metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.215D6, C3⋊C8.3Q8, C4⋊C4.74D6, C4.33(S3×Q8), C6.29(C4⋊Q8), C12.33(C2×Q8), C3⋊4(C8.5Q8), (C2×C12).274D4, C42.C2.4S3, C6.108(C4○D8), C6.Q16.14C2, C12.6Q8.7C2, (C2×C12).383C23, (C4×C12).113C22, C12.Q8.15C2, C2.9(Dic3⋊Q8), C2.27(Q8.13D6), C4⋊Dic3.153C22, (C4×C3⋊C8).10C2, (C2×C6).514(C2×D4), (C2×C3⋊C8).254C22, (C3×C42.C2).3C2, (C2×C4).111(C3⋊D4), (C3×C4⋊C4).121C22, (C2×C4).481(C22×S3), C22.187(C2×C3⋊D4), SmallGroup(192,624)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.215D6
G = < a,b,c,d | a4=b4=1, c6=d2=a2, ab=ba, cac-1=dad-1=a-1b2, cbc-1=dbd-1=b-1, dcd-1=bc5 >
Subgroups: 192 in 86 conjugacy classes, 43 normal (21 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C8, C2×C4, C2×C4, C2×C4, Dic3, C12, C12, C2×C6, C42, C4⋊C4, C4⋊C4, C2×C8, C3⋊C8, C2×Dic3, C2×C12, C2×C12, C2×C12, C4×C8, C4.Q8, C2.D8, C42.C2, C42.C2, C2×C3⋊C8, Dic3⋊C4, C4⋊Dic3, C4×C12, C3×C4⋊C4, C3×C4⋊C4, C8.5Q8, C4×C3⋊C8, C6.Q16, C12.Q8, C12.6Q8, C3×C42.C2, C42.215D6
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, C3⋊D4, C22×S3, C4⋊Q8, C4○D8, S3×Q8, C2×C3⋊D4, C8.5Q8, Dic3⋊Q8, Q8.13D6, C42.215D6
►Regular action on 192 points
Generators in S192
(1 137 7 143)(2 44 8 38)(3 139 9 133)(4 46 10 40)(5 141 11 135)(6 48 12 42)(13 180 19 174)(14 191 20 185)(15 170 21 176)(16 181 22 187)(17 172 23 178)(18 183 24 189)(25 66 31 72)(26 89 32 95)(27 68 33 62)(28 91 34 85)(29 70 35 64)(30 93 36 87)(37 57 43 51)(39 59 45 53)(41 49 47 55)(50 142 56 136)(52 144 58 138)(54 134 60 140)(61 162 67 168)(63 164 69 158)(65 166 71 160)(73 154 79 148)(74 108 80 102)(75 156 81 150)(76 98 82 104)(77 146 83 152)(78 100 84 106)(86 165 92 159)(88 167 94 161)(90 157 96 163)(97 122 103 128)(99 124 105 130)(101 126 107 132)(109 177 115 171)(110 188 116 182)(111 179 117 173)(112 190 118 184)(113 169 119 175)(114 192 120 186)(121 149 127 155)(123 151 129 145)(125 153 131 147)
(1 29 51 159)(2 160 52 30)(3 31 53 161)(4 162 54 32)(5 33 55 163)(6 164 56 34)(7 35 57 165)(8 166 58 36)(9 25 59 167)(10 168 60 26)(11 27 49 157)(12 158 50 28)(13 127 118 80)(14 81 119 128)(15 129 120 82)(16 83 109 130)(17 131 110 84)(18 73 111 132)(19 121 112 74)(20 75 113 122)(21 123 114 76)(22 77 115 124)(23 125 116 78)(24 79 117 126)(37 86 137 70)(38 71 138 87)(39 88 139 72)(40 61 140 89)(41 90 141 62)(42 63 142 91)(43 92 143 64)(44 65 144 93)(45 94 133 66)(46 67 134 95)(47 96 135 68)(48 69 136 85)(97 191 150 175)(98 176 151 192)(99 181 152 177)(100 178 153 182)(101 183 154 179)(102 180 155 184)(103 185 156 169)(104 170 145 186)(105 187 146 171)(106 172 147 188)(107 189 148 173)(108 174 149 190)
(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)(181 182 183 184 185 186 187 188 189 190 191 192)
(1 75 7 81)(2 118 8 112)(3 73 9 79)(4 116 10 110)(5 83 11 77)(6 114 12 120)(13 58 19 52)(14 29 20 35)(15 56 21 50)(16 27 22 33)(17 54 23 60)(18 25 24 31)(26 84 32 78)(28 82 34 76)(30 80 36 74)(37 150 43 156)(38 180 44 174)(39 148 45 154)(40 178 46 172)(41 146 47 152)(42 176 48 170)(49 124 55 130)(51 122 57 128)(53 132 59 126)(61 100 67 106)(62 171 68 177)(63 98 69 104)(64 169 70 175)(65 108 71 102)(66 179 72 173)(85 145 91 151)(86 191 92 185)(87 155 93 149)(88 189 94 183)(89 153 95 147)(90 187 96 181)(97 143 103 137)(99 141 105 135)(101 139 107 133)(109 157 115 163)(111 167 117 161)(113 165 119 159)(121 160 127 166)(123 158 129 164)(125 168 131 162)(134 188 140 182)(136 186 142 192)(138 184 144 190)
