direct product, metabelian, supersoluble, monomial, A-group
Aliases: C3⋊S3×C20, C60⋊6S3, C30.63D6, C12⋊2(C5×S3), C3⋊2(S3×C20), C15⋊15(C4×S3), (C3×C60)⋊10C2, (C3×C12)⋊4C10, C32⋊5(C2×C20), C6.13(S3×C10), C3⋊Dic3⋊4C10, (C3×C30).53C22, (C3×C15)⋊30(C2×C4), C2.1(C10×C3⋊S3), (C10×C3⋊S3).6C2, (C2×C3⋊S3).3C10, C10.14(C2×C3⋊S3), (C5×C3⋊Dic3)⋊9C2, (C3×C6).12(C2×C10), SmallGroup(360,106)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — C3⋊S3×C20 |
Generators and relations for C3⋊S3×C20
G = < a,b,c,d | a20=b3=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >
Subgroups: 256 in 96 conjugacy classes, 46 normal (18 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C2×C4, C32, C10, C10, Dic3, C12, D6, C15, C3⋊S3, C3×C6, C20, C20, C2×C10, C4×S3, C5×S3, C30, C3⋊Dic3, C3×C12, C2×C3⋊S3, C2×C20, C3×C15, C5×Dic3, C60, S3×C10, C4×C3⋊S3, C5×C3⋊S3, C3×C30, S3×C20, C5×C3⋊Dic3, C3×C60, C10×C3⋊S3, C3⋊S3×C20
Quotients: C1, C2, C4, C22, C5, S3, C2×C4, C10, D6, C3⋊S3, C20, C2×C10, C4×S3, C5×S3, C2×C3⋊S3, C2×C20, S3×C10, C4×C3⋊S3, C5×C3⋊S3, S3×C20, C10×C3⋊S3, C3⋊S3×C20
►On 180 points
Generators in S180
(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)(141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160)(161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180)
(1 71 164)(2 72 165)(3 73 166)(4 74 167)(5 75 168)(6 76 169)(7 77 170)(8 78 171)(9 79 172)(10 80 173)(11 61 174)(12 62 175)(13 63 176)(14 64 177)(15 65 178)(16 66 179)(17 67 180)(18 68 161)(19 69 162)(20 70 163)(21 155 111)(22 156 112)(23 157 113)(24 158 114)(25 159 115)(26 160 116)(27 141 117)(28 142 118)(29 143 119)(30 144 120)(31 145 101)(32 146 102)(33 147 103)(34 148 104)(35 149 105)(36 150 106)(37 151 107)(38 152 108)(39 153 109)(40 154 110)(41 128 88)(42 129 89)(43 130 90)(44 131 91)(45 132 92)(46 133 93)(47 134 94)(48 135 95)(49 136 96)(50 137 97)(51 138 98)(52 139 99)(53 140 100)(54 121 81)(55 122 82)(56 123 83)(57 124 84)(58 125 85)(59 126 86)(60 127 87)
(1 32 99)(2 33 100)(3 34 81)(4 35 82)(5 36 83)(6 37 84)(7 38 85)(8 39 86)(9 40 87)(10 21 88)(11 22 89)(12 23 90)(13 24 91)(14 25 92)(15 26 93)(16 27 94)(17 28 95)(18 29 96)(19 30 97)(20 31 98)(41 80 155)(42 61 156)(43 62 157)(44 63 158)(45 64 159)(46 65 160)(47 66 141)(48 67 142)(49 68 143)(50 69 144)(51 70 145)(52 71 146)(53 72 147)(54 73 148)(55 74 149)(56 75 150)(57 76 151)(58 77 152)(59 78 153)(60 79 154)(101 138 163)(102 139 164)(103 140 165)(104 121 166)(105 122 167)(106 123 168)(107 124 169)(108 125 170)(109 126 171)(110 127 172)(111 128 173)(112 129 174)(113 130 175)(114 131 176)(115 132 177)(116 133 178)(117 134 179)(118 135 180)(119 136 161)(120 137 162)
(21 88)(22 89)(23 90)(24 91)(25 92)(26 93)(27 94)(28 95)(29 96)(30 97)(31 98)(32 99)(33 100)(34 81)(35 82)(36 83)(37 84)(38 85)(39 86)(40 87)(41 111)(42 112)(43 113)(44 114)(45 115)(46 116)(47 117)(48 118)(49 119)(50 120)(51 101)(52 102)(53 103)(54 104)(55 105)(56 106)(57 107)(58 108)(59 109)(60 110)(61 174)(62 175)(63 176)(64 177)(65 178)(66 179)(67 180)(68 161)(69 162)(70 163)(71 164)(72 165)(73 166)(74 167)(75 168)(76 169)(77 170)(78 171)(79 172)(80 173)(121 148)(122 149)(123 150)(124 151)(125 152)(126 153)(127 154)(128 155)(129 156)(130 157)(131 158)(132 159)(133 160)(134 141)(135 142)(136 143)(137 144)(138 145)(139 146)(140 147)
G:=sub<Sym(180)| (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)(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160)(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180), (1,71,164)(2,72,165)(3,73,166)(4,74,167)(5,75,168)(6,76,169)(7,77,170)(8,78,171)(9,79,172)(10,80,173)(11,61,174)(12,62,175)(13,63,176)(14,64,177)(15,65,178)(16,66,179)(17,67,180)(18,68,161)(19,69,162)(20,70,163)(21,155,111)(22,156,112)(23,157,113)(24,158,114)(25,159,115)(26,160,116)(27,141,117)(28,142,118)(29,143,119)(30,144,120)(31,145,101)(32,146,102)(33,147,103)(34,148,104)(35,149,105)(36,150,106)(37,151,107)(38,152,108)(39,153,109)(40,154,110)(41,128,88)(42,129,89)(43,130,90)(44,131,91)(45,132,92)(46,133,93)(47,134,94)(48,135,95)(49,136,96)(50,137,97)(51,138,98)(52,139,99)(53,140,100)(54,121,81)(55,122,82)(56,123,83)(57,124,84)(58,125,85)(59,126,86)(60,127,87), (1,32,99)(2,33,100)(3,34,81)(4,35,82)(5,36,83)(6,37,84)(7,38,85)(8,39,86)(9,40,87)(10,21,88)(11,22,89)(12,23,90)(13,24,91)(14,25,92)(15,26,93)(16,27,94)(17,28,95)(18,29,96)(19,30,97)(20,31,98)(41,80,155)(42,61,156)(43,62,157)(44,63,158)(45,64,159)(46,65,160)(47,66,141)(48,67,142)(49,68,143)(50,69,144)(51,70,145)(52,71,146)(53,72,147)(54,73,148)(55,74,149)(56,75,150)(57,76,151)(58,77,152)(59,78,153)(60,79,154)(101,138,163)(102,139,164)(103,140,165)(104,121,166)(105,122,167)(106,123,168)(107,124,169)(108,125,170)(109,126,171)(110,127,172)(111,128,173)(112,129,174)(113,130,175)(114,131,176)(115,132,177)(116,133,178)(117,134,179)(118,135,180)(119,136,161)(120,137,162), (21,88)(22,89)(23,90)(24,91)(25,92)(26,93)(27,94)(28,95)(29,96)(30,97)(31,98)(32,99)(33,100)(34,81)(35,82)(36,83)(37,84)(38,85)(39,86)(40,87)(41,111)(42,112)(43,113)(44,114)(45,115)(46,116)(47,117)(48,118)(49,119)(50,120)(51,101)(52,102)(53,103)(54,104)(55,105)(56,106)(57,107)(58,108)(59,109)(60,110)(61,174)(62,175)(63,176)(64,177)(65,178)(66,179)(67,180)(68,161)(69,162)(70,163)(71,164)(72,165)(73,166)(74,167)(75,168)(76,169)(77,170)(78,171)(79,172)(80,173)(121,148)(122,149)(123,150)(124,151)(125,152)(126,153)(127,154)(128,155)(129,156)(130,157)(131,158)(132,159)(133,160)(134,141)(135,142)(136,143)(137,144)(138,145)(139,146)(140,147)>;
G:=Group( (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)(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160)(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180), (1,71,164)(2,72,165)(3,73,166)(4,74,167)(5,75,168)(6,76,169)(7,77,170)(8,78,171)(9,79,172)(10,80,173)(11,61,174)(12,62,175)(13,63,176)(14,64,177)(15,65,178)(16,66,179)(17,67,180)(18,68,161)(19,69,162)(20,70,163)(21,155,111)(22,156,112)(23,157,113)(24,158,114)(25,159,115)(26,160,116)(27,141,117)(28,142,118)(29,143,119)(30,144,120)(31,145,101)(32,146,102)(33,147,103)(34,148,104)(35,149,105)(36,150,106)(37,151,107)(38,152,108)(39,153,109)(40,154,110)(41,128,88)(42,129,89)(43,130,90)(44,131,91)(45,132,92)(46,133,93)(47,134,94)(48,135,95)(49,136,96)(50,137,97)(51,138,98)(52,139,99)(53,140,100)(54,121,81)(55,122,82)(56,123,83)(57,124,84)(58,125,85)(59,126,86)(60,127,87), (1,32,99)(2,33,100)(3,34,81)(4,35,82)(5,36,83)(6,37,84)(7,38,85)(8,39,86)(9,40,87)(10,21,88)(11,22,89)(12,23,90)(13,24,91)(14,25,92)(15,26,93)(16,27,94)(17,28,95)(18,29,96)(19,30,97)(20,31,98)(41,80,155)(42,61,156)(43,62,157)(44,63,158)(45,64,159)(46,65,160)(47,66,141)(48,67,142)(49,68,143)(50,69,144)(51,70,145)(52,71,146)(53,72,147)(54,73,148)(55,74,149)(56,75,150)(57,76,151)(58,77,152)(59,78,153)(60,79,154)(101,138,163)(102,139,164)(103,140,165)(104,121,166)(105,122,167)(106,123,168)(107,124,169)(108,125,170)(109,126,171)(110,127,172)(111,128,173)(112,129,174)(113,130,175)(114,131,176)(115,132,177)(116,133,178)(117,134,179)(118,135,180)(119,136,161)(120,137,162), (21,88)(22,89)(23,90)(24,91)(25,92)(26,93)(27,94)(28,95)(29,96)(30,97)(31,98)(32,99)(33,100)(34,81)(35,82)(36,83)(37,84)(38,85)(39,86)(40,87)(41,111)(42,112)(43,113)(44,114)(45,115)(46,116)(47,117)(48,118)(49,119)(50,120)(51,101)(52,102)(53,103)(54,104)(55,105)(56,106)(57,107)(58,108)(59,109)(60,110)(61,174)(62,175)(63,176)(64,177)(65,178)(66,179)(67,180)(68,161)(69,162)(70,163)(71,164)(72,165)(73,166)(74,167)(75,168)(76,169)(77,170)(78,171)(79,172)(80,173)(121,148)(122,149)(123,150)(124,151)(125,152)(126,153)(127,154)(128,155)(129,156)(130,157)(131,158)(132,159)(133,160)(134,141)(135,142)(136,143)(137,144)(138,145)(139,146)(140,147) );
G=PermutationGroup([[(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),(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160),(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180)], [(1,71,164),(2,72,165),(3,73,166),(4,74,167),(5,75,168),(6,76,169),(7,77,170),(8,78,171),(9,79,172),(10,80,173),(11,61,174),(12,62,175),(13,63,176),(14,64,177),(15,65,178),(16,66,179),(17,67,180),(18,68,161),(19,69,162),(20,70,163),(21,155,111),(22,156,112),(23,157,113),(24,158,114),(25,159,115),(26,160,116),(27,141,117),(28,142,118),(29,143,119),(30,144,120),(31,145,101),(32,146,102),(33,147,103),(34,148,104),(35,149,105),(36,150,106),(37,151,107),(38,152,108),(39,153,109),(40,154,110),(41,128,88),(42,129,89),(43,130,90),(44,131,91),(45,132,92),(46,133,93),(47,134,94),(48,135,95),(49,136,96),(50,137,97),(51,138,98),(52,139,99),(53,140,100),(54,121,81),(55,122,82),(56,123,83),(57,124,84),(58,125,85),(59,126,86),(60,127,87)], [(1,32,99),(2,33,100),(3,34,81),(4,35,82),(5,36,83),(6,37,84),(7,38,85),(8,39,86),(9,40,87),(10,21,88),(11,22,89),(12,23,90),(13,24,91),(14,25,92),(15,26,93),(16,27,94),(17,28,95),(18,29,96),(19,30,97),(20,31,98),(41,80,155),(42,61,156),(43,62,157),(44,63,158),(45,64,159),(46,65,160),(47,66,141),(48,67,142),(49,68,143),(50,69,144),(51,70,145),(52,71,146),(53,72,147),(54,73,148),(55,74,149),(56,75,150),(57,76,151),(58,77,152),(59,78,153),(60,79,154),(101,138,163),(102,139,164),(103,140,165),(104,121,166),(105,122,167),(106,123,168),(107,124,169),(108,125,170),(109,126,171),(110,127,172),(111,128,173),(112,129,174),(113,130,175),(114,131,176),(115,132,177),(116,133,178),(117,134,179),(118,135,180),(119,136,161),(120,137,162)], [(21,88),(22,89),(23,90),(24,91),(25,92),(26,93),(27,94),(28,95),(29,96),(30,97),(31,98),(32,99),(33,100),(34,81),(35,82),(36,83),(37,84),(38,85),(39,86),(40,87),(41,111),(42,112),(43,113),(44,114),(45,115),(46,116),(47,117),(48,118),(49,119),(50,120),(51,101),(52,102),(53,103),(54,104),(55,105),(56,106),(57,107),(58,108),(59,109),(60,110),(61,174),(62,175),(63,176),(64,177),(65,178),(66,179),(67,180),(68,161),(69,162),(70,163),(71,164),(72,165),(73,166),(74,167),(75,168),(76,169),(77,170),(78,171),(79,172),(80,173),(121,148),(122,149),(123,150),(124,151),(125,152),(126,153),(127,154),(128,155),(129,156),(130,157),(131,158),(132,159),(133,160),(134,141),(135,142),(136,143),(137,144),(138,145),(139,146),(140,147)]])
120 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 6D | 10A | 10B | 10C | 10D | 10E | ··· | 10L | 12A | ··· | 12H | 15A | ··· | 15P | 20A | ··· | 20H | 20I | ··· | 20P | 30A | ··· | 30P | 60A | ··· | 60AF |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 6 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | ··· | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 9 | 9 | 2 | 2 | 2 | 2 | 1 | 1 | 9 | 9 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 9 | ··· | 9 | 2 | ··· | 2 | 2 | ··· | 2 | 1 | ··· | 1 | 9 | ··· | 9 | 2 | ··· | 2 | 2 | ··· | 2 |
120 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C5 | C10 | C10 | C10 | C20 | S3 | D6 | C4×S3 | C5×S3 | S3×C10 | S3×C20 |
| kernel | C3⋊S3×C20 | C5×C3⋊Dic3 | C3×C60 | C10×C3⋊S3 | C5×C3⋊S3 | C4×C3⋊S3 | C3⋊Dic3 | C3×C12 | C2×C3⋊S3 | C3⋊S3 | C60 | C30 | C15 | C12 | C6 | C3 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 16 | 4 | 4 | 8 | 16 | 16 | 32 |
Matrix representation of C3⋊S3×C20 ►in GL4(𝔽61) generated by
| 38 | 0 | 0 | 0 |
| 0 | 38 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 60 |
| 0 | 1 | 0 | 0 |
| 60 | 60 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 60 | 60 |
| 0 | 1 | 0 | 0 |
| 60 | 60 | 0 | 0 |
| 0 | 0 | 60 | 60 |
| 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 60 | 60 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 1 | 1 |
G:=sub<GL(4,GF(61))| [38,0,0,0,0,38,0,0,0,0,60,0,0,0,0,60],[0,60,0,0,1,60,0,0,0,0,0,60,0,0,1,60],[0,60,0,0,1,60,0,0,0,0,60,1,0,0,60,0],[1,60,0,0,0,60,0,0,0,0,60,1,0,0,0,1] >;
C3⋊S3×C20 in GAP, Magma, Sage, TeX
C_3\rtimes S_3\times C_{20} % in TeX
G:=Group("C3:S3xC20"); // GroupNames label
G:=SmallGroup(360,106);
// by ID
G=gap.SmallGroup(360,106);
# by ID
G:=PCGroup([6,-2,-2,-5,-2,-3,-3,127,2404,8645]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^3=c^3=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations