direct product, metabelian, supersoluble, monomial
Aliases: C3×C32⋊5Q16, C33⋊11Q16, C32⋊7Dic12, C24.7(C3×S3), C24.7(C3⋊S3), C12.82(S3×C6), (C3×C24).16S3, (C3×C24).11C6, C3⋊1(C3×Dic12), (C3×C6).63D12, C6.21(C3×D12), C32⋊8(C3×Q16), (C3×C12).209D6, (C32×C24).3C2, (C32×C6).59D4, C32⋊4Q8.3C6, C6.24(C12⋊S3), (C32×C12).84C22, C8.(C3×C3⋊S3), C4.10(C6×C3⋊S3), C12.86(C2×C3⋊S3), (C3×C6).51(C3×D4), C2.5(C3×C12⋊S3), (C3×C12).74(C2×C6), (C3×C32⋊4Q8).6C2, SmallGroup(432,484)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C32⋊5Q16
G = < a,b,c,d,e | a3=b3=c3=d8=1, e2=d4, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=b-1, cd=dc, ece-1=c-1, ede-1=d-1 >
Subgroups: 420 in 140 conjugacy classes, 54 normal (18 characteristic)
C1, C2, C3, C3, C3, C4, C4, C6, C6, C6, C8, Q8, C32, C32, C32, Dic3, C12, C12, C12, Q16, C3×C6, C3×C6, C3×C6, C24, C24, C24, Dic6, C3×Q8, C33, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, C3×C12, Dic12, C3×Q16, C32×C6, C3×C24, C3×C24, C3×C24, C3×Dic6, C32⋊4Q8, C3×C3⋊Dic3, C32×C12, C3×Dic12, C32⋊5Q16, C32×C24, C3×C32⋊4Q8, C3×C32⋊5Q16
Quotients: C1, C2, C3, C22, S3, C6, D4, D6, C2×C6, Q16, C3×S3, C3⋊S3, D12, C3×D4, S3×C6, C2×C3⋊S3, Dic12, C3×Q16, C3×C3⋊S3, C3×D12, C12⋊S3, C6×C3⋊S3, C3×Dic12, C32⋊5Q16, C3×C12⋊S3, C3×C32⋊5Q16
►On 144 points
Generators in S144
(1 79 121)(2 80 122)(3 73 123)(4 74 124)(5 75 125)(6 76 126)(7 77 127)(8 78 128)(9 52 20)(10 53 21)(11 54 22)(12 55 23)(13 56 24)(14 49 17)(15 50 18)(16 51 19)(25 58 90)(26 59 91)(27 60 92)(28 61 93)(29 62 94)(30 63 95)(31 64 96)(32 57 89)(33 45 141)(34 46 142)(35 47 143)(36 48 144)(37 41 137)(38 42 138)(39 43 139)(40 44 140)(65 129 88)(66 130 81)(67 131 82)(68 132 83)(69 133 84)(70 134 85)(71 135 86)(72 136 87)(97 117 112)(98 118 105)(99 119 106)(100 120 107)(101 113 108)(102 114 109)(103 115 110)(104 116 111)
(1 68 100)(2 69 101)(3 70 102)(4 71 103)(5 72 104)(6 65 97)(7 66 98)(8 67 99)(9 140 25)(10 141 26)(11 142 27)(12 143 28)(13 144 29)(14 137 30)(15 138 31)(16 139 32)(17 41 95)(18 42 96)(19 43 89)(20 44 90)(21 45 91)(22 46 92)(23 47 93)(24 48 94)(33 59 53)(34 60 54)(35 61 55)(36 62 56)(37 63 49)(38 64 50)(39 57 51)(40 58 52)(73 134 114)(74 135 115)(75 136 116)(76 129 117)(77 130 118)(78 131 119)(79 132 120)(80 133 113)(81 105 127)(82 106 128)(83 107 121)(84 108 122)(85 109 123)(86 110 124)(87 111 125)(88 112 126)
(1 132 107)(2 133 108)(3 134 109)(4 135 110)(5 136 111)(6 129 112)(7 130 105)(8 131 106)(9 44 58)(10 45 59)(11 46 60)(12 47 61)(13 48 62)(14 41 63)(15 42 64)(16 43 57)(17 37 30)(18 38 31)(19 39 32)(20 40 25)(21 33 26)(22 34 27)(23 35 28)(24 36 29)(49 137 95)(50 138 96)(51 139 89)(52 140 90)(53 141 91)(54 142 92)(55 143 93)(56 144 94)(65 117 126)(66 118 127)(67 119 128)(68 120 121)(69 113 122)(70 114 123)(71 115 124)(72 116 125)(73 85 102)(74 86 103)(75 87 104)(76 88 97)(77 81 98)(78 82 99)(79 83 100)(80 84 101)
