direct product, metabelian, supersoluble, monomial
Aliases: C2×C32⋊9SD16, C62.132D4, (C3×C6)⋊9SD16, C6⋊3(D4.S3), (C3×D4).42D6, (C6×D4).15S3, (C3×C12).99D4, (C2×C12).151D6, C32⋊18(C2×SD16), C12.58(C3⋊D4), C12.99(C22×S3), C4.6(C32⋊7D4), (C3×C12).103C23, (C6×C12).142C22, C32⋊4C8⋊23C22, C32⋊4Q8⋊18C22, (D4×C32).27C22, C22.22(C32⋊7D4), (D4×C3×C6).8C2, D4.7(C2×C3⋊S3), C3⋊4(C2×D4.S3), (C2×D4).4(C3⋊S3), (C3×C6).280(C2×D4), C6.121(C2×C3⋊D4), C4.13(C22×C3⋊S3), (C2×C32⋊4C8)⋊10C2, (C2×C32⋊4Q8)⋊15C2, C2.10(C2×C32⋊7D4), (C2×C6).100(C3⋊D4), (C2×C4).47(C2×C3⋊S3), SmallGroup(288,790)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C32⋊9SD16
G = < a,b,c,d,e | a2=b3=c3=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=b-1, be=eb, dcd-1=c-1, ce=ec, ede=d3 >
Subgroups: 628 in 204 conjugacy classes, 77 normal (17 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C32, Dic3, C12, C2×C6, C2×C6, C2×C8, SD16, C2×D4, C2×Q8, C3×C6, C3×C6, C3×C6, C3⋊C8, Dic6, C2×Dic3, C2×C12, C3×D4, C3×D4, C22×C6, C2×SD16, C3⋊Dic3, C3×C12, C62, C62, C2×C3⋊C8, D4.S3, C2×Dic6, C6×D4, C32⋊4C8, C32⋊4Q8, C32⋊4Q8, C2×C3⋊Dic3, C6×C12, D4×C32, D4×C32, C2×C62, C2×D4.S3, C2×C32⋊4C8, C32⋊9SD16, C2×C32⋊4Q8, D4×C3×C6, C2×C32⋊9SD16
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, C2×D4, C3⋊S3, C3⋊D4, C22×S3, C2×SD16, C2×C3⋊S3, D4.S3, C2×C3⋊D4, C32⋊7D4, C22×C3⋊S3, C2×D4.S3, C32⋊9SD16, C2×C32⋊7D4, C2×C32⋊9SD16
►On 144 points
Generators in S144
(1 99)(2 100)(3 101)(4 102)(5 103)(6 104)(7 97)(8 98)(9 115)(10 116)(11 117)(12 118)(13 119)(14 120)(15 113)(16 114)(17 129)(18 130)(19 131)(20 132)(21 133)(22 134)(23 135)(24 136)(25 110)(26 111)(27 112)(28 105)(29 106)(30 107)(31 108)(32 109)(33 80)(34 73)(35 74)(36 75)(37 76)(38 77)(39 78)(40 79)(41 84)(42 85)(43 86)(44 87)(45 88)(46 81)(47 82)(48 83)(49 92)(50 93)(51 94)(52 95)(53 96)(54 89)(55 90)(56 91)(57 68)(58 69)(59 70)(60 71)(61 72)(62 65)(63 66)(64 67)(121 141)(122 142)(123 143)(124 144)(125 137)(126 138)(127 139)(128 140)
(1 141 48)(2 41 142)(3 143 42)(4 43 144)(5 137 44)(6 45 138)(7 139 46)(8 47 140)(9 136 55)(10 56 129)(11 130 49)(12 50 131)(13 132 51)(14 52 133)(15 134 53)(16 54 135)(17 116 91)(18 92 117)(19 118 93)(20 94 119)(21 120 95)(22 96 113)(23 114 89)(24 90 115)(25 69 39)(26 40 70)(27 71 33)(28 34 72)(29 65 35)(30 36 66)(31 67 37)(32 38 68)(57 109 77)(58 78 110)(59 111 79)(60 80 112)(61 105 73)(62 74 106)(63 107 75)(64 76 108)(81 97 127)(82 128 98)(83 99 121)(84 122 100)(85 101 123)(86 124 102)(87 103 125)(88 126 104)
(1 136 108)(2 109 129)(3 130 110)(4 111 131)(5 132 112)(6 105 133)(7 134 106)(8 107 135)(9 76 48)(10 41 77)(11 78 42)(12 43 79)(13 80 44)(14 45 73)(15 74 46)(16 47 75)(17 100 32)(18 25 101)(19 102 26)(20 27 103)(21 104 28)(22 29 97)(23 98 30)(24 31 99)(33 87 119)(34 120 88)(35 81 113)(36 114 82)(37 83 115)(38 116 84)(39 85 117)(40 118 86)(49 58 143)(50 144 59)(51 60 137)(52 138 61)(53 62 139)(54 140 63)(55 64 141)(56 142 57)(65 127 96)(66 89 128)(67 121 90)(68 91 122)(69 123 92)(70 93 124)(71 125 94)(72 95 126)
(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)
(2 4)(3 7)(6 8)(10 12)(11 15)(14 16)(17 19)(18 22)(21 23)(25 29)(26 32)(28 30)(34 36)(35 39)(38 40)(41 43)(42 46)(45 47)(49 53)(50 56)(52 54)(57 59)(58 62)(61 63)(65 69)(66 72)(68 70)(73 75)(74 78)(77 79)(81 85)(82 88)(84 86)(89 95)(91 93)(92 96)(97 101)(98 104)(100 102)(105 107)(106 110)(109 111)(113 117)(114 120)(116 118)(122 124)(123 127)(126 128)(129 131)(130 134)(133 135)(138 140)(139 143)(142 144)
G:=sub<Sym(144)| (1,99)(2,100)(3,101)(4,102)(5,103)(6,104)(7,97)(8,98)(9,115)(10,116)(11,117)(12,118)(13,119)(14,120)(15,113)(16,114)(17,129)(18,130)(19,131)(20,132)(21,133)(22,134)(23,135)(24,136)(25,110)(26,111)(27,112)(28,105)(29,106)(30,107)(31,108)(32,109)(33,80)(34,73)(35,74)(36,75)(37,76)(38,77)(39,78)(40,79)(41,84)(42,85)(43,86)(44,87)(45,88)(46,81)(47,82)(48,83)(49,92)(50,93)(51,94)(52,95)(53,96)(54,89)(55,90)(56,91)(57,68)(58,69)(59,70)(60,71)(61,72)(62,65)(63,66)(64,67)(121,141)(122,142)(123,143)(124,144)(125,137)(126,138)(127,139)(128,140), (1,141,48)(2,41,142)(3,143,42)(4,43,144)(5,137,44)(6,45,138)(7,139,46)(8,47,140)(9,136,55)(10,56,129)(11,130,49)(12,50,131)(13,132,51)(14,52,133)(15,134,53)(16,54,135)(17,116,91)(18,92,117)(19,118,93)(20,94,119)(21,120,95)(22,96,113)(23,114,89)(24,90,115)(25,69,39)(26,40,70)(27,71,33)(28,34,72)(29,65,35)(30,36,66)(31,67,37)(32,38,68)(57,109,77)(58,78,110)(59,111,79)(60,80,112)(61,105,73)(62,74,106)(63,107,75)(64,76,108)(81,97,127)(82,128,98)(83,99,121)(84,122,100)(85,101,123)(86,124,102)(87,103,125)(88,126,104), (1,136,108)(2,109,129)(3,130,110)(4,111,131)(5,132,112)(6,105,133)(7,134,106)(8,107,135)(9,76,48)(10,41,77)(11,78,42)(12,43,79)(13,80,44)(14,45,73)(15,74,46)(16,47,75)(17,100,32)(18,25,101)(19,102,26)(20,27,103)(21,104,28)(22,29,97)(23,98,30)(24,31,99)(33,87,119)(34,120,88)(35,81,113)(36,114,82)(37,83,115)(38,116,84)(39,85,117)(40,118,86)(49,58,143)(50,144,59)(51,60,137)(52,138,61)(53,62,139)(54,140,63)(55,64,141)(56,142,57)(65,127,96)(66,89,128)(67,121,90)(68,91,122)(69,123,92)(70,93,124)(71,125,94)(72,95,126), (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), (2,4)(3,7)(6,8)(10,12)(11,15)(14,16)(17,19)(18,22)(21,23)(25,29)(26,32)(28,30)(34,36)(35,39)(38,40)(41,43)(42,46)(45,47)(49,53)(50,56)(52,54)(57,59)(58,62)(61,63)(65,69)(66,72)(68,70)(73,75)(74,78)(77,79)(81,85)(82,88)(84,86)(89,95)(91,93)(92,96)(97,101)(98,104)(100,102)(105,107)(106,110)(109,111)(113,117)(114,120)(116,118)(122,124)(123,127)(126,128)(129,131)(130,134)(133,135)(138,140)(139,143)(142,144)>;
