metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊11Dic3, (C4×C12)⋊11C4, (C2×C4).91D12, C12.32(C4⋊C4), (C2×C12).54Q8, C3⋊2(C42⋊8C4), (C2×C12).384D4, (C2×C42).18S3, (C2×C4).46Dic6, C4.15(C4⋊Dic3), C22.37(C2×D12), (C22×C4).413D6, C6.4(C42.C2), C2.2(C42⋊7S3), C6.11(C4.4D4), C2.3(C12.6Q8), C22.22(C2×Dic6), C6.42(C42⋊C2), C22.45(C4○D12), C6.C42.12C2, (C22×C6).311C23, C23.279(C22×S3), (C22×C12).475C22, C2.6(C23.26D6), C22.37(C22×Dic3), (C22×Dic3).30C22, C6.28(C2×C4⋊C4), (C2×C4×C12).12C2, C2.6(C2×C4⋊Dic3), (C2×C6).29(C2×Q8), (C2×C6).147(C2×D4), (C2×C12).297(C2×C4), (C2×C6).70(C4○D4), (C2×C4⋊Dic3).16C2, (C2×C4).62(C2×Dic3), (C2×C6).176(C22×C4), SmallGroup(192,495)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42⋊11Dic3
G = < a,b,c,d | a4=b4=c6=1, d2=c3, ab=ba, ac=ca, dad-1=ab2, bc=cb, dbd-1=a2b, dcd-1=c-1 >
Subgroups: 344 in 154 conjugacy classes, 87 normal (17 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C2×C4, C2×C4, C23, Dic3, C12, C12, C2×C6, C2×C6, C42, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×Dic3, C2×C12, C2×C12, C22×C6, C2.C42, C2×C42, C2×C4⋊C4, C4⋊Dic3, C4×C12, C22×Dic3, C22×C12, C22×C12, C42⋊8C4, C6.C42, C2×C4⋊Dic3, C2×C4×C12, C42⋊11Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, C23, Dic3, D6, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic6, D12, C2×Dic3, C22×S3, C2×C4⋊C4, C42⋊C2, C4.4D4, C42.C2, C4⋊Dic3, C2×Dic6, C2×D12, C4○D12, C22×Dic3, C42⋊8C4, C12.6Q8, C42⋊7S3, C2×C4⋊Dic3, C23.26D6, C42⋊11Dic3
►Regular action on 192 points
Generators in S192
(1 66 8 56)(2 61 9 57)(3 62 10 58)(4 63 11 59)(5 64 12 60)(6 65 7 55)(13 135 190 130)(14 136 191 131)(15 137 192 132)(16 138 187 127)(17 133 188 128)(18 134 189 129)(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 89 41 79)(32 90 42 80)(33 85 37 81)(34 86 38 82)(35 87 39 83)(36 88 40 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)(103 173 114 163)(104 174 109 164)(105 169 110 165)(106 170 111 166)(107 171 112 167)(108 172 113 168)(115 162 125 151)(116 157 126 152)(117 158 121 153)(118 159 122 154)(119 160 123 155)(120 161 124 156)(139 186 149 175)(140 181 150 176)(141 182 145 177)(142 183 146 178)(143 184 147 179)(144 185 148 180)
(1 47 23 32)(2 48 24 33)(3 43 19 34)(4 44 20 35)(5 45 21 36)(6 46 22 31)(7 51 27 41)(8 52 28 42)(9 53 29 37)(10 54 30 38)(11 49 25 39)(12 50 26 40)(13 166 183 154)(14 167 184 155)(15 168 185 156)(16 163 186 151)(17 164 181 152)(18 165 182 153)(55 94 70 79)(56 95 71 80)(57 96 72 81)(58 91 67 82)(59 92 68 83)(60 93 69 84)(61 100 77 85)(62 101 78 86)(63 102 73 87)(64 97 74 88)(65 98 75 89)(66 99 76 90)(103 149 115 138)(104 150 116 133)(105 145 117 134)(106 146 118 135)(107 147 119 136)(108 148 120 137)(109 140 126 128)(110 141 121 129)(111 142 122 130)(112 143 123 131)(113 144 124 132)(114 139 125 127)(157 188 174 176)(158 189 169 177)(159 190 170 178)(160 191 171 179)(161 192 172 180)(162 187 173 175)