(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 29 5 25)(2 28 6 32)(3 27 7 31)(4 26 8 30)(9 100 13 104)(10 99 14 103)(11 98 15 102)(12 97 16 101)(17 110 21 106)(18 109 22 105)(19 108 23 112)(20 107 24 111)(33 131 37 135)(34 130 38 134)(35 129 39 133)(36 136 40 132)(41 86 45 82)(42 85 46 81)(43 84 47 88)(44 83 48 87)(49 115 53 119)(50 114 54 118)(51 113 55 117)(52 120 56 116)(57 80 61 76)(58 79 62 75)(59 78 63 74)(60 77 64 73)(65 139 69 143)(66 138 70 142)(67 137 71 141)(68 144 72 140)(89 122 93 126)(90 121 94 125)(91 128 95 124)(92 127 96 123)
G:=sub<Sym(144)| (1,79,121)(2,80,122)(3,73,123)(4,74,124)(5,75,125)(6,76,126)(7,77,127)(8,78,128)(9,52,20)(10,53,21)(11,54,22)(12,55,23)(13,56,24)(14,49,17)(15,50,18)(16,51,19)(25,58,90)(26,59,91)(27,60,92)(28,61,93)(29,62,94)(30,63,95)(31,64,96)(32,57,89)(33,45,141)(34,46,142)(35,47,143)(36,48,144)(37,41,137)(38,42,138)(39,43,139)(40,44,140)(65,129,88)(66,130,81)(67,131,82)(68,132,83)(69,133,84)(70,134,85)(71,135,86)(72,136,87)(97,117,112)(98,118,105)(99,119,106)(100,120,107)(101,113,108)(102,114,109)(103,115,110)(104,116,111), (1,68,100)(2,69,101)(3,70,102)(4,71,103)(5,72,104)(6,65,97)(7,66,98)(8,67,99)(9,140,25)(10,141,26)(11,142,27)(12,143,28)(13,144,29)(14,137,30)(15,138,31)(16,139,32)(17,41,95)(18,42,96)(19,43,89)(20,44,90)(21,45,91)(22,46,92)(23,47,93)(24,48,94)(33,59,53)(34,60,54)(35,61,55)(36,62,56)(37,63,49)(38,64,50)(39,57,51)(40,58,52)(73,134,114)(74,135,115)(75,136,116)(76,129,117)(77,130,118)(78,131,119)(79,132,120)(80,133,113)(81,105,127)(82,106,128)(83,107,121)(84,108,122)(85,109,123)(86,110,124)(87,111,125)(88,112,126), (1,132,107)(2,133,108)(3,134,109)(4,135,110)(5,136,111)(6,129,112)(7,130,105)(8,131,106)(9,44,58)(10,45,59)(11,46,60)(12,47,61)(13,48,62)(14,41,63)(15,42,64)(16,43,57)(17,37,30)(18,38,31)(19,39,32)(20,40,25)(21,33,26)(22,34,27)(23,35,28)(24,36,29)(49,137,95)(50,138,96)(51,139,89)(52,140,90)(53,141,91)(54,142,92)(55,143,93)(56,144,94)(65,117,126)(66,118,127)(67,119,128)(68,120,121)(69,113,122)(70,114,123)(71,115,124)(72,116,125)(73,85,102)(74,86,103)(75,87,104)(76,88,97)(77,81,98)(78,82,99)(79,83,100)(80,84,101), (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), (1,29,5,25)(2,28,6,32)(3,27,7,31)(4,26,8,30)(9,100,13,104)(10,99,14,103)(11,98,15,102)(12,97,16,101)(17,110,21,106)(18,109,22,105)(19,108,23,112)(20,107,24,111)(33,131,37,135)(34,130,38,134)(35,129,39,133)(36,136,40,132)(41,86,45,82)(42,85,46,81)(43,84,47,88)(44,83,48,87)(49,115,53,119)(50,114,54,118)(51,113,55,117)(52,120,56,116)(57,80,61,76)(58,79,62,75)(59,78,63,74)(60,77,64,73)(65,139,69,143)(66,138,70,142)(67,137,71,141)(68,144,72,140)(89,122,93,126)(90,121,94,125)(91,128,95,124)(92,127,96,123)>;
G:=Group( (1,79,121)(2,80,122)(3,73,123)(4,74,124)(5,75,125)(6,76,126)(7,77,127)(8,78,128)(9,52,20)(10,53,21)(11,54,22)(12,55,23)(13,56,24)(14,49,17)(15,50,18)(16,51,19)(25,58,90)(26,59,91)(27,60,92)(28,61,93)(29,62,94)(30,63,95)(31,64,96)(32,57,89)(33,45,141)(34,46,142)(35,47,143)(36,48,144)(37,41,137)(38,42,138)(39,43,139)(40,44,140)(65,129,88)(66,130,81)(67,131,82)(68,132,83)(69,133,84)(70,134,85)(71,135,86)(72,136,87)(97,117,112)(98,118,105)(99,119,106)(100,120,107)(101,113,108)(102,114,109)(103,115,110)(104,116,111), (1,68,100)(2,69,101)(3,70,102)(4,71,103)(5,72,104)(6,65,97)(7,66,98)(8,67,99)(9,140,25)(10,141,26)(11,142,27)(12,143,28)(13,144,29)(14,137,30)(15,138,31)(16,139,32)(17,41,95)(18,42,96)(19,43,89)(20,44,90)(21,45,91)(22,46,92)(23,47,93)(24,48,94)(33,59,53)(34,60,54)(35,61,55)(36,62,56)(37,63,49)(38,64,50)(39,57,51)(40,58,52)(73,134,114)(74,135,115)(75,136,116)(76,129,117)(77,130,118)(78,131,119)(79,132,120)(80,133,113)(81,105,127)(82,106,128)(83,107,121)(84,108,122)(85,109,123)(86,110,124)(87,111,125)(88,112,126), (1,132,107)(2,133,108)(3,134,109)(4,135,110)(5,136,111)(6,129,112)(7,130,105)(8,131,106)(9,44,58)(10,45,59)(11,46,60)(12,47,61)(13,48,62)(14,41,63)(15,42,64)(16,43,57)(17,37,30)(18,38,31)(19,39,32)(20,40,25)(21,33,26)(22,34,27)(23,35,28)(24,36,29)(49,137,95)(50,138,96)(51,139,89)(52,140,90)(53,141,91)(54,142,92)(55,143,93)(56,144,94)(65,117,126)(66,118,127)(67,119,128)(68,120,121)(69,113,122)(70,114,123)(71,115,124)(72,116,125)(73,85,102)(74,86,103)(75,87,104)(76,88,97)(77,81,98)(78,82,99)(79,83,100)(80,84,101), (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), (1,29,5,25)(2,28,6,32)(3,27,7,31)(4,26,8,30)(9,100,13,104)(10,99,14,103)(11,98,15,102)(12,97,16,101)(17,110,21,106)(18,109,22,105)(19,108,23,112)(20,107,24,111)(33,131,37,135)(34,130,38,134)(35,129,39,133)(36,136,40,132)(41,86,45,82)(42,85,46,81)(43,84,47,88)(44,83,48,87)(49,115,53,119)(50,114,54,118)(51,113,55,117)(52,120,56,116)(57,80,61,76)(58,79,62,75)(59,78,63,74)(60,77,64,73)(65,139,69,143)(66,138,70,142)(67,137,71,141)(68,144,72,140)(89,122,93,126)(90,121,94,125)(91,128,95,124)(92,127,96,123) );
G=PermutationGroup([[(1,79,121),(2,80,122),(3,73,123),(4,74,124),(5,75,125),(6,76,126),(7,77,127),(8,78,128),(9,52,20),(10,53,21),(11,54,22),(12,55,23),(13,56,24),(14,49,17),(15,50,18),(16,51,19),(25,58,90),(26,59,91),(27,60,92),(28,61,93),(29,62,94),(30,63,95),(31,64,96),(32,57,89),(33,45,141),(34,46,142),(35,47,143),(36,48,144),(37,41,137),(38,42,138),(39,43,139),(40,44,140),(65,129,88),(66,130,81),(67,131,82),(68,132,83),(69,133,84),(70,134,85),(71,135,86),(72,136,87),(97,117,112),(98,118,105),(99,119,106),(100,120,107),(101,113,108),(102,114,109),(103,115,110),(104,116,111)], [(1,68,100),(2,69,101),(3,70,102),(4,71,103),(5,72,104),(6,65,97),(7,66,98),(8,67,99),(9,140,25),(10,141,26),(11,142,27),(12,143,28),(13,144,29),(14,137,30),(15,138,31),(16,139,32),(17,41,95),(18,42,96),(19,43,89),(20,44,90),(21,45,91),(22,46,92),(23,47,93),(24,48,94),(33,59,53),(34,60,54),(35,61,55),(36,62,56),(37,63,49),(38,64,50),(39,57,51),(40,58,52),(73,134,114),(74,135,115),(75,136,116),(76,129,117),(77,130,118),(78,131,119),(79,132,120),(80,133,113),(81,105,127),(82,106,128),(83,107,121),(84,108,122),(85,109,123),(86,110,124),(87,111,125),(88,112,126)], [(1,132,107),(2,133,108),(3,134,109),(4,135,110),(5,136,111),(6,129,112),(7,130,105),(8,131,106),(9,44,58),(10,45,59),(11,46,60),(12,47,61),(13,48,62),(14,41,63),(15,42,64),(16,43,57),(17,37,30),(18,38,31),(19,39,32),(20,40,25),(21,33,26),(22,34,27),(23,35,28),(24,36,29),(49,137,95),(50,138,96),(51,139,89),(52,140,90),(53,141,91),(54,142,92),(55,143,93),(56,144,94),(65,117,126),(66,118,127),(67,119,128),(68,120,121),(69,113,122),(70,114,123),(71,115,124),(72,116,125),(73,85,102),(74,86,103),(75,87,104),(76,88,97),(77,81,98),(78,82,99),(79,83,100),(80,84,101)], [(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)], [(1,29,5,25),(2,28,6,32),(3,27,7,31),(4,26,8,30),(9,100,13,104),(10,99,14,103),(11,98,15,102),(12,97,16,101),(17,110,21,106),(18,109,22,105),(19,108,23,112),(20,107,24,111),(33,131,37,135),(34,130,38,134),(35,129,39,133),(36,136,40,132),(41,86,45,82),(42,85,46,81),(43,84,47,88),(44,83,48,87),(49,115,53,119),(50,114,54,118),(51,113,55,117),(52,120,56,116),(57,80,61,76),(58,79,62,75),(59,78,63,74),(60,77,64,73),(65,139,69,143),(66,138,70,142),(67,137,71,141),(68,144,72,140),(89,122,93,126),(90,121,94,125),(91,128,95,124),(92,127,96,123)]])
117 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | ··· | 3N | 4A | 4B | 4C | 6A | 6B | 6C | ··· | 6N | 8A | 8B | 12A | ··· | 12Z | 12AA | 12AB | 12AC | 12AD | 24A | ··· | 24AZ |
| order | 1 | 2 | 3 | 3 | 3 | ··· | 3 | 4 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 8 | 8 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 2 | 36 | 36 | 1 | 1 | 2 | ··· | 2 | 2 | 2 | 2 | ··· | 2 | 36 | 36 | 36 | 36 | 2 | ··· | 2 |
117 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | - | |||||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | S3 | D4 | D6 | Q16 | C3×S3 | D12 | C3×D4 | S3×C6 | Dic12 | C3×Q16 | C3×D12 | C3×Dic12 |
| kernel | C3×C32⋊5Q16 | C32×C24 | C3×C32⋊4Q8 | C32⋊5Q16 | C3×C24 | C32⋊4Q8 | C3×C24 | C32×C6 | C3×C12 | C33 | C24 | C3×C6 | C3×C6 | C12 | C32 | C32 | C6 | C3 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 1 | 4 | 2 | 8 | 8 | 2 | 8 | 16 | 4 | 16 | 32 |
Matrix representation of C3×C32⋊5Q16 ►in GL4(𝔽73) generated by
| 64 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 64 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 8 |
| 8 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 22 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 |
| 0 | 0 | 46 | 0 |
| 0 | 0 | 0 | 27 |
| 0 | 1 | 0 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(73))| [64,0,0,0,0,64,0,0,0,0,8,0,0,0,0,8],[64,0,0,0,0,8,0,0,0,0,64,0,0,0,0,8],[8,0,0,0,0,64,0,0,0,0,1,0,0,0,0,1],[22,0,0,0,0,10,0,0,0,0,46,0,0,0,0,27],[0,72,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;
C3×C32⋊5Q16 in GAP, Magma, Sage, TeX
C_3\times C_3^2\rtimes_5Q_{16} % in TeX
G:=Group("C3xC3^2:5Q16"); // GroupNames label
G:=SmallGroup(432,484);
// by ID
G=gap.SmallGroup(432,484);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,168,197,260,1011,80,4037,14118]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^8=1,e^2=d^4,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e^-1=b^-1,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=d^-1>;
// generators/relations