G:=sub<Sym(192)| (1,137,7,143)(2,44,8,38)(3,139,9,133)(4,46,10,40)(5,141,11,135)(6,48,12,42)(13,180,19,174)(14,191,20,185)(15,170,21,176)(16,181,22,187)(17,172,23,178)(18,183,24,189)(25,66,31,72)(26,89,32,95)(27,68,33,62)(28,91,34,85)(29,70,35,64)(30,93,36,87)(37,57,43,51)(39,59,45,53)(41,49,47,55)(50,142,56,136)(52,144,58,138)(54,134,60,140)(61,162,67,168)(63,164,69,158)(65,166,71,160)(73,154,79,148)(74,108,80,102)(75,156,81,150)(76,98,82,104)(77,146,83,152)(78,100,84,106)(86,165,92,159)(88,167,94,161)(90,157,96,163)(97,122,103,128)(99,124,105,130)(101,126,107,132)(109,177,115,171)(110,188,116,182)(111,179,117,173)(112,190,118,184)(113,169,119,175)(114,192,120,186)(121,149,127,155)(123,151,129,145)(125,153,131,147), (1,29,51,159)(2,160,52,30)(3,31,53,161)(4,162,54,32)(5,33,55,163)(6,164,56,34)(7,35,57,165)(8,166,58,36)(9,25,59,167)(10,168,60,26)(11,27,49,157)(12,158,50,28)(13,127,118,80)(14,81,119,128)(15,129,120,82)(16,83,109,130)(17,131,110,84)(18,73,111,132)(19,121,112,74)(20,75,113,122)(21,123,114,76)(22,77,115,124)(23,125,116,78)(24,79,117,126)(37,86,137,70)(38,71,138,87)(39,88,139,72)(40,61,140,89)(41,90,141,62)(42,63,142,91)(43,92,143,64)(44,65,144,93)(45,94,133,66)(46,67,134,95)(47,96,135,68)(48,69,136,85)(97,191,150,175)(98,176,151,192)(99,181,152,177)(100,178,153,182)(101,183,154,179)(102,180,155,184)(103,185,156,169)(104,170,145,186)(105,187,146,171)(106,172,147,188)(107,189,148,173)(108,174,149,190), (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)(181,182,183,184,185,186,187,188,189,190,191,192), (1,75,7,81)(2,118,8,112)(3,73,9,79)(4,116,10,110)(5,83,11,77)(6,114,12,120)(13,58,19,52)(14,29,20,35)(15,56,21,50)(16,27,22,33)(17,54,23,60)(18,25,24,31)(26,84,32,78)(28,82,34,76)(30,80,36,74)(37,150,43,156)(38,180,44,174)(39,148,45,154)(40,178,46,172)(41,146,47,152)(42,176,48,170)(49,124,55,130)(51,122,57,128)(53,132,59,126)(61,100,67,106)(62,171,68,177)(63,98,69,104)(64,169,70,175)(65,108,71,102)(66,179,72,173)(85,145,91,151)(86,191,92,185)(87,155,93,149)(88,189,94,183)(89,153,95,147)(90,187,96,181)(97,143,103,137)(99,141,105,135)(101,139,107,133)(109,157,115,163)(111,167,117,161)(113,165,119,159)(121,160,127,166)(123,158,129,164)(125,168,131,162)(134,188,140,182)(136,186,142,192)(138,184,144,190)>;
G:=Group( (1,137,7,143)(2,44,8,38)(3,139,9,133)(4,46,10,40)(5,141,11,135)(6,48,12,42)(13,180,19,174)(14,191,20,185)(15,170,21,176)(16,181,22,187)(17,172,23,178)(18,183,24,189)(25,66,31,72)(26,89,32,95)(27,68,33,62)(28,91,34,85)(29,70,35,64)(30,93,36,87)(37,57,43,51)(39,59,45,53)(41,49,47,55)(50,142,56,136)(52,144,58,138)(54,134,60,140)(61,162,67,168)(63,164,69,158)(65,166,71,160)(73,154,79,148)(74,108,80,102)(75,156,81,150)(76,98,82,104)(77,146,83,152)(78,100,84,106)(86,165,92,159)(88,167,94,161)(90,157,96,163)(97,122,103,128)(99,124,105,130)(101,126,107,132)(109,177,115,171)(110,188,116,182)(111,179,117,173)(112,190,118,184)(113,169,119,175)(114,192,120,186)(121,149,127,155)(123,151,129,145)(125,153,131,147), (1,29,51,159)(2,160,52,30)(3,31,53,161)(4,162,54,32)(5,33,55,163)(6,164,56,34)(7,35,57,165)(8,166,58,36)(9,25,59,167)(10,168,60,26)(11,27,49,157)(12,158,50,28)(13,127,118,80)(14,81,119,128)(15,129,120,82)(16,83,109,130)(17,131,110,84)(18,73,111,132)(19,121,112,74)(20,75,113,122)(21,123,114,76)(22,77,115,124)(23,125,116,78)(24,79,117,126)(37,86,137,70)(38,71,138,87)(39,88,139,72)(40,61,140,89)(41,90,141,62)(42,63,142,91)(43,92,143,64)(44,65,144,93)(45,94,133,66)(46,67,134,95)(47,96,135,68)(48,69,136,85)(97,191,150,175)(98,176,151,192)(99,181,152,177)(100,178,153,182)(101,183,154,179)(102,180,155,184)(103,185,156,169)(104,170,145,186)(105,187,146,171)(106,172,147,188)(107,189,148,173)(108,174,149,190), (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)(181,182,183,184,185,186,187,188,189,190,191,192), (1,75,7,81)(2,118,8,112)(3,73,9,79)(4,116,10,110)(5,83,11,77)(6,114,12,120)(13,58,19,52)(14,29,20,35)(15,56,21,50)(16,27,22,33)(17,54,23,60)(18,25,24,31)(26,84,32,78)(28,82,34,76)(30,80,36,74)(37,150,43,156)(38,180,44,174)(39,148,45,154)(40,178,46,172)(41,146,47,152)(42,176,48,170)(49,124,55,130)(51,122,57,128)(53,132,59,126)(61,100,67,106)(62,171,68,177)(63,98,69,104)(64,169,70,175)(65,108,71,102)(66,179,72,173)(85,145,91,151)(86,191,92,185)(87,155,93,149)(88,189,94,183)(89,153,95,147)(90,187,96,181)(97,143,103,137)(99,141,105,135)(101,139,107,133)(109,157,115,163)(111,167,117,161)(113,165,119,159)(121,160,127,166)(123,158,129,164)(125,168,131,162)(134,188,140,182)(136,186,142,192)(138,184,144,190) );
G=PermutationGroup([[(1,137,7,143),(2,44,8,38),(3,139,9,133),(4,46,10,40),(5,141,11,135),(6,48,12,42),(13,180,19,174),(14,191,20,185),(15,170,21,176),(16,181,22,187),(17,172,23,178),(18,183,24,189),(25,66,31,72),(26,89,32,95),(27,68,33,62),(28,91,34,85),(29,70,35,64),(30,93,36,87),(37,57,43,51),(39,59,45,53),(41,49,47,55),(50,142,56,136),(52,144,58,138),(54,134,60,140),(61,162,67,168),(63,164,69,158),(65,166,71,160),(73,154,79,148),(74,108,80,102),(75,156,81,150),(76,98,82,104),(77,146,83,152),(78,100,84,106),(86,165,92,159),(88,167,94,161),(90,157,96,163),(97,122,103,128),(99,124,105,130),(101,126,107,132),(109,177,115,171),(110,188,116,182),(111,179,117,173),(112,190,118,184),(113,169,119,175),(114,192,120,186),(121,149,127,155),(123,151,129,145),(125,153,131,147)], [(1,29,51,159),(2,160,52,30),(3,31,53,161),(4,162,54,32),(5,33,55,163),(6,164,56,34),(7,35,57,165),(8,166,58,36),(9,25,59,167),(10,168,60,26),(11,27,49,157),(12,158,50,28),(13,127,118,80),(14,81,119,128),(15,129,120,82),(16,83,109,130),(17,131,110,84),(18,73,111,132),(19,121,112,74),(20,75,113,122),(21,123,114,76),(22,77,115,124),(23,125,116,78),(24,79,117,126),(37,86,137,70),(38,71,138,87),(39,88,139,72),(40,61,140,89),(41,90,141,62),(42,63,142,91),(43,92,143,64),(44,65,144,93),(45,94,133,66),(46,67,134,95),(47,96,135,68),(48,69,136,85),(97,191,150,175),(98,176,151,192),(99,181,152,177),(100,178,153,182),(101,183,154,179),(102,180,155,184),(103,185,156,169),(104,170,145,186),(105,187,146,171),(106,172,147,188),(107,189,148,173),(108,174,149,190)], [(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),(181,182,183,184,185,186,187,188,189,190,191,192)], [(1,75,7,81),(2,118,8,112),(3,73,9,79),(4,116,10,110),(5,83,11,77),(6,114,12,120),(13,58,19,52),(14,29,20,35),(15,56,21,50),(16,27,22,33),(17,54,23,60),(18,25,24,31),(26,84,32,78),(28,82,34,76),(30,80,36,74),(37,150,43,156),(38,180,44,174),(39,148,45,154),(40,178,46,172),(41,146,47,152),(42,176,48,170),(49,124,55,130),(51,122,57,128),(53,132,59,126),(61,100,67,106),(62,171,68,177),(63,98,69,104),(64,169,70,175),(65,108,71,102),(66,179,72,173),(85,145,91,151),(86,191,92,185),(87,155,93,149),(88,189,94,183),(89,153,95,147),(90,187,96,181),(97,143,103,137),(99,141,105,135),(101,139,107,133),(109,157,115,163),(111,167,117,161),(113,165,119,159),(121,160,127,166),(123,158,129,164),(125,168,131,162),(134,188,140,182),(136,186,142,192),(138,184,144,190)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 6A | 6B | 6C | 8A | ··· | 8H | 12A | ··· | 12F | 12G | 12H | 12I | 12J |
| order | 1 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | ··· | 2 | 8 | 8 | 24 | 24 | 2 | 2 | 2 | 6 | ··· | 6 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | Q8 | D4 | D6 | D6 | C3⋊D4 | C4○D8 | S3×Q8 | Q8.13D6 |
| kernel | C42.215D6 | C4×C3⋊C8 | C6.Q16 | C12.Q8 | C12.6Q8 | C3×C42.C2 | C42.C2 | C3⋊C8 | C2×C12 | C42 | C4⋊C4 | C2×C4 | C6 | C4 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 2 | 1 | 2 | 4 | 8 | 2 | 4 |
Matrix representation of C42.215D6 ►in GL6(𝔽73)
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 27 | 0 | 0 |
| 0 | 0 | 46 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 27 | 0 |
| 0 | 0 | 0 | 0 | 0 | 27 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 3 |
| 0 | 0 | 0 | 0 | 48 | 1 |
| 30 | 43 | 0 | 0 | 0 | 0 |
| 30 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 26 | 37 | 0 | 0 |
| 0 | 0 | 37 | 47 | 0 | 0 |
| 0 | 0 | 0 | 0 | 51 | 62 |
| 0 | 0 | 0 | 0 | 64 | 22 |
| 59 | 66 | 0 | 0 | 0 | 0 |
| 7 | 14 | 0 | 0 | 0 | 0 |
| 0 | 0 | 30 | 14 | 0 | 0 |
| 0 | 0 | 14 | 43 | 0 | 0 |
| 0 | 0 | 0 | 0 | 53 | 21 |
| 0 | 0 | 0 | 0 | 40 | 20 |
G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,46,0,0,0,0,27,0,0,0,0,0,0,0,27,0,0,0,0,0,0,27],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,1,0,0,0,0,0,0,0,72,48,0,0,0,0,3,1],[30,30,0,0,0,0,43,60,0,0,0,0,0,0,26,37,0,0,0,0,37,47,0,0,0,0,0,0,51,64,0,0,0,0,62,22],[59,7,0,0,0,0,66,14,0,0,0,0,0,0,30,14,0,0,0,0,14,43,0,0,0,0,0,0,53,40,0,0,0,0,21,20] >;
C42.215D6 in GAP, Magma, Sage, TeX
C_4^2._{215}D_6 % in TeX
G:=Group("C4^2.215D6"); // GroupNames label
G:=SmallGroup(192,624);
// by ID
G=gap.SmallGroup(192,624);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,64,422,471,58,438,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=d^2=a^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1*b^2,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b*c^5>;
// generators/relations