(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)
(1 106 4 103)(2 105 5 108)(3 104 6 107)(7 112 10 109)(8 111 11 114)(9 110 12 113)(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)(31 131 34 128)(32 130 35 127)(33 129 36 132)(37 134 40 137)(38 133 41 136)(39 138 42 135)(43 140 46 143)(44 139 47 142)(45 144 48 141)(49 149 52 146)(50 148 53 145)(51 147 54 150)(55 155 58 152)(56 154 59 151)(57 153 60 156)(61 158 64 161)(62 157 65 160)(63 162 66 159)(67 164 70 167)(68 163 71 166)(69 168 72 165)(73 173 76 170)(74 172 77 169)(75 171 78 174)(79 179 82 176)(80 178 83 175)(81 177 84 180)(85 182 88 185)(86 181 89 184)(87 186 90 183)(91 188 94 191)(92 187 95 190)(93 192 96 189)
G:=sub<Sym(192)| (1,66,8,56)(2,61,9,57)(3,62,10,58)(4,63,11,59)(5,64,12,60)(6,65,7,55)(13,135,190,130)(14,136,191,131)(15,137,192,132)(16,138,187,127)(17,133,188,128)(18,134,189,129)(19,78,30,67)(20,73,25,68)(21,74,26,69)(22,75,27,70)(23,76,28,71)(24,77,29,72)(31,89,41,79)(32,90,42,80)(33,85,37,81)(34,86,38,82)(35,87,39,83)(36,88,40,84)(43,101,54,91)(44,102,49,92)(45,97,50,93)(46,98,51,94)(47,99,52,95)(48,100,53,96)(103,173,114,163)(104,174,109,164)(105,169,110,165)(106,170,111,166)(107,171,112,167)(108,172,113,168)(115,162,125,151)(116,157,126,152)(117,158,121,153)(118,159,122,154)(119,160,123,155)(120,161,124,156)(139,186,149,175)(140,181,150,176)(141,182,145,177)(142,183,146,178)(143,184,147,179)(144,185,148,180), (1,47,23,32)(2,48,24,33)(3,43,19,34)(4,44,20,35)(5,45,21,36)(6,46,22,31)(7,51,27,41)(8,52,28,42)(9,53,29,37)(10,54,30,38)(11,49,25,39)(12,50,26,40)(13,166,183,154)(14,167,184,155)(15,168,185,156)(16,163,186,151)(17,164,181,152)(18,165,182,153)(55,94,70,79)(56,95,71,80)(57,96,72,81)(58,91,67,82)(59,92,68,83)(60,93,69,84)(61,100,77,85)(62,101,78,86)(63,102,73,87)(64,97,74,88)(65,98,75,89)(66,99,76,90)(103,149,115,138)(104,150,116,133)(105,145,117,134)(106,146,118,135)(107,147,119,136)(108,148,120,137)(109,140,126,128)(110,141,121,129)(111,142,122,130)(112,143,123,131)(113,144,124,132)(114,139,125,127)(157,188,174,176)(158,189,169,177)(159,190,170,178)(160,191,171,179)(161,192,172,180)(162,187,173,175), (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), (1,106,4,103)(2,105,5,108)(3,104,6,107)(7,112,10,109)(8,111,11,114)(9,110,12,113)(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)(31,131,34,128)(32,130,35,127)(33,129,36,132)(37,134,40,137)(38,133,41,136)(39,138,42,135)(43,140,46,143)(44,139,47,142)(45,144,48,141)(49,149,52,146)(50,148,53,145)(51,147,54,150)(55,155,58,152)(56,154,59,151)(57,153,60,156)(61,158,64,161)(62,157,65,160)(63,162,66,159)(67,164,70,167)(68,163,71,166)(69,168,72,165)(73,173,76,170)(74,172,77,169)(75,171,78,174)(79,179,82,176)(80,178,83,175)(81,177,84,180)(85,182,88,185)(86,181,89,184)(87,186,90,183)(91,188,94,191)(92,187,95,190)(93,192,96,189)>;
