(35 139 104)(36 97 140)(37 141 98)(38 99 142)(39 143 100)(40 101 144)(41 72 75)(42 76 65)(43 66 77)(44 78 67)(45 68 79)(46 80 69)(47 70 73)(48 74 71)(49 130 120)(50 113 131)(51 132 114)(52 115 133)(53 134 116)(54 117 135)(55 136 118)(56 119 129)(89 110 124)(90 125 111)(91 112 126)(92 127 105)(93 106 128)(94 121 107)(95 108 122)(96 123 109)
(1 15 31)(2 32 16)(3 9 25)(4 26 10)(5 11 27)(6 28 12)(7 13 29)(8 30 14)(17 81 58)(18 59 82)(19 83 60)(20 61 84)(21 85 62)(22 63 86)(23 87 64)(24 57 88)(33 137 102)(34 103 138)(35 139 104)(36 97 140)(37 141 98)(38 99 142)(39 143 100)(40 101 144)(41 72 75)(42 76 65)(43 66 77)(44 78 67)(45 68 79)(46 80 69)(47 70 73)(48 74 71)(49 120 130)(50 131 113)(51 114 132)(52 133 115)(53 116 134)(54 135 117)(55 118 136)(56 129 119)(89 110 124)(90 125 111)(91 112 126)(92 127 105)(93 106 128)(94 121 107)(95 108 122)(96 123 109)
(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 70)(2 65)(3 68)(4 71)(5 66)(6 69)(7 72)(8 67)(9 79)(10 74)(11 77)(12 80)(13 75)(14 78)(15 73)(16 76)(17 94)(18 89)(19 92)(20 95)(21 90)(22 93)(23 96)(24 91)(25 45)(26 48)(27 43)(28 46)(29 41)(30 44)(31 47)(32 42)(33 113)(34 116)(35 119)(36 114)(37 117)(38 120)(39 115)(40 118)(49 142)(50 137)(51 140)(52 143)(53 138)(54 141)(55 144)(56 139)(57 112)(58 107)(59 110)(60 105)(61 108)(62 111)(63 106)(64 109)(81 121)(82 124)(83 127)(84 122)(85 125)(86 128)(87 123)(88 126)(97 132)(98 135)(99 130)(100 133)(101 136)(102 131)(103 134)(104 129)

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

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

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

48 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 ··· 12M 12N ··· 12U 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 36 2 ··· 2 4 4 4 4 2 12 2 ··· 2 4 4 4 4 36 36 54 54 4 ··· 4 12 ··· 12

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⋊13SD16 C33⋊7C8 C32×Dic6 C3×C12⋊S3 C3×Dic6 C12⋊S3 C32×C6 C3×C12 C33 C3×C6 C12 C32 C32 C6 C3 # reps 1 1 1 1 4 1 1 5 2 10 4 1 4 4 8

Matrix representation of C3313SD16 in GL8(𝔽73)

 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 70 0 0 0 0 0 0 1 71 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
,
 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
,
 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 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 0 18 0 0 0 0 0 0 69 61 0 0 0 0 0 0 0 0 1 70 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0
,
 1 3 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 72 3 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 0 1 0 0 0 0 0 0 1 0

G:=sub<GL(8,GF(73))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,70,71,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72],[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],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,69,0,0,0,0,0,0,18,61,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,70,72,0,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,3,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,3,1,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;

C3313SD16 in GAP, Magma, Sage, TeX

C_3^3\rtimes_{13}{\rm SD}_{16}
% in TeX

G:=Group("C3^3:13SD16");
// GroupNames label

G:=SmallGroup(432,440);
// by ID

G=gap.SmallGroup(432,440);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,85,254,135,58,571,2028,14118]);
// 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=e*a*e=a^-1,b*c=c*b,d*b*d^-1=e*b*e=b^-1,d*c*d^-1=c^-1,c*e=e*c,e*d*e=d^3>;
// generators/relations

׿
×
𝔽