G:=Group( (1,99)(2,100)(3,101)(4,102)(5,103)(6,104)(7,97)(8,98)(9,115)(10,116)(11,117)(12,118)(13,119)(14,120)(15,113)(16,114)(17,129)(18,130)(19,131)(20,132)(21,133)(22,134)(23,135)(24,136)(25,110)(26,111)(27,112)(28,105)(29,106)(30,107)(31,108)(32,109)(33,80)(34,73)(35,74)(36,75)(37,76)(38,77)(39,78)(40,79)(41,84)(42,85)(43,86)(44,87)(45,88)(46,81)(47,82)(48,83)(49,92)(50,93)(51,94)(52,95)(53,96)(54,89)(55,90)(56,91)(57,68)(58,69)(59,70)(60,71)(61,72)(62,65)(63,66)(64,67)(121,141)(122,142)(123,143)(124,144)(125,137)(126,138)(127,139)(128,140), (1,141,48)(2,41,142)(3,143,42)(4,43,144)(5,137,44)(6,45,138)(7,139,46)(8,47,140)(9,136,55)(10,56,129)(11,130,49)(12,50,131)(13,132,51)(14,52,133)(15,134,53)(16,54,135)(17,116,91)(18,92,117)(19,118,93)(20,94,119)(21,120,95)(22,96,113)(23,114,89)(24,90,115)(25,69,39)(26,40,70)(27,71,33)(28,34,72)(29,65,35)(30,36,66)(31,67,37)(32,38,68)(57,109,77)(58,78,110)(59,111,79)(60,80,112)(61,105,73)(62,74,106)(63,107,75)(64,76,108)(81,97,127)(82,128,98)(83,99,121)(84,122,100)(85,101,123)(86,124,102)(87,103,125)(88,126,104), (1,136,108)(2,109,129)(3,130,110)(4,111,131)(5,132,112)(6,105,133)(7,134,106)(8,107,135)(9,76,48)(10,41,77)(11,78,42)(12,43,79)(13,80,44)(14,45,73)(15,74,46)(16,47,75)(17,100,32)(18,25,101)(19,102,26)(20,27,103)(21,104,28)(22,29,97)(23,98,30)(24,31,99)(33,87,119)(34,120,88)(35,81,113)(36,114,82)(37,83,115)(38,116,84)(39,85,117)(40,118,86)(49,58,143)(50,144,59)(51,60,137)(52,138,61)(53,62,139)(54,140,63)(55,64,141)(56,142,57)(65,127,96)(66,89,128)(67,121,90)(68,91,122)(69,123,92)(70,93,124)(71,125,94)(72,95,126), (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), (2,4)(3,7)(6,8)(10,12)(11,15)(14,16)(17,19)(18,22)(21,23)(25,29)(26,32)(28,30)(34,36)(35,39)(38,40)(41,43)(42,46)(45,47)(49,53)(50,56)(52,54)(57,59)(58,62)(61,63)(65,69)(66,72)(68,70)(73,75)(74,78)(77,79)(81,85)(82,88)(84,86)(89,95)(91,93)(92,96)(97,101)(98,104)(100,102)(105,107)(106,110)(109,111)(113,117)(114,120)(116,118)(122,124)(123,127)(126,128)(129,131)(130,134)(133,135)(138,140)(139,143)(142,144) );
G=PermutationGroup([[(1,99),(2,100),(3,101),(4,102),(5,103),(6,104),(7,97),(8,98),(9,115),(10,116),(11,117),(12,118),(13,119),(14,120),(15,113),(16,114),(17,129),(18,130),(19,131),(20,132),(21,133),(22,134),(23,135),(24,136),(25,110),(26,111),(27,112),(28,105),(29,106),(30,107),(31,108),(32,109),(33,80),(34,73),(35,74),(36,75),(37,76),(38,77),(39,78),(40,79),(41,84),(42,85),(43,86),(44,87),(45,88),(46,81),(47,82),(48,83),(49,92),(50,93),(51,94),(52,95),(53,96),(54,89),(55,90),(56,91),(57,68),(58,69),(59,70),(60,71),(61,72),(62,65),(63,66),(64,67),(121,141),(122,142),(123,143),(124,144),(125,137),(126,138),(127,139),(128,140)], [(1,141,48),(2,41,142),(3,143,42),(4,43,144),(5,137,44),(6,45,138),(7,139,46),(8,47,140),(9,136,55),(10,56,129),(11,130,49),(12,50,131),(13,132,51),(14,52,133),(15,134,53),(16,54,135),(17,116,91),(18,92,117),(19,118,93),(20,94,119),(21,120,95),(22,96,113),(23,114,89),(24,90,115),(25,69,39),(26,40,70),(27,71,33),(28,34,72),(29,65,35),(30,36,66),(31,67,37),(32,38,68),(57,109,77),(58,78,110),(59,111,79),(60,80,112),(61,105,73),(62,74,106),(63,107,75),(64,76,108),(81,97,127),(82,128,98),(83,99,121),(84,122,100),(85,101,123),(86,124,102),(87,103,125),(88,126,104)], [(1,136,108),(2,109,129),(3,130,110),(4,111,131),(5,132,112),(6,105,133),(7,134,106),(8,107,135),(9,76,48),(10,41,77),(11,78,42),(12,43,79),(13,80,44),(14,45,73),(15,74,46),(16,47,75),(17,100,32),(18,25,101),(19,102,26),(20,27,103),(21,104,28),(22,29,97),(23,98,30),(24,31,99),(33,87,119),(34,120,88),(35,81,113),(36,114,82),(37,83,115),(38,116,84),(39,85,117),(40,118,86),(49,58,143),(50,144,59),(51,60,137),(52,138,61),(53,62,139),(54,140,63),(55,64,141),(56,142,57),(65,127,96),(66,89,128),(67,121,90),(68,91,122),(69,123,92),(70,93,124),(71,125,94),(72,95,126)], [(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)], [(2,4),(3,7),(6,8),(10,12),(11,15),(14,16),(17,19),(18,22),(21,23),(25,29),(26,32),(28,30),(34,36),(35,39),(38,40),(41,43),(42,46),(45,47),(49,53),(50,56),(52,54),(57,59),(58,62),(61,63),(65,69),(66,72),(68,70),(73,75),(74,78),(77,79),(81,85),(82,88),(84,86),(89,95),(91,93),(92,96),(97,101),(98,104),(100,102),(105,107),(106,110),(109,111),(113,117),(114,120),(116,118),(122,124),(123,127),(126,128),(129,131),(130,134),(133,135),(138,140),(139,143),(142,144)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 6A | ··· | 6L | 6M | ··· | 6AB | 8A | 8B | 8C | 8D | 12A | ··· | 12H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 36 | 36 | 2 | ··· | 2 | 4 | ··· | 4 | 18 | 18 | 18 | 18 | 4 | ··· | 4 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | SD16 | C3⋊D4 | C3⋊D4 | D4.S3 |
| kernel | C2×C32⋊9SD16 | C2×C32⋊4C8 | C32⋊9SD16 | C2×C32⋊4Q8 | D4×C3×C6 | C6×D4 | C3×C12 | C62 | C2×C12 | C3×D4 | C3×C6 | C12 | C2×C6 | C6 |
| # reps | 1 | 1 | 4 | 1 | 1 | 4 | 1 | 1 | 4 | 8 | 4 | 8 | 8 | 8 |
Matrix representation of C2×C32⋊9SD16 ►in GL6(𝔽73)
| 1 | 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 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 40 | 64 |
| 64 | 0 | 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 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 60 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 10 |
| 0 | 0 | 0 | 0 | 31 | 45 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 10 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 9 | 72 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[8,0,0,0,0,0,0,64,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,40,0,0,0,0,0,64],[64,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,72,0,0,0,0,1,0,0,0,0,0,0,0,12,28,0,0,0,0,60,0,0,0,0,0,0,0,28,31,0,0,0,0,10,45],[1,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,10,72,0,0,0,0,0,0,1,9,0,0,0,0,0,72] >;
C2×C32⋊9SD16 in GAP, Magma, Sage, TeX
C_2\times C_3^2\rtimes_9{\rm SD}_{16} % in TeX
G:=Group("C2xC3^2:9SD16"); // GroupNames label
G:=SmallGroup(288,790);
// by ID
G=gap.SmallGroup(288,790);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,254,675,185,80,2693,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=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=b^-1,b*e=e*b,d*c*d^-1=c^-1,c*e=e*c,e*d*e=d^3>;
// generators/relations