direct product, metabelian, supersoluble, monomial
Aliases: C32×C3⋊Q16, C33⋊13Q16, C12.4C62, C12.60(S3×C6), C3⋊2(C32×Q16), C32⋊9(C3×Q16), (C3×C12).187D6, Dic6.2(C3×C6), (C3×Dic6).7C6, C6.10(D4×C32), (C32×C6).69D4, Q8.3(S3×C32), (Q8×C33).1C2, (Q8×C32).22C6, (Q8×C32).28S3, (C32×Dic6).5C2, (C32×C12).26C22, C3⋊C8.(C3×C6), C4.4(S3×C3×C6), (C3×C3⋊C8).4C6, (C3×C6).65(C3×D4), C6.56(C3×C3⋊D4), (C32×C3⋊C8).3C2, (C3×C12).45(C2×C6), (C3×Q8).13(C3×C6), (C3×Q8).36(C3×S3), C2.7(C32×C3⋊D4), (C3×C6).125(C3⋊D4), SmallGroup(432,478)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C32×C3⋊Q16
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, be=eb, dcd-1=c-1, ce=ec, ede-1=d-1 >
Subgroups: 288 in 156 conjugacy classes, 66 normal (22 characteristic)
C1, C2, C3, C3, C3, C4, C4, C6, C6, C6, C8, Q8, Q8, C32, C32, C32, Dic3, C12, C12, C12, Q16, C3×C6, C3×C6, C3×C6, C3⋊C8, C24, Dic6, C3×Q8, C3×Q8, C3×Q8, C33, C3×Dic3, C3×C12, C3×C12, C3×C12, C3⋊Q16, C3×Q16, C32×C6, C3×C3⋊C8, C3×C24, C3×Dic6, Q8×C32, Q8×C32, Q8×C32, C32×Dic3, C32×C12, C32×C12, C3×C3⋊Q16, C32×Q16, C32×C3⋊C8, C32×Dic6, Q8×C33, C32×C3⋊Q16
Quotients: C1, C2, C3, C22, S3, C6, D4, C32, D6, C2×C6, Q16, C3×S3, C3×C6, C3⋊D4, C3×D4, S3×C6, C62, C3⋊Q16, C3×Q16, S3×C32, C3×C3⋊D4, D4×C32, S3×C3×C6, C3×C3⋊Q16, C32×Q16, C32×C3⋊D4, C32×C3⋊Q16
►On 144 points
Generators in S144
(1 118 9)(2 119 10)(3 120 11)(4 113 12)(5 114 13)(6 115 14)(7 116 15)(8 117 16)(17 37 28)(18 38 29)(19 39 30)(20 40 31)(21 33 32)(22 34 25)(23 35 26)(24 36 27)(41 106 49)(42 107 50)(43 108 51)(44 109 52)(45 110 53)(46 111 54)(47 112 55)(48 105 56)(57 129 124)(58 130 125)(59 131 126)(60 132 127)(61 133 128)(62 134 121)(63 135 122)(64 136 123)(65 100 90)(66 101 91)(67 102 92)(68 103 93)(69 104 94)(70 97 95)(71 98 96)(72 99 89)(73 143 81)(74 144 82)(75 137 83)(76 138 84)(77 139 85)(78 140 86)(79 141 87)(80 142 88)
(1 44 139)(2 45 140)(3 46 141)(4 47 142)(5 48 143)(6 41 144)(7 42 137)(8 43 138)(9 52 77)(10 53 78)(11 54 79)(12 55 80)(13 56 73)(14 49 74)(15 50 75)(16 51 76)(17 60 96)(18 61 89)(19 62 90)(20 63 91)(21 64 92)(22 57 93)(23 58 94)(24 59 95)(25 124 103)(26 125 104)(27 126 97)(28 127 98)(29 128 99)(30 121 100)(31 122 101)(32 123 102)(33 136 67)(34 129 68)(35 130 69)(36 131 70)(37 132 71)(38 133 72)(39 134 65)(40 135 66)(81 114 105)(82 115 106)(83 116 107)(84 117 108)(85 118 109)(86 119 110)(87 120 111)(88 113 112)
(1 44 139)(2 140 45)(3 46 141)(4 142 47)(5 48 143)(6 144 41)(7 42 137)(8 138 43)(9 52 77)(10 78 53)(11 54 79)(12 80 55)(13 56 73)(14 74 49)(15 50 75)(16 76 51)(17 60 96)(18 89 61)(19 62 90)(20 91 63)(21 64 92)(22 93 57)(23 58 94)(24 95 59)(25 103 124)(26 125 104)(27 97 126)(28 127 98)(29 99 128)(30 121 100)(31 101 122)(32 123 102)(33 136 67)(34 68 129)(35 130 69)(36 70 131)(37 132 71)(38 72 133)(39 134 65)(40 66 135)(81 114 105)(82 106 115)(83 116 107)(84 108 117)(85 118 109)(86 110 119)(87 120 111)(88 112 113)
(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 94 5 90)(2 93 6 89)(3 92 7 96)(4 91 8 95)(9 104 13 100)(10 103 14 99)(11 102 15 98)(12 101 16 97)(17 46 21 42)(18 45 22 41)(19 44 23 48)(20 43 24 47)(25 49 29 53)(26 56 30 52)(27 55 31 51)(28 54 32 50)(33 107 37 111)(34 106 38 110)(35 105 39 109)(36 112 40 108)(57 144 61 140)(58 143 62 139)(59 142 63 138)(60 141 64 137)(65 118 69 114)(66 117 70 113)(67 116 71 120)(68 115 72 119)(73 121 77 125)(74 128 78 124)(75 127 79 123)(76 126 80 122)(81 134 85 130)(82 133 86 129)(83 132 87 136)(84 131 88 135)
G:=sub<Sym(144)| (1,118,9)(2,119,10)(3,120,11)(4,113,12)(5,114,13)(6,115,14)(7,116,15)(8,117,16)(17,37,28)(18,38,29)(19,39,30)(20,40,31)(21,33,32)(22,34,25)(23,35,26)(24,36,27)(41,106,49)(42,107,50)(43,108,51)(44,109,52)(45,110,53)(46,111,54)(47,112,55)(48,105,56)(57,129,124)(58,130,125)(59,131,126)(60,132,127)(61,133,128)(62,134,121)(63,135,122)(64,136,123)(65,100,90)(66,101,91)(67,102,92)(68,103,93)(69,104,94)(70,97,95)(71,98,96)(72,99,89)(73,143,81)(74,144,82)(75,137,83)(76,138,84)(77,139,85)(78,140,86)(79,141,87)(80,142,88), (1,44,139)(2,45,140)(3,46,141)(4,47,142)(5,48,143)(6,41,144)(7,42,137)(8,43,138)(9,52,77)(10,53,78)(11,54,79)(12,55,80)(13,56,73)(14,49,74)(15,50,75)(16,51,76)(17,60,96)(18,61,89)(19,62,90)(20,63,91)(21,64,92)(22,57,93)(23,58,94)(24,59,95)(25,124,103)(26,125,104)(27,126,97)(28,127,98)(29,128,99)(30,121,100)(31,122,101)(32,123,102)(33,136,67)(34,129,68)(35,130,69)(36,131,70)(37,132,71)(38,133,72)(39,134,65)(40,135,66)(81,114,105)(82,115,106)(83,116,107)(84,117,108)(85,118,109)(86,119,110)(87,120,111)(88,113,112), (1,44,139)(2,140,45)(3,46,141)(4,142,47)(5,48,143)(6,144,41)(7,42,137)(8,138,43)(9,52,77)(10,78,53)(11,54,79)(12,80,55)(13,56,73)(14,74,49)(15,50,75)(16,76,51)(17,60,96)(18,89,61)(19,62,90)(20,91,63)(21,64,92)(22,93,57)(23,58,94)(24,95,59)(25,103,124)(26,125,104)(27,97,126)(28,127,98)(29,99,128)(30,121,100)(31,101,122)(32,123,102)(33,136,67)(34,68,129)(35,130,69)(36,70,131)(37,132,71)(38,72,133)(39,134,65)(40,66,135)(81,114,105)(82,106,115)(83,116,107)(84,108,117)(85,118,109)(86,110,119)(87,120,111)(88,112,113), (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), (1,94,5,90)(2,93,6,89)(3,92,7,96)(4,91,8,95)(9,104,13,100)(10,103,14,99)(11,102,15,98)(12,101,16,97)(17,46,21,42)(18,45,22,41)(19,44,23,48)(20,43,24,47)(25,49,29,53)(26,56,30,52)(27,55,31,51)(28,54,32,50)(33,107,37,111)(34,106,38,110)(35,105,39,109)(36,112,40,108)(57,144,61,140)(58,143,62,139)(59,142,63,138)(60,141,64,137)(65,118,69,114)(66,117,70,113)(67,116,71,120)(68,115,72,119)(73,121,77,125)(74,128,78,124)(75,127,79,123)(76,126,80,122)(81,134,85,130)(82,133,86,129)(83,132,87,136)(84,131,88,135)>;
G:=Group( (1,118,9)(2,119,10)(3,120,11)(4,113,12)(5,114,13)(6,115,14)(7,116,15)(8,117,16)(17,37,28)(18,38,29)(19,39,30)(20,40,31)(21,33,32)(22,34,25)(23,35,26)(24,36,27)(41,106,49)(42,107,50)(43,108,51)(44,109,52)(45,110,53)(46,111,54)(47,112,55)(48,105,56)(57,129,124)(58,130,125)(59,131,126)(60,132,127)(61,133,128)(62,134,121)(63,135,122)(64,136,123)(65,100,90)(66,101,91)(67,102,92)(68,103,93)(69,104,94)(70,97,95)(71,98,96)(72,99,89)(73,143,81)(74,144,82)(75,137,83)(76,138,84)(77,139,85)(78,140,86)(79,141,87)(80,142,88), (1,44,139)(2,45,140)(3,46,141)(4,47,142)(5,48,143)(6,41,144)(7,42,137)(8,43,138)(9,52,77)(10,53,78)(11,54,79)(12,55,80)(13,56,73)(14,49,74)(15,50,75)(16,51,76)(17,60,96)(18,61,89)(19,62,90)(20,63,91)(21,64,92)(22,57,93)(23,58,94)(24,59,95)(25,124,103)(26,125,104)(27,126,97)(28,127,98)(29,128,99)(30,121,100)(31,122,101)(32,123,102)(33,136,67)(34,129,68)(35,130,69)(36,131,70)(37,132,71)(38,133,72)(39,134,65)(40,135,66)(81,114,105)(82,115,106)(83,116,107)(84,117,108)(85,118,109)(86,119,110)(87,120,111)(88,113,112), (1,44,139)(2,140,45)(3,46,141)(4,142,47)(5,48,143)(6,144,41)(7,42,137)(8,138,43)(9,52,77)(10,78,53)(11,54,79)(12,80,55)(13,56,73)(14,74,49)(15,50,75)(16,76,51)(17,60,96)(18,89,61)(19,62,90)(20,91,63)(21,64,92)(22,93,57)(23,58,94)(24,95,59)(25,103,124)(26,125,104)(27,97,126)(28,127,98)(29,99,128)(30,121,100)(31,101,122)(32,123,102)(33,136,67)(34,68,129)(35,130,69)(36,70,131)(37,132,71)(38,72,133)(39,134,65)(40,66,135)(81,114,105)(82,106,115)(83,116,107)(84,108,117)(85,118,109)(86,110,119)(87,120,111)(88,112,113), (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), (1,94,5,90)(2,93,6,89)(3,92,7,96)(4,91,8,95)(9,104,13,100)(10,103,14,99)(11,102,15,98)(12,101,16,97)(17,46,21,42)(18,45,22,41)(19,44,23,48)(20,43,24,47)(25,49,29,53)(26,56,30,52)(27,55,31,51)(28,54,32,50)(33,107,37,111)(34,106,38,110)(35,105,39,109)(36,112,40,108)(57,144,61,140)(58,143,62,139)(59,142,63,138)(60,141,64,137)(65,118,69,114)(66,117,70,113)(67,116,71,120)(68,115,72,119)(73,121,77,125)(74,128,78,124)(75,127,79,123)(76,126,80,122)(81,134,85,130)(82,133,86,129)(83,132,87,136)(84,131,88,135) );
G=PermutationGroup([[(1,118,9),(2,119,10),(3,120,11),(4,113,12),(5,114,13),(6,115,14),(7,116,15),(8,117,16),(17,37,28),(18,38,29),(19,39,30),(20,40,31),(21,33,32),(22,34,25),(23,35,26),(24,36,27),(41,106,49),(42,107,50),(43,108,51),(44,109,52),(45,110,53),(46,111,54),(47,112,55),(48,105,56),(57,129,124),(58,130,125),(59,131,126),(60,132,127),(61,133,128),(62,134,121),(63,135,122),(64,136,123),(65,100,90),(66,101,91),(67,102,92),(68,103,93),(69,104,94),(70,97,95),(71,98,96),(72,99,89),(73,143,81),(74,144,82),(75,137,83),(76,138,84),(77,139,85),(78,140,86),(79,141,87),(80,142,88)], [(1,44,139),(2,45,140),(3,46,141),(4,47,142),(5,48,143),(6,41,144),(7,42,137),(8,43,138),(9,52,77),(10,53,78),(11,54,79),(12,55,80),(13,56,73),(14,49,74),(15,50,75),(16,51,76),(17,60,96),(18,61,89),(19,62,90),(20,63,91),(21,64,92),(22,57,93),(23,58,94),(24,59,95),(25,124,103),(26,125,104),(27,126,97),(28,127,98),(29,128,99),(30,121,100),(31,122,101),(32,123,102),(33,136,67),(34,129,68),(35,130,69),(36,131,70),(37,132,71),(38,133,72),(39,134,65),(40,135,66),(81,114,105),(82,115,106),(83,116,107),(84,117,108),(85,118,109),(86,119,110),(87,120,111),(88,113,112)], [(1,44,139),(2,140,45),(3,46,141),(4,142,47),(5,48,143),(6,144,41),(7,42,137),(8,138,43),(9,52,77),(10,78,53),(11,54,79),(12,80,55),(13,56,73),(14,74,49),(15,50,75),(16,76,51),(17,60,96),(18,89,61),(19,62,90),(20,91,63),(21,64,92),(22,93,57),(23,58,94),(24,95,59),(25,103,124),(26,125,104),(27,97,126),(28,127,98),(29,99,128),(30,121,100),(31,101,122),(32,123,102),(33,136,67),(34,68,129),(35,130,69),(36,70,131),(37,132,71),(38,72,133),(39,134,65),(40,66,135),(81,114,105),(82,106,115),(83,116,107),(84,108,117),(85,118,109),(86,110,119),(87,120,111),(88,112,113)], [(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)], [(1,94,5,90),(2,93,6,89),(3,92,7,96),(4,91,8,95),(9,104,13,100),(10,103,14,99),(11,102,15,98),(12,101,16,97),(17,46,21,42),(18,45,22,41),(19,44,23,48),(20,43,24,47),(25,49,29,53),(26,56,30,52),(27,55,31,51),(28,54,32,50),(33,107,37,111),(34,106,38,110),(35,105,39,109),(36,112,40,108),(57,144,61,140),(58,143,62,139),(59,142,63,138),(60,141,64,137),(65,118,69,114),(66,117,70,113),(67,116,71,120),(68,115,72,119),(73,121,77,125),(74,128,78,124),(75,127,79,123),(76,126,80,122),(81,134,85,130),(82,133,86,129),(83,132,87,136),(84,131,88,135)]])
108 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3H | 3I | ··· | 3Q | 4A | 4B | 4C | 6A | ··· | 6H | 6I | ··· | 6Q | 8A | 8B | 12A | ··· | 12H | 12I | ··· | 12AQ | 12AR | ··· | 12AY | 24A | ··· | 24P |
| 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 | 4 | 12 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | 6 | 2 | ··· | 2 | 4 | ··· | 4 | 12 | ··· | 12 | 6 | ··· | 6 |
108 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 | Q16 | C3×S3 | C3⋊D4 | C3×D4 | S3×C6 | C3×Q16 | C3×C3⋊D4 | C3⋊Q16 | C3×C3⋊Q16 |
| kernel | C32×C3⋊Q16 | C32×C3⋊C8 | C32×Dic6 | Q8×C33 | C3×C3⋊Q16 | C3×C3⋊C8 | C3×Dic6 | 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 | 8 | 8 | 8 | 8 | 1 | 1 | 1 | 2 | 8 | 2 | 8 | 8 | 16 | 16 | 1 | 8 |
Matrix representation of C32×C3⋊Q16 ►in GL4(𝔽73) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 8 | 0 | 0 | 0 |
| 52 | 64 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 51 | 10 | 0 | 0 |
| 32 | 22 | 0 | 0 |
| 0 | 0 | 32 | 32 |
| 0 | 0 | 57 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 6 | 59 |
| 0 | 0 | 60 | 67 |
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,8,0,0,0,0,8],[8,0,0,0,0,8,0,0,0,0,1,0,0,0,0,1],[8,52,0,0,0,64,0,0,0,0,1,0,0,0,0,1],[51,32,0,0,10,22,0,0,0,0,32,57,0,0,32,0],[72,0,0,0,0,72,0,0,0,0,6,60,0,0,59,67] >;
C32×C3⋊Q16 in GAP, Magma, Sage, TeX
C_3^2\times C_3\rtimes Q_{16} % in TeX
G:=Group("C3^2xC3:Q16"); // GroupNames label
G:=SmallGroup(432,478);
// by ID
G=gap.SmallGroup(432,478);
# by ID
G:=PCGroup([7,-2,-2,-3,-3,-2,-2,-3,504,533,512,3784,1901,102,14118]);
// 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,b*e=e*b,d*c*d^-1=c^-1,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations