metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊7Dic3, (C4×C12)⋊7C4, (C2×C42).7S3, C3⋊2(C42⋊5C4), (C22×C4).414D6, C2.2(C42⋊3S3), C6.4(C42⋊2C2), C6.43(C42⋊C2), C22.46(C4○D12), C6.C42.13C2, (C22×C6).312C23, C23.280(C22×S3), (C22×C12).476C22, C2.7(C23.26D6), C22.38(C22×Dic3), (C22×Dic3).31C22, (C2×C4×C12).3C2, (C2×C12).278(C2×C4), (C2×C6).71(C4○D4), (C2×C4).63(C2×Dic3), (C2×C6).177(C22×C4), SmallGroup(192,496)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42⋊7Dic3
G = < a,b,c,d | a4=b4=c6=1, d2=c3, ab=ba, ac=ca, dad-1=ab2, bc=cb, dbd-1=a2b-1, dcd-1=c-1 >
Subgroups: 312 in 138 conjugacy classes, 71 normal (9 characteristic)
C1, C2, C3, C4, C22, C22, C6, C2×C4, C2×C4, C23, Dic3, C12, C2×C6, C2×C6, C42, C22×C4, C22×C4, C2×Dic3, C2×C12, C2×C12, C22×C6, C2.C42, C2×C42, C4×C12, C22×Dic3, C22×C12, C42⋊5C4, C6.C42, C2×C4×C12, C42⋊7Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, Dic3, D6, C22×C4, C4○D4, C2×Dic3, C22×S3, C42⋊C2, C42⋊2C2, C4○D12, C22×Dic3, C42⋊5C4, C42⋊3S3, C23.26D6, C42⋊7Dic3
►Regular action on 192 points
Generators in S192
(1 55 7 63)(2 56 8 64)(3 57 9 65)(4 58 10 66)(5 59 11 61)(6 60 12 62)(13 129 190 137)(14 130 191 138)(15 131 192 133)(16 132 187 134)(17 127 188 135)(18 128 189 136)(19 78 30 67)(20 73 25 68)(21 74 26 69)(22 75 27 70)(23 76 28 71)(24 77 29 72)(31 87 39 79)(32 88 40 80)(33 89 41 81)(34 90 42 82)(35 85 37 83)(36 86 38 84)(43 101 54 91)(44 102 49 92)(45 97 50 93)(46 98 51 94)(47 99 52 95)(48 100 53 96)(103 164 111 174)(104 165 112 169)(105 166 113 170)(106 167 114 171)(107 168 109 172)(108 163 110 173)(115 156 125 158)(116 151 126 159)(117 152 121 160)(118 153 122 161)(119 154 123 162)(120 155 124 157)(139 180 149 182)(140 175 150 183)(141 176 145 184)(142 177 146 185)(143 178 147 186)(144 179 148 181)
(1 47 23 39)(2 48 24 40)(3 43 19 41)(4 44 20 42)(5 45 21 37)(6 46 22 38)(7 52 28 31)(8 53 29 32)(9 54 30 33)(10 49 25 34)(11 50 26 35)(12 51 27 36)(13 161 177 166)(14 162 178 167)(15 157 179 168)(16 158 180 163)(17 159 175 164)(18 160 176 165)(55 99 76 79)(56 100 77 80)(57 101 78 81)(58 102 73 82)(59 97 74 83)(60 98 75 84)(61 93 69 85)(62 94 70 86)(63 95 71 87)(64 96 72 88)(65 91 67 89)(66 92 68 90)(103 135 126 140)(104 136 121 141)(105 137 122 142)(106 138 123 143)(107 133 124 144)(108 134 125 139)(109 131 120 148)(110 132 115 149)(111 127 116 150)(112 128 117 145)(113 129 118 146)(114 130 119 147)(151 183 174 188)(152 184 169 189)(153 185 170 190)(154 186 171 191)(155 181 172 192)(156 182 173 187)
(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)
(1 113 4 110)(2 112 5 109)(3 111 6 114)(7 105 10 108)(8 104 11 107)(9 103 12 106)(13 102 16 99)(14 101 17 98)(15 100 18 97)(19 116 22 119)(20 115 23 118)(21 120 24 117)(25 125 28 122)(26 124 29 121)(27 123 30 126)(31 129 34 132)(32 128 35 131)(33 127 36 130)(37 133 40 136)(38 138 41 135)(39 137 42 134)(43 140 46 143)(44 139 47 142)(45 144 48 141)(49 149 52 146)(50 148 53 145)(51 147 54 150)(55 153 58 156)(56 152 59 155)(57 151 60 154)(61 157 64 160)(62 162 65 159)(63 161 66 158)(67 164 70 167)(68 163 71 166)(69 168 72 165)(73 173 76 170)(74 172 77 169)(75 171 78 174)(79 177 82 180)(80 176 83 179)(81 175 84 178)(85 181 88 184)(86 186 89 183)(87 185 90 182)(91 188 94 191)(92 187 95 190)(93 192 96 189)
G:=sub<Sym(192)| (1,55,7,63)(2,56,8,64)(3,57,9,65)(4,58,10,66)(5,59,11,61)(6,60,12,62)(13,129,190,137)(14,130,191,138)(15,131,192,133)(16,132,187,134)(17,127,188,135)(18,128,189,136)(19,78,30,67)(20,73,25,68)(21,74,26,69)(22,75,27,70)(23,76,28,71)(24,77,29,72)(31,87,39,79)(32,88,40,80)(33,89,41,81)(34,90,42,82)(35,85,37,83)(36,86,38,84)(43,101,54,91)(44,102,49,92)(45,97,50,93)(46,98,51,94)(47,99,52,95)(48,100,53,96)(103,164,111,174)(104,165,112,169)(105,166,113,170)(106,167,114,171)(107,168,109,172)(108,163,110,173)(115,156,125,158)(116,151,126,159)(117,152,121,160)(118,153,122,161)(119,154,123,162)(120,155,124,157)(139,180,149,182)(140,175,150,183)(141,176,145,184)(142,177,146,185)(143,178,147,186)(144,179,148,181), (1,47,23,39)(2,48,24,40)(3,43,19,41)(4,44,20,42)(5,45,21,37)(6,46,22,38)(7,52,28,31)(8,53,29,32)(9,54,30,33)(10,49,25,34)(11,50,26,35)(12,51,27,36)(13,161,177,166)(14,162,178,167)(15,157,179,168)(16,158,180,163)(17,159,175,164)(18,160,176,165)(55,99,76,79)(56,100,77,80)(57,101,78,81)(58,102,73,82)(59,97,74,83)(60,98,75,84)(61,93,69,85)(62,94,70,86)(63,95,71,87)(64,96,72,88)(65,91,67,89)(66,92,68,90)(103,135,126,140)(104,136,121,141)(105,137,122,142)(106,138,123,143)(107,133,124,144)(108,134,125,139)(109,131,120,148)(110,132,115,149)(111,127,116,150)(112,128,117,145)(113,129,118,146)(114,130,119,147)(151,183,174,188)(152,184,169,189)(153,185,170,190)(154,186,171,191)(155,181,172,192)(156,182,173,187), (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), (1,113,4,110)(2,112,5,109)(3,111,6,114)(7,105,10,108)(8,104,11,107)(9,103,12,106)(13,102,16,99)(14,101,17,98)(15,100,18,97)(19,116,22,119)(20,115,23,118)(21,120,24,117)(25,125,28,122)(26,124,29,121)(27,123,30,126)(31,129,34,132)(32,128,35,131)(33,127,36,130)(37,133,40,136)(38,138,41,135)(39,137,42,134)(43,140,46,143)(44,139,47,142)(45,144,48,141)(49,149,52,146)(50,148,53,145)(51,147,54,150)(55,153,58,156)(56,152,59,155)(57,151,60,154)(61,157,64,160)(62,162,65,159)(63,161,66,158)(67,164,70,167)(68,163,71,166)(69,168,72,165)(73,173,76,170)(74,172,77,169)(75,171,78,174)(79,177,82,180)(80,176,83,179)(81,175,84,178)(85,181,88,184)(86,186,89,183)(87,185,90,182)(91,188,94,191)(92,187,95,190)(93,192,96,189)>;
G:=Group( (1,55,7,63)(2,56,8,64)(3,57,9,65)(4,58,10,66)(5,59,11,61)(6,60,12,62)(13,129,190,137)(14,130,191,138)(15,131,192,133)(16,132,187,134)(17,127,188,135)(18,128,189,136)(19,78,30,67)(20,73,25,68)(21,74,26,69)(22,75,27,70)(23,76,28,71)(24,77,29,72)(31,87,39,79)(32,88,40,80)(33,89,41,81)(34,90,42,82)(35,85,37,83)(36,86,38,84)(43,101,54,91)(44,102,49,92)(45,97,50,93)(46,98,51,94)(47,99,52,95)(48,100,53,96)(103,164,111,174)(104,165,112,169)(105,166,113,170)(106,167,114,171)(107,168,109,172)(108,163,110,173)(115,156,125,158)(116,151,126,159)(117,152,121,160)(118,153,122,161)(119,154,123,162)(120,155,124,157)(139,180,149,182)(140,175,150,183)(141,176,145,184)(142,177,146,185)(143,178,147,186)(144,179,148,181), (1,47,23,39)(2,48,24,40)(3,43,19,41)(4,44,20,42)(5,45,21,37)(6,46,22,38)(7,52,28,31)(8,53,29,32)(9,54,30,33)(10,49,25,34)(11,50,26,35)(12,51,27,36)(13,161,177,166)(14,162,178,167)(15,157,179,168)(16,158,180,163)(17,159,175,164)(18,160,176,165)(55,99,76,79)(56,100,77,80)(57,101,78,81)(58,102,73,82)(59,97,74,83)(60,98,75,84)(61,93,69,85)(62,94,70,86)(63,95,71,87)(64,96,72,88)(65,91,67,89)(66,92,68,90)(103,135,126,140)(104,136,121,141)(105,137,122,142)(106,138,123,143)(107,133,124,144)(108,134,125,139)(109,131,120,148)(110,132,115,149)(111,127,116,150)(112,128,117,145)(113,129,118,146)(114,130,119,147)(151,183,174,188)(152,184,169,189)(153,185,170,190)(154,186,171,191)(155,181,172,192)(156,182,173,187), (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), (1,113,4,110)(2,112,5,109)(3,111,6,114)(7,105,10,108)(8,104,11,107)(9,103,12,106)(13,102,16,99)(14,101,17,98)(15,100,18,97)(19,116,22,119)(20,115,23,118)(21,120,24,117)(25,125,28,122)(26,124,29,121)(27,123,30,126)(31,129,34,132)(32,128,35,131)(33,127,36,130)(37,133,40,136)(38,138,41,135)(39,137,42,134)(43,140,46,143)(44,139,47,142)(45,144,48,141)(49,149,52,146)(50,148,53,145)(51,147,54,150)(55,153,58,156)(56,152,59,155)(57,151,60,154)(61,157,64,160)(62,162,65,159)(63,161,66,158)(67,164,70,167)(68,163,71,166)(69,168,72,165)(73,173,76,170)(74,172,77,169)(75,171,78,174)(79,177,82,180)(80,176,83,179)(81,175,84,178)(85,181,88,184)(86,186,89,183)(87,185,90,182)(91,188,94,191)(92,187,95,190)(93,192,96,189) );
G=PermutationGroup([[(1,55,7,63),(2,56,8,64),(3,57,9,65),(4,58,10,66),(5,59,11,61),(6,60,12,62),(13,129,190,137),(14,130,191,138),(15,131,192,133),(16,132,187,134),(17,127,188,135),(18,128,189,136),(19,78,30,67),(20,73,25,68),(21,74,26,69),(22,75,27,70),(23,76,28,71),(24,77,29,72),(31,87,39,79),(32,88,40,80),(33,89,41,81),(34,90,42,82),(35,85,37,83),(36,86,38,84),(43,101,54,91),(44,102,49,92),(45,97,50,93),(46,98,51,94),(47,99,52,95),(48,100,53,96),(103,164,111,174),(104,165,112,169),(105,166,113,170),(106,167,114,171),(107,168,109,172),(108,163,110,173),(115,156,125,158),(116,151,126,159),(117,152,121,160),(118,153,122,161),(119,154,123,162),(120,155,124,157),(139,180,149,182),(140,175,150,183),(141,176,145,184),(142,177,146,185),(143,178,147,186),(144,179,148,181)], [(1,47,23,39),(2,48,24,40),(3,43,19,41),(4,44,20,42),(5,45,21,37),(6,46,22,38),(7,52,28,31),(8,53,29,32),(9,54,30,33),(10,49,25,34),(11,50,26,35),(12,51,27,36),(13,161,177,166),(14,162,178,167),(15,157,179,168),(16,158,180,163),(17,159,175,164),(18,160,176,165),(55,99,76,79),(56,100,77,80),(57,101,78,81),(58,102,73,82),(59,97,74,83),(60,98,75,84),(61,93,69,85),(62,94,70,86),(63,95,71,87),(64,96,72,88),(65,91,67,89),(66,92,68,90),(103,135,126,140),(104,136,121,141),(105,137,122,142),(106,138,123,143),(107,133,124,144),(108,134,125,139),(109,131,120,148),(110,132,115,149),(111,127,116,150),(112,128,117,145),(113,129,118,146),(114,130,119,147),(151,183,174,188),(152,184,169,189),(153,185,170,190),(154,186,171,191),(155,181,172,192),(156,182,173,187)], [(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)], [(1,113,4,110),(2,112,5,109),(3,111,6,114),(7,105,10,108),(8,104,11,107),(9,103,12,106),(13,102,16,99),(14,101,17,98),(15,100,18,97),(19,116,22,119),(20,115,23,118),(21,120,24,117),(25,125,28,122),(26,124,29,121),(27,123,30,126),(31,129,34,132),(32,128,35,131),(33,127,36,130),(37,133,40,136),(38,138,41,135),(39,137,42,134),(43,140,46,143),(44,139,47,142),(45,144,48,141),(49,149,52,146),(50,148,53,145),(51,147,54,150),(55,153,58,156),(56,152,59,155),(57,151,60,154),(61,157,64,160),(62,162,65,159),(63,161,66,158),(67,164,70,167),(68,163,71,166),(69,168,72,165),(73,173,76,170),(74,172,77,169),(75,171,78,174),(79,177,82,180),(80,176,83,179),(81,175,84,178),(85,181,88,184),(86,186,89,183),(87,185,90,182),(91,188,94,191),(92,187,95,190),(93,192,96,189)]])
60 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3 | 4A | ··· | 4L | 4M | ··· | 4T | 6A | ··· | 6G | 12A | ··· | 12X |
| order | 1 | 2 | ··· | 2 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | ··· | 2 | 12 | ··· | 12 | 2 | ··· | 2 | 2 | ··· | 2 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | |||
| image | C1 | C2 | C2 | C4 | S3 | Dic3 | D6 | C4○D4 | C4○D12 |
| kernel | C42⋊7Dic3 | C6.C42 | C2×C4×C12 | C4×C12 | C2×C42 | C42 | C22×C4 | C2×C6 | C22 |
| # reps | 1 | 6 | 1 | 8 | 1 | 4 | 3 | 12 | 24 |
Matrix representation of C42⋊7Dic3 ►in GL6(𝔽13)
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 9 |
| 0 | 0 | 0 | 0 | 4 | 11 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 6 |
| 0 | 0 | 0 | 0 | 7 | 3 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 12 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 12 | 11 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 11 | 0 | 0 |
| 0 | 0 | 12 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 4 |
| 0 | 0 | 0 | 0 | 2 | 2 |
G:=sub<GL(6,GF(13))| [5,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,2,4,0,0,0,0,9,11],[1,12,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,1,12,0,0,0,0,0,0,10,7,0,0,0,0,6,3],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,1,0,0,0,0,12,0],[12,0,0,0,0,0,11,1,0,0,0,0,0,0,7,12,0,0,0,0,11,6,0,0,0,0,0,0,11,2,0,0,0,0,4,2] >;
C42⋊7Dic3 in GAP, Magma, Sage, TeX
C_4^2\rtimes_7{\rm Dic}_3 % in TeX
G:=Group("C4^2:7Dic3"); // GroupNames label
G:=SmallGroup(192,496);
// by ID
G=gap.SmallGroup(192,496);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,477,120,1094,184,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=1,d^2=c^3,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^2,b*c=c*b,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^-1>;
// generators/relations