metabelian, supersoluble, monomial
Aliases: C32⋊7D20, C30.13D6, (C3×C15)⋊8D4, (C6×D5)⋊2S3, C3⋊Dic3⋊1D5, C6.20(S3×D5), D10⋊2(C3⋊S3), C15⋊2(C3⋊D4), C3⋊2(C3⋊D20), (C3×C6).24D10, C5⋊1(C32⋊7D4), (C3×C30).12C22, (D5×C3×C6)⋊4C2, C2.5(D5×C3⋊S3), (C2×C3⋊D15)⋊3C2, C10.5(C2×C3⋊S3), (C5×C3⋊Dic3)⋊3C2, SmallGroup(360,69)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C32⋊7D20
G = < a,b,c,d | a3=b3=c20=d2=1, ab=ba, cac-1=dad=a-1, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 704 in 96 conjugacy classes, 34 normal (14 characteristic)
C1, C2, C2, C3, C4, C22, C5, S3, C6, C6, D4, C32, D5, C10, Dic3, D6, C2×C6, C15, C3⋊S3, C3×C6, C3×C6, C20, D10, D10, C3⋊D4, C3×D5, D15, C30, C3⋊Dic3, C2×C3⋊S3, C62, D20, C3×C15, C5×Dic3, C6×D5, D30, C32⋊7D4, C32×D5, C3⋊D15, C3×C30, C3⋊D20, C5×C3⋊Dic3, D5×C3×C6, C2×C3⋊D15, C32⋊7D20
Quotients: C1, C2, C22, S3, D4, D5, D6, C3⋊S3, D10, C3⋊D4, C2×C3⋊S3, D20, S3×D5, C32⋊7D4, C3⋊D20, D5×C3⋊S3, C32⋊7D20
►On 180 points
Generators in S180
(1 68 119)(2 120 69)(3 70 101)(4 102 71)(5 72 103)(6 104 73)(7 74 105)(8 106 75)(9 76 107)(10 108 77)(11 78 109)(12 110 79)(13 80 111)(14 112 61)(15 62 113)(16 114 63)(17 64 115)(18 116 65)(19 66 117)(20 118 67)(21 90 141)(22 142 91)(23 92 143)(24 144 93)(25 94 145)(26 146 95)(27 96 147)(28 148 97)(29 98 149)(30 150 99)(31 100 151)(32 152 81)(33 82 153)(34 154 83)(35 84 155)(36 156 85)(37 86 157)(38 158 87)(39 88 159)(40 160 89)(41 138 165)(42 166 139)(43 140 167)(44 168 121)(45 122 169)(46 170 123)(47 124 171)(48 172 125)(49 126 173)(50 174 127)(51 128 175)(52 176 129)(53 130 177)(54 178 131)(55 132 179)(56 180 133)(57 134 161)(58 162 135)(59 136 163)(60 164 137)
(1 123 86)(2 87 124)(3 125 88)(4 89 126)(5 127 90)(6 91 128)(7 129 92)(8 93 130)(9 131 94)(10 95 132)(11 133 96)(12 97 134)(13 135 98)(14 99 136)(15 137 100)(16 81 138)(17 139 82)(18 83 140)(19 121 84)(20 85 122)(21 103 174)(22 175 104)(23 105 176)(24 177 106)(25 107 178)(26 179 108)(27 109 180)(28 161 110)(29 111 162)(30 163 112)(31 113 164)(32 165 114)(33 115 166)(34 167 116)(35 117 168)(36 169 118)(37 119 170)(38 171 120)(39 101 172)(40 173 102)(41 63 152)(42 153 64)(43 65 154)(44 155 66)(45 67 156)(46 157 68)(47 69 158)(48 159 70)(49 71 160)(50 141 72)(51 73 142)(52 143 74)(53 75 144)(54 145 76)(55 77 146)(56 147 78)(57 79 148)(58 149 80)(59 61 150)(60 151 62)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 20)(17 19)(21 56)(22 55)(23 54)(24 53)(25 52)(26 51)(27 50)(28 49)(29 48)(30 47)(31 46)(32 45)(33 44)(34 43)(35 42)(36 41)(37 60)(38 59)(39 58)(40 57)(61 120)(62 119)(63 118)(64 117)(65 116)(66 115)(67 114)(68 113)(69 112)(70 111)(71 110)(72 109)(73 108)(74 107)(75 106)(76 105)(77 104)(78 103)(79 102)(80 101)(81 122)(82 121)(83 140)(84 139)(85 138)(86 137)(87 136)(88 135)(89 134)(90 133)(91 132)(92 131)(93 130)(94 129)(95 128)(96 127)(97 126)(98 125)(99 124)(100 123)(141 180)(142 179)(143 178)(144 177)(145 176)(146 175)(147 174)(148 173)(149 172)(150 171)(151 170)(152 169)(153 168)(154 167)(155 166)(156 165)(157 164)(158 163)(159 162)(160 161)
