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## G = C3×C32⋊11SD16order 432 = 24·33

### Direct product of C3 and C32⋊11SD16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — C3×C32⋊11SD16
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C32×C12 — C3×C12⋊S3 — C3×C32⋊11SD16
 Lower central C32 — C3×C6 — C3×C12 — C3×C32⋊11SD16
 Upper central C1 — C6 — C12 — C3×Q8

Generators and relations for C3×C3211SD16
G = < a,b,c,d,e | a3=b3=c3=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=ebe=b-1, dcd-1=ece=c-1, ede=d3 >

Subgroups: 564 in 168 conjugacy classes, 54 normal (22 characteristic)
C1, C2, C2, C3, C3, C3, C4, C4, C22, S3, C6, C6, C6, C8, D4, Q8, C32, C32, C32, C12, C12, C12, D6, C2×C6, SD16, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, C3⋊C8, C24, D12, C3×D4, C3×Q8, C3×Q8, C3×Q8, C33, C3×C12, C3×C12, C3×C12, S3×C6, C2×C3⋊S3, Q82S3, C3×SD16, C3×C3⋊S3, C32×C6, C3×C3⋊C8, C324C8, C3×D12, C12⋊S3, Q8×C32, Q8×C32, Q8×C32, C32×C12, C32×C12, C6×C3⋊S3, C3×Q82S3, C3211SD16, C3×C324C8, C3×C12⋊S3, Q8×C33, C3×C3211SD16
Quotients: C1, C2, C3, C22, S3, C6, D4, D6, C2×C6, SD16, C3×S3, C3⋊S3, C3⋊D4, C3×D4, S3×C6, C2×C3⋊S3, Q82S3, C3×SD16, C3×C3⋊S3, C3×C3⋊D4, C327D4, C6×C3⋊S3, C3×Q82S3, C3211SD16, C3×C327D4, C3×C3211SD16

Smallest permutation representation of C3×C3211SD16
On 144 points
Generators in S144
(1 47 135)(2 48 136)(3 41 129)(4 42 130)(5 43 131)(6 44 132)(7 45 133)(8 46 134)(9 33 54)(10 34 55)(11 35 56)(12 36 49)(13 37 50)(14 38 51)(15 39 52)(16 40 53)(17 121 120)(18 122 113)(19 123 114)(20 124 115)(21 125 116)(22 126 117)(23 127 118)(24 128 119)(25 59 89)(26 60 90)(27 61 91)(28 62 92)(29 63 93)(30 64 94)(31 57 95)(32 58 96)(65 80 97)(66 73 98)(67 74 99)(68 75 100)(69 76 101)(70 77 102)(71 78 103)(72 79 104)(81 105 138)(82 106 139)(83 107 140)(84 108 141)(85 109 142)(86 110 143)(87 111 144)(88 112 137)
(1 91 124)(2 125 92)(3 93 126)(4 127 94)(5 95 128)(6 121 96)(7 89 122)(8 123 90)(9 88 100)(10 101 81)(11 82 102)(12 103 83)(13 84 104)(14 97 85)(15 86 98)(16 99 87)(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 45)(26 46 114)(27 115 47)(28 48 116)(29 117 41)(30 42 118)(31 119 43)(32 44 120)(33 112 68)(34 69 105)(35 106 70)(36 71 107)(37 108 72)(38 65 109)(39 110 66)(40 67 111)(49 78 140)(50 141 79)(51 80 142)(52 143 73)(53 74 144)(54 137 75)(55 76 138)(56 139 77)
(1 27 20)(2 21 28)(3 29 22)(4 23 30)(5 31 24)(6 17 32)(7 25 18)(8 19 26)(9 112 75)(10 76 105)(11 106 77)(12 78 107)(13 108 79)(14 80 109)(15 110 73)(16 74 111)(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 63 126)(42 127 64)(43 57 128)(44 121 58)(45 59 122)(46 123 60)(47 61 124)(48 125 62)(49 71 83)(50 84 72)(51 65 85)(52 86 66)(53 67 87)(54 88 68)(55 69 81)(56 82 70)(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 46)(10 41)(11 44)(12 47)(13 42)(14 45)(15 48)(16 43)(17 70)(18 65)(19 68)(20 71)(21 66)(22 69)(23 72)(24 67)(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)(57 111)(58 106)(59 109)(60 112)(61 107)(62 110)(63 105)(64 108)(73 125)(74 128)(75 123)(76 126)(77 121)(78 124)(79 127)(80 122)(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)| (1,47,135)(2,48,136)(3,41,129)(4,42,130)(5,43,131)(6,44,132)(7,45,133)(8,46,134)(9,33,54)(10,34,55)(11,35,56)(12,36,49)(13,37,50)(14,38,51)(15,39,52)(16,40,53)(17,121,120)(18,122,113)(19,123,114)(20,124,115)(21,125,116)(22,126,117)(23,127,118)(24,128,119)(25,59,89)(26,60,90)(27,61,91)(28,62,92)(29,63,93)(30,64,94)(31,57,95)(32,58,96)(65,80,97)(66,73,98)(67,74,99)(68,75,100)(69,76,101)(70,77,102)(71,78,103)(72,79,104)(81,105,138)(82,106,139)(83,107,140)(84,108,141)(85,109,142)(86,110,143)(87,111,144)(88,112,137), (1,91,124)(2,125,92)(3,93,126)(4,127,94)(5,95,128)(6,121,96)(7,89,122)(8,123,90)(9,88,100)(10,101,81)(11,82,102)(12,103,83)(13,84,104)(14,97,85)(15,86,98)(16,99,87)(17,58,132)(18,133,59)(19,60,134)(20,135,61)(21,62,136)(22,129,63)(23,64,130)(24,131,57)(25,113,45)(26,46,114)(27,115,47)(28,48,116)(29,117,41)(30,42,118)(31,119,43)(32,44,120)(33,112,68)(34,69,105)(35,106,70)(36,71,107)(37,108,72)(38,65,109)(39,110,66)(40,67,111)(49,78,140)(50,141,79)(51,80,142)(52,143,73)(53,74,144)(54,137,75)(55,76,138)(56,139,77), (1,27,20)(2,21,28)(3,29,22)(4,23,30)(5,31,24)(6,17,32)(7,25,18)(8,19,26)(9,112,75)(10,76,105)(11,106,77)(12,78,107)(13,108,79)(14,80,109)(15,110,73)(16,74,111)(33,137,100)(34,101,138)(35,139,102)(36,103,140)(37,141,104)(38,97,142)(39,143,98)(40,99,144)(41,63,126)(42,127,64)(43,57,128)(44,121,58)(45,59,122)(46,123,60)(47,61,124)(48,125,62)(49,71,83)(50,84,72)(51,65,85)(52,86,66)(53,67,87)(54,88,68)(55,69,81)(56,82,70)(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)(9,46)(10,41)(11,44)(12,47)(13,42)(14,45)(15,48)(16,43)(17,70)(18,65)(19,68)(20,71)(21,66)(22,69)(23,72)(24,67)(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)(57,111)(58,106)(59,109)(60,112)(61,107)(62,110)(63,105)(64,108)(73,125)(74,128)(75,123)(76,126)(77,121)(78,124)(79,127)(80,122)(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( (1,47,135)(2,48,136)(3,41,129)(4,42,130)(5,43,131)(6,44,132)(7,45,133)(8,46,134)(9,33,54)(10,34,55)(11,35,56)(12,36,49)(13,37,50)(14,38,51)(15,39,52)(16,40,53)(17,121,120)(18,122,113)(19,123,114)(20,124,115)(21,125,116)(22,126,117)(23,127,118)(24,128,119)(25,59,89)(26,60,90)(27,61,91)(28,62,92)(29,63,93)(30,64,94)(31,57,95)(32,58,96)(65,80,97)(66,73,98)(67,74,99)(68,75,100)(69,76,101)(70,77,102)(71,78,103)(72,79,104)(81,105,138)(82,106,139)(83,107,140)(84,108,141)(85,109,142)(86,110,143)(87,111,144)(88,112,137), (1,91,124)(2,125,92)(3,93,126)(4,127,94)(5,95,128)(6,121,96)(7,89,122)(8,123,90)(9,88,100)(10,101,81)(11,82,102)(12,103,83)(13,84,104)(14,97,85)(15,86,98)(16,99,87)(17,58,132)(18,133,59)(19,60,134)(20,135,61)(21,62,136)(22,129,63)(23,64,130)(24,131,57)(25,113,45)(26,46,114)(27,115,47)(28,48,116)(29,117,41)(30,42,118)(31,119,43)(32,44,120)(33,112,68)(34,69,105)(35,106,70)(36,71,107)(37,108,72)(38,65,109)(39,110,66)(40,67,111)(49,78,140)(50,141,79)(51,80,142)(52,143,73)(53,74,144)(54,137,75)(55,76,138)(56,139,77), (1,27,20)(2,21,28)(3,29,22)(4,23,30)(5,31,24)(6,17,32)(7,25,18)(8,19,26)(9,112,75)(10,76,105)(11,106,77)(12,78,107)(13,108,79)(14,80,109)(15,110,73)(16,74,111)(33,137,100)(34,101,138)(35,139,102)(36,103,140)(37,141,104)(38,97,142)(39,143,98)(40,99,144)(41,63,126)(42,127,64)(43,57,128)(44,121,58)(45,59,122)(46,123,60)(47,61,124)(48,125,62)(49,71,83)(50,84,72)(51,65,85)(52,86,66)(53,67,87)(54,88,68)(55,69,81)(56,82,70)(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)(9,46)(10,41)(11,44)(12,47)(13,42)(14,45)(15,48)(16,43)(17,70)(18,65)(19,68)(20,71)(21,66)(22,69)(23,72)(24,67)(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)(57,111)(58,106)(59,109)(60,112)(61,107)(62,110)(63,105)(64,108)(73,125)(74,128)(75,123)(76,126)(77,121)(78,124)(79,127)(80,122)(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([[(1,47,135),(2,48,136),(3,41,129),(4,42,130),(5,43,131),(6,44,132),(7,45,133),(8,46,134),(9,33,54),(10,34,55),(11,35,56),(12,36,49),(13,37,50),(14,38,51),(15,39,52),(16,40,53),(17,121,120),(18,122,113),(19,123,114),(20,124,115),(21,125,116),(22,126,117),(23,127,118),(24,128,119),(25,59,89),(26,60,90),(27,61,91),(28,62,92),(29,63,93),(30,64,94),(31,57,95),(32,58,96),(65,80,97),(66,73,98),(67,74,99),(68,75,100),(69,76,101),(70,77,102),(71,78,103),(72,79,104),(81,105,138),(82,106,139),(83,107,140),(84,108,141),(85,109,142),(86,110,143),(87,111,144),(88,112,137)], [(1,91,124),(2,125,92),(3,93,126),(4,127,94),(5,95,128),(6,121,96),(7,89,122),(8,123,90),(9,88,100),(10,101,81),(11,82,102),(12,103,83),(13,84,104),(14,97,85),(15,86,98),(16,99,87),(17,58,132),(18,133,59),(19,60,134),(20,135,61),(21,62,136),(22,129,63),(23,64,130),(24,131,57),(25,113,45),(26,46,114),(27,115,47),(28,48,116),(29,117,41),(30,42,118),(31,119,43),(32,44,120),(33,112,68),(34,69,105),(35,106,70),(36,71,107),(37,108,72),(38,65,109),(39,110,66),(40,67,111),(49,78,140),(50,141,79),(51,80,142),(52,143,73),(53,74,144),(54,137,75),(55,76,138),(56,139,77)], [(1,27,20),(2,21,28),(3,29,22),(4,23,30),(5,31,24),(6,17,32),(7,25,18),(8,19,26),(9,112,75),(10,76,105),(11,106,77),(12,78,107),(13,108,79),(14,80,109),(15,110,73),(16,74,111),(33,137,100),(34,101,138),(35,139,102),(36,103,140),(37,141,104),(38,97,142),(39,143,98),(40,99,144),(41,63,126),(42,127,64),(43,57,128),(44,121,58),(45,59,122),(46,123,60),(47,61,124),(48,125,62),(49,71,83),(50,84,72),(51,65,85),(52,86,66),(53,67,87),(54,88,68),(55,69,81),(56,82,70),(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),(9,46),(10,41),(11,44),(12,47),(13,42),(14,45),(15,48),(16,43),(17,70),(18,65),(19,68),(20,71),(21,66),(22,69),(23,72),(24,67),(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),(57,111),(58,106),(59,109),(60,112),(61,107),(62,110),(63,105),(64,108),(73,125),(74,128),(75,123),(76,126),(77,121),(78,124),(79,127),(80,122),(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)]])

81 conjugacy classes

 class 1 2A 2B 3A 3B 3C ··· 3N 4A 4B 6A 6B 6C ··· 6N 6O 6P 8A 8B 12A 12B 12C ··· 12AN 24A 24B 24C 24D order 1 2 2 3 3 3 ··· 3 4 4 6 6 6 ··· 6 6 6 8 8 12 12 12 ··· 12 24 24 24 24 size 1 1 36 1 1 2 ··· 2 2 4 1 1 2 ··· 2 36 36 18 18 2 2 4 ··· 4 18 18 18 18

81 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 S3 D4 D6 SD16 C3×S3 C3⋊D4 C3×D4 S3×C6 C3×SD16 C3×C3⋊D4 Q8⋊2S3 C3×Q8⋊2S3 kernel C3×C32⋊11SD16 C3×C32⋊4C8 C3×C12⋊S3 Q8×C33 C32⋊11SD16 C32⋊4C8 C12⋊S3 Q8×C32 Q8×C32 C32×C6 C3×C12 C33 C3×Q8 C3×C6 C3×C6 C12 C32 C6 C32 C3 # reps 1 1 1 1 2 2 2 2 4 1 4 2 8 8 2 8 4 16 4 8

Matrix representation of C3×C3211SD16 in GL6(𝔽73)

 64 0 0 0 0 0 0 64 0 0 0 0 0 0 64 0 0 0 0 0 0 64 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 8 0 0 0 0 0 0 64 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 64 0 0 0 0 0 56 8 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 72 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 61 45 0 0 0 0 13 0
,
 72 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 11 49 0 0 0 0 5 62

G:=sub<GL(6,GF(73))| [64,0,0,0,0,0,0,64,0,0,0,0,0,0,64,0,0,0,0,0,0,64,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,0,0,0,0,0,0,64,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[64,56,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,0,0,0,0,0,1,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,61,13,0,0,0,0,45,0],[72,0,0,0,0,0,1,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,11,5,0,0,0,0,49,62] >;

C3×C3211SD16 in GAP, Magma, Sage, TeX

C_3\times C_3^2\rtimes_{11}{\rm SD}_{16}
% in TeX

G:=Group("C3xC3^2:11SD16");
// GroupNames label

G:=SmallGroup(432,493);
// by ID

G=gap.SmallGroup(432,493);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,197,176,1011,514,80,4037,14118]);
// 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,a*e=e*a,b*c=c*b,d*b*d^-1=e*b*e=b^-1,d*c*d^-1=e*c*e=c^-1,e*d*e=d^3>;
// generators/relations

׿
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