G:=Group( (1,66,8,56)(2,61,9,57)(3,62,10,58)(4,63,11,59)(5,64,12,60)(6,65,7,55)(13,135,190,130)(14,136,191,131)(15,137,192,132)(16,138,187,127)(17,133,188,128)(18,134,189,129)(19,78,30,67)(20,73,25,68)(21,74,26,69)(22,75,27,70)(23,76,28,71)(24,77,29,72)(31,89,41,79)(32,90,42,80)(33,85,37,81)(34,86,38,82)(35,87,39,83)(36,88,40,84)(43,101,54,91)(44,102,49,92)(45,97,50,93)(46,98,51,94)(47,99,52,95)(48,100,53,96)(103,173,114,163)(104,174,109,164)(105,169,110,165)(106,170,111,166)(107,171,112,167)(108,172,113,168)(115,162,125,151)(116,157,126,152)(117,158,121,153)(118,159,122,154)(119,160,123,155)(120,161,124,156)(139,186,149,175)(140,181,150,176)(141,182,145,177)(142,183,146,178)(143,184,147,179)(144,185,148,180), (1,47,23,32)(2,48,24,33)(3,43,19,34)(4,44,20,35)(5,45,21,36)(6,46,22,31)(7,51,27,41)(8,52,28,42)(9,53,29,37)(10,54,30,38)(11,49,25,39)(12,50,26,40)(13,166,183,154)(14,167,184,155)(15,168,185,156)(16,163,186,151)(17,164,181,152)(18,165,182,153)(55,94,70,79)(56,95,71,80)(57,96,72,81)(58,91,67,82)(59,92,68,83)(60,93,69,84)(61,100,77,85)(62,101,78,86)(63,102,73,87)(64,97,74,88)(65,98,75,89)(66,99,76,90)(103,149,115,138)(104,150,116,133)(105,145,117,134)(106,146,118,135)(107,147,119,136)(108,148,120,137)(109,140,126,128)(110,141,121,129)(111,142,122,130)(112,143,123,131)(113,144,124,132)(114,139,125,127)(157,188,174,176)(158,189,169,177)(159,190,170,178)(160,191,171,179)(161,192,172,180)(162,187,173,175), (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), (1,106,4,103)(2,105,5,108)(3,104,6,107)(7,112,10,109)(8,111,11,114)(9,110,12,113)(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)(31,131,34,128)(32,130,35,127)(33,129,36,132)(37,134,40,137)(38,133,41,136)(39,138,42,135)(43,140,46,143)(44,139,47,142)(45,144,48,141)(49,149,52,146)(50,148,53,145)(51,147,54,150)(55,155,58,152)(56,154,59,151)(57,153,60,156)(61,158,64,161)(62,157,65,160)(63,162,66,159)(67,164,70,167)(68,163,71,166)(69,168,72,165)(73,173,76,170)(74,172,77,169)(75,171,78,174)(79,179,82,176)(80,178,83,175)(81,177,84,180)(85,182,88,185)(86,181,89,184)(87,186,90,183)(91,188,94,191)(92,187,95,190)(93,192,96,189) );
G=PermutationGroup([[(1,66,8,56),(2,61,9,57),(3,62,10,58),(4,63,11,59),(5,64,12,60),(6,65,7,55),(13,135,190,130),(14,136,191,131),(15,137,192,132),(16,138,187,127),(17,133,188,128),(18,134,189,129),(19,78,30,67),(20,73,25,68),(21,74,26,69),(22,75,27,70),(23,76,28,71),(24,77,29,72),(31,89,41,79),(32,90,42,80),(33,85,37,81),(34,86,38,82),(35,87,39,83),(36,88,40,84),(43,101,54,91),(44,102,49,92),(45,97,50,93),(46,98,51,94),(47,99,52,95),(48,100,53,96),(103,173,114,163),(104,174,109,164),(105,169,110,165),(106,170,111,166),(107,171,112,167),(108,172,113,168),(115,162,125,151),(116,157,126,152),(117,158,121,153),(118,159,122,154),(119,160,123,155),(120,161,124,156),(139,186,149,175),(140,181,150,176),(141,182,145,177),(142,183,146,178),(143,184,147,179),(144,185,148,180)], [(1,47,23,32),(2,48,24,33),(3,43,19,34),(4,44,20,35),(5,45,21,36),(6,46,22,31),(7,51,27,41),(8,52,28,42),(9,53,29,37),(10,54,30,38),(11,49,25,39),(12,50,26,40),(13,166,183,154),(14,167,184,155),(15,168,185,156),(16,163,186,151),(17,164,181,152),(18,165,182,153),(55,94,70,79),(56,95,71,80),(57,96,72,81),(58,91,67,82),(59,92,68,83),(60,93,69,84),(61,100,77,85),(62,101,78,86),(63,102,73,87),(64,97,74,88),(65,98,75,89),(66,99,76,90),(103,149,115,138),(104,150,116,133),(105,145,117,134),(106,146,118,135),(107,147,119,136),(108,148,120,137),(109,140,126,128),(110,141,121,129),(111,142,122,130),(112,143,123,131),(113,144,124,132),(114,139,125,127),(157,188,174,176),(158,189,169,177),(159,190,170,178),(160,191,171,179),(161,192,172,180),(162,187,173,175)], [(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)], [(1,106,4,103),(2,105,5,108),(3,104,6,107),(7,112,10,109),(8,111,11,114),(9,110,12,113),(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),(31,131,34,128),(32,130,35,127),(33,129,36,132),(37,134,40,137),(38,133,41,136),(39,138,42,135),(43,140,46,143),(44,139,47,142),(45,144,48,141),(49,149,52,146),(50,148,53,145),(51,147,54,150),(55,155,58,152),(56,154,59,151),(57,153,60,156),(61,158,64,161),(62,157,65,160),(63,162,66,159),(67,164,70,167),(68,163,71,166),(69,168,72,165),(73,173,76,170),(74,172,77,169),(75,171,78,174),(79,179,82,176),(80,178,83,175),(81,177,84,180),(85,182,88,185),(86,181,89,184),(87,186,90,183),(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 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | - | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D4 | Q8 | Dic3 | D6 | C4○D4 | Dic6 | D12 | C4○D12 |
| kernel | C42⋊11Dic3 | C6.C42 | C2×C4⋊Dic3 | C2×C4×C12 | C4×C12 | C2×C42 | C2×C12 | C2×C12 | C42 | C22×C4 | C2×C6 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 4 | 2 | 1 | 8 | 1 | 2 | 2 | 4 | 3 | 8 | 4 | 4 | 16 |
Matrix representation of C42⋊11Dic3 ►in GL5(𝔽13)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 7 | 3 |
| 0 | 0 | 0 | 5 | 6 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 11 | 9 | 0 | 0 |
| 0 | 4 | 2 | 0 | 0 |
| 0 | 0 | 0 | 7 | 3 |
| 0 | 0 | 0 | 5 | 6 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 8 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 6 | 10 |
| 0 | 0 | 0 | 3 | 7 |
G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,5,0,0,0,0,0,5,0,0,0,0,0,7,5,0,0,0,3,6],[12,0,0,0,0,0,11,4,0,0,0,9,2,0,0,0,0,0,7,5,0,0,0,3,6],[12,0,0,0,0,0,0,12,0,0,0,1,12,0,0,0,0,0,1,0,0,0,0,0,1],[8,0,0,0,0,0,12,1,0,0,0,0,1,0,0,0,0,0,6,3,0,0,0,10,7] >;
C42⋊11Dic3 in GAP, Magma, Sage, TeX
C_4^2\rtimes_{11}{\rm Dic}_3 % in TeX
G:=Group("C4^2:11Dic3"); // GroupNames label
G:=SmallGroup(192,495);
// by ID
G=gap.SmallGroup(192,495);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,477,120,422,184,6278]);
// 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,d*c*d^-1=c^-1>;
// generators/relations