G:=sub<Sym(180)| (1,68,119)(2,120,69)(3,70,101)(4,102,71)(5,72,103)(6,104,73)(7,74,105)(8,106,75)(9,76,107)(10,108,77)(11,78,109)(12,110,79)(13,80,111)(14,112,61)(15,62,113)(16,114,63)(17,64,115)(18,116,65)(19,66,117)(20,118,67)(21,90,141)(22,142,91)(23,92,143)(24,144,93)(25,94,145)(26,146,95)(27,96,147)(28,148,97)(29,98,149)(30,150,99)(31,100,151)(32,152,81)(33,82,153)(34,154,83)(35,84,155)(36,156,85)(37,86,157)(38,158,87)(39,88,159)(40,160,89)(41,138,165)(42,166,139)(43,140,167)(44,168,121)(45,122,169)(46,170,123)(47,124,171)(48,172,125)(49,126,173)(50,174,127)(51,128,175)(52,176,129)(53,130,177)(54,178,131)(55,132,179)(56,180,133)(57,134,161)(58,162,135)(59,136,163)(60,164,137), (1,123,86)(2,87,124)(3,125,88)(4,89,126)(5,127,90)(6,91,128)(7,129,92)(8,93,130)(9,131,94)(10,95,132)(11,133,96)(12,97,134)(13,135,98)(14,99,136)(15,137,100)(16,81,138)(17,139,82)(18,83,140)(19,121,84)(20,85,122)(21,103,174)(22,175,104)(23,105,176)(24,177,106)(25,107,178)(26,179,108)(27,109,180)(28,161,110)(29,111,162)(30,163,112)(31,113,164)(32,165,114)(33,115,166)(34,167,116)(35,117,168)(36,169,118)(37,119,170)(38,171,120)(39,101,172)(40,173,102)(41,63,152)(42,153,64)(43,65,154)(44,155,66)(45,67,156)(46,157,68)(47,69,158)(48,159,70)(49,71,160)(50,141,72)(51,73,142)(52,143,74)(53,75,144)(54,145,76)(55,77,146)(56,147,78)(57,79,148)(58,149,80)(59,61,150)(60,151,62), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(21,56)(22,55)(23,54)(24,53)(25,52)(26,51)(27,50)(28,49)(29,48)(30,47)(31,46)(32,45)(33,44)(34,43)(35,42)(36,41)(37,60)(38,59)(39,58)(40,57)(61,120)(62,119)(63,118)(64,117)(65,116)(66,115)(67,114)(68,113)(69,112)(70,111)(71,110)(72,109)(73,108)(74,107)(75,106)(76,105)(77,104)(78,103)(79,102)(80,101)(81,122)(82,121)(83,140)(84,139)(85,138)(86,137)(87,136)(88,135)(89,134)(90,133)(91,132)(92,131)(93,130)(94,129)(95,128)(96,127)(97,126)(98,125)(99,124)(100,123)(141,180)(142,179)(143,178)(144,177)(145,176)(146,175)(147,174)(148,173)(149,172)(150,171)(151,170)(152,169)(153,168)(154,167)(155,166)(156,165)(157,164)(158,163)(159,162)(160,161)>;
G:=Group( (1,68,119)(2,120,69)(3,70,101)(4,102,71)(5,72,103)(6,104,73)(7,74,105)(8,106,75)(9,76,107)(10,108,77)(11,78,109)(12,110,79)(13,80,111)(14,112,61)(15,62,113)(16,114,63)(17,64,115)(18,116,65)(19,66,117)(20,118,67)(21,90,141)(22,142,91)(23,92,143)(24,144,93)(25,94,145)(26,146,95)(27,96,147)(28,148,97)(29,98,149)(30,150,99)(31,100,151)(32,152,81)(33,82,153)(34,154,83)(35,84,155)(36,156,85)(37,86,157)(38,158,87)(39,88,159)(40,160,89)(41,138,165)(42,166,139)(43,140,167)(44,168,121)(45,122,169)(46,170,123)(47,124,171)(48,172,125)(49,126,173)(50,174,127)(51,128,175)(52,176,129)(53,130,177)(54,178,131)(55,132,179)(56,180,133)(57,134,161)(58,162,135)(59,136,163)(60,164,137), (1,123,86)(2,87,124)(3,125,88)(4,89,126)(5,127,90)(6,91,128)(7,129,92)(8,93,130)(9,131,94)(10,95,132)(11,133,96)(12,97,134)(13,135,98)(14,99,136)(15,137,100)(16,81,138)(17,139,82)(18,83,140)(19,121,84)(20,85,122)(21,103,174)(22,175,104)(23,105,176)(24,177,106)(25,107,178)(26,179,108)(27,109,180)(28,161,110)(29,111,162)(30,163,112)(31,113,164)(32,165,114)(33,115,166)(34,167,116)(35,117,168)(36,169,118)(37,119,170)(38,171,120)(39,101,172)(40,173,102)(41,63,152)(42,153,64)(43,65,154)(44,155,66)(45,67,156)(46,157,68)(47,69,158)(48,159,70)(49,71,160)(50,141,72)(51,73,142)(52,143,74)(53,75,144)(54,145,76)(55,77,146)(56,147,78)(57,79,148)(58,149,80)(59,61,150)(60,151,62), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(21,56)(22,55)(23,54)(24,53)(25,52)(26,51)(27,50)(28,49)(29,48)(30,47)(31,46)(32,45)(33,44)(34,43)(35,42)(36,41)(37,60)(38,59)(39,58)(40,57)(61,120)(62,119)(63,118)(64,117)(65,116)(66,115)(67,114)(68,113)(69,112)(70,111)(71,110)(72,109)(73,108)(74,107)(75,106)(76,105)(77,104)(78,103)(79,102)(80,101)(81,122)(82,121)(83,140)(84,139)(85,138)(86,137)(87,136)(88,135)(89,134)(90,133)(91,132)(92,131)(93,130)(94,129)(95,128)(96,127)(97,126)(98,125)(99,124)(100,123)(141,180)(142,179)(143,178)(144,177)(145,176)(146,175)(147,174)(148,173)(149,172)(150,171)(151,170)(152,169)(153,168)(154,167)(155,166)(156,165)(157,164)(158,163)(159,162)(160,161) );
G=PermutationGroup([[(1,68,119),(2,120,69),(3,70,101),(4,102,71),(5,72,103),(6,104,73),(7,74,105),(8,106,75),(9,76,107),(10,108,77),(11,78,109),(12,110,79),(13,80,111),(14,112,61),(15,62,113),(16,114,63),(17,64,115),(18,116,65),(19,66,117),(20,118,67),(21,90,141),(22,142,91),(23,92,143),(24,144,93),(25,94,145),(26,146,95),(27,96,147),(28,148,97),(29,98,149),(30,150,99),(31,100,151),(32,152,81),(33,82,153),(34,154,83),(35,84,155),(36,156,85),(37,86,157),(38,158,87),(39,88,159),(40,160,89),(41,138,165),(42,166,139),(43,140,167),(44,168,121),(45,122,169),(46,170,123),(47,124,171),(48,172,125),(49,126,173),(50,174,127),(51,128,175),(52,176,129),(53,130,177),(54,178,131),(55,132,179),(56,180,133),(57,134,161),(58,162,135),(59,136,163),(60,164,137)], [(1,123,86),(2,87,124),(3,125,88),(4,89,126),(5,127,90),(6,91,128),(7,129,92),(8,93,130),(9,131,94),(10,95,132),(11,133,96),(12,97,134),(13,135,98),(14,99,136),(15,137,100),(16,81,138),(17,139,82),(18,83,140),(19,121,84),(20,85,122),(21,103,174),(22,175,104),(23,105,176),(24,177,106),(25,107,178),(26,179,108),(27,109,180),(28,161,110),(29,111,162),(30,163,112),(31,113,164),(32,165,114),(33,115,166),(34,167,116),(35,117,168),(36,169,118),(37,119,170),(38,171,120),(39,101,172),(40,173,102),(41,63,152),(42,153,64),(43,65,154),(44,155,66),(45,67,156),(46,157,68),(47,69,158),(48,159,70),(49,71,160),(50,141,72),(51,73,142),(52,143,74),(53,75,144),(54,145,76),(55,77,146),(56,147,78),(57,79,148),(58,149,80),(59,61,150),(60,151,62)], [(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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,20),(17,19),(21,56),(22,55),(23,54),(24,53),(25,52),(26,51),(27,50),(28,49),(29,48),(30,47),(31,46),(32,45),(33,44),(34,43),(35,42),(36,41),(37,60),(38,59),(39,58),(40,57),(61,120),(62,119),(63,118),(64,117),(65,116),(66,115),(67,114),(68,113),(69,112),(70,111),(71,110),(72,109),(73,108),(74,107),(75,106),(76,105),(77,104),(78,103),(79,102),(80,101),(81,122),(82,121),(83,140),(84,139),(85,138),(86,137),(87,136),(88,135),(89,134),(90,133),(91,132),(92,131),(93,130),(94,129),(95,128),(96,127),(97,126),(98,125),(99,124),(100,123),(141,180),(142,179),(143,178),(144,177),(145,176),(146,175),(147,174),(148,173),(149,172),(150,171),(151,170),(152,169),(153,168),(154,167),(155,166),(156,165),(157,164),(158,163),(159,162),(160,161)]])
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 4 | 5A | 5B | 6A | 6B | 6C | 6D | 6E | ··· | 6L | 10A | 10B | 15A | ··· | 15H | 20A | 20B | 20C | 20D | 30A | ··· | 30H |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 10 | 10 | 15 | ··· | 15 | 20 | 20 | 20 | 20 | 30 | ··· | 30 |
| size | 1 | 1 | 10 | 90 | 2 | 2 | 2 | 2 | 18 | 2 | 2 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 2 | 2 | 4 | ··· | 4 | 18 | 18 | 18 | 18 | 4 | ··· | 4 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | S3 | D4 | D5 | D6 | D10 | C3⋊D4 | D20 | S3×D5 | C3⋊D20 |
| kernel | C32⋊7D20 | C5×C3⋊Dic3 | D5×C3×C6 | C2×C3⋊D15 | C6×D5 | C3×C15 | C3⋊Dic3 | C30 | C3×C6 | C15 | C32 | C6 | C3 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 2 | 4 | 2 | 8 | 4 | 8 | 8 |
Matrix representation of C32⋊7D20 ►in GL6(𝔽61)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 60 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 41 | 0 | 0 |
| 0 | 0 | 52 | 59 | 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 | 59 | 20 | 0 | 0 |
| 0 | 0 | 9 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 52 | 43 | 0 | 0 | 0 | 0 |
| 52 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 9 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 43 | 1 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 9 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 43 |
| 0 | 0 | 0 | 0 | 0 | 60 |
G:=sub<GL(6,GF(61))| [0,60,0,0,0,0,1,60,0,0,0,0,0,0,1,52,0,0,0,0,41,59,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,59,9,0,0,0,0,20,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[52,52,0,0,0,0,43,9,0,0,0,0,0,0,60,9,0,0,0,0,0,1,0,0,0,0,0,0,43,60,0,0,0,0,1,0],[60,1,0,0,0,0,0,1,0,0,0,0,0,0,60,9,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,43,60] >;
C32⋊7D20 in GAP, Magma, Sage, TeX
C_3^2\rtimes_7D_{20} % in TeX
G:=Group("C3^2:7D20"); // GroupNames label
G:=SmallGroup(360,69);
// by ID
G=gap.SmallGroup(360,69);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-3,-5,73,31,201,730,10373]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^20=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a^-1,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations