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## G = C42.59D6order 192 = 26·3

### 59th non-split extension by C42 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C42.59D6
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C2×Dic6 — C2×C3⋊Q16 — C42.59D6
 Lower central C3 — C6 — C12 — C42.59D6
 Upper central C1 — C22 — C42 — C4×Q8

Generators and relations for C42.59D6
G = < a,b,c,d | a4=b4=1, c6=b2, d2=cbc-1=b-1, ab=ba, cac-1=dad-1=ab2, bd=db, dcd-1=b-1c5 >

Subgroups: 232 in 108 conjugacy classes, 51 normal (39 characteristic)
C1, C2 [×3], C3, C4 [×2], C4 [×8], C22, C6 [×3], C8 [×3], C2×C4 [×3], C2×C4 [×4], Q8 [×2], Q8 [×4], Dic3 [×3], C12 [×2], C12 [×5], C2×C6, C42, C42 [×2], C4⋊C4, C4⋊C4 [×3], C2×C8 [×2], Q16 [×4], C2×Q8, C2×Q8, C3⋊C8 [×2], C3⋊C8, Dic6 [×2], Dic6, C2×Dic3 [×2], C2×C12 [×3], C2×C12 [×2], C3×Q8 [×2], C3×Q8, C8⋊C4, Q8⋊C4 [×2], C4.Q8, C4×Q8, C4×Q8, C2×Q16, C2×C3⋊C8 [×2], C4×Dic3, Dic3⋊C4, C4⋊Dic3, C3⋊Q16 [×4], C4×C12, C4×C12, C3×C4⋊C4, C3×C4⋊C4, C2×Dic6, C6×Q8, Q16⋊C4, C42.S3, C12.Q8, C6.SD16, Q82Dic3, C4×Dic6, C2×C3⋊Q16, Q8×C12, C42.59D6
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×2], C23, D6 [×3], C22×C4, C2×D4, C4○D4, C4×S3 [×2], C3⋊D4 [×2], C22×S3, C4×D4, C8.C22 [×2], S3×C2×C4, C4○D12, C2×C3⋊D4, Q16⋊C4, C4×C3⋊D4, Q8.11D6, Q8.14D6, C42.59D6

Smallest permutation representation of C42.59D6
Regular action on 192 points
Generators in S192
```(1 16 111 62)(2 23 112 69)(3 18 113 64)(4 13 114 71)(5 20 115 66)(6 15 116 61)(7 22 117 68)(8 17 118 63)(9 24 119 70)(10 19 120 65)(11 14 109 72)(12 21 110 67)(25 82 182 172)(26 77 183 179)(27 84 184 174)(28 79 185 169)(29 74 186 176)(30 81 187 171)(31 76 188 178)(32 83 189 173)(33 78 190 180)(34 73 191 175)(35 80 192 170)(36 75 181 177)(37 161 156 53)(38 168 145 60)(39 163 146 55)(40 158 147 50)(41 165 148 57)(42 160 149 52)(43 167 150 59)(44 162 151 54)(45 157 152 49)(46 164 153 56)(47 159 154 51)(48 166 155 58)(85 143 101 125)(86 138 102 132)(87 133 103 127)(88 140 104 122)(89 135 105 129)(90 142 106 124)(91 137 107 131)(92 144 108 126)(93 139 97 121)(94 134 98 128)(95 141 99 123)(96 136 100 130)
(1 127 7 121)(2 122 8 128)(3 129 9 123)(4 124 10 130)(5 131 11 125)(6 126 12 132)(13 90 19 96)(14 85 20 91)(15 92 21 86)(16 87 22 93)(17 94 23 88)(18 89 24 95)(25 157 31 163)(26 164 32 158)(27 159 33 165)(28 166 34 160)(29 161 35 167)(30 168 36 162)(37 170 43 176)(38 177 44 171)(39 172 45 178)(40 179 46 173)(41 174 47 180)(42 169 48 175)(49 188 55 182)(50 183 56 189)(51 190 57 184)(52 185 58 191)(53 192 59 186)(54 187 60 181)(61 108 67 102)(62 103 68 97)(63 98 69 104)(64 105 70 99)(65 100 71 106)(66 107 72 101)(73 149 79 155)(74 156 80 150)(75 151 81 145)(76 146 82 152)(77 153 83 147)(78 148 84 154)(109 143 115 137)(110 138 116 144)(111 133 117 139)(112 140 118 134)(113 135 119 141)(114 142 120 136)
(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 47 121 174 7 41 127 180)(2 173 128 46 8 179 122 40)(3 45 123 172 9 39 129 178)(4 171 130 44 10 177 124 38)(5 43 125 170 11 37 131 176)(6 169 132 42 12 175 126 48)(13 36 96 168 19 30 90 162)(14 167 91 35 20 161 85 29)(15 34 86 166 21 28 92 160)(16 165 93 33 22 159 87 27)(17 32 88 164 23 26 94 158)(18 163 95 31 24 157 89 25)(49 105 182 64 55 99 188 70)(50 63 189 104 56 69 183 98)(51 103 184 62 57 97 190 68)(52 61 191 102 58 67 185 108)(53 101 186 72 59 107 192 66)(54 71 181 100 60 65 187 106)(73 144 155 116 79 138 149 110)(74 115 150 143 80 109 156 137)(75 142 145 114 81 136 151 120)(76 113 152 141 82 119 146 135)(77 140 147 112 83 134 153 118)(78 111 154 139 84 117 148 133)```

`G:=sub<Sym(192)| (1,16,111,62)(2,23,112,69)(3,18,113,64)(4,13,114,71)(5,20,115,66)(6,15,116,61)(7,22,117,68)(8,17,118,63)(9,24,119,70)(10,19,120,65)(11,14,109,72)(12,21,110,67)(25,82,182,172)(26,77,183,179)(27,84,184,174)(28,79,185,169)(29,74,186,176)(30,81,187,171)(31,76,188,178)(32,83,189,173)(33,78,190,180)(34,73,191,175)(35,80,192,170)(36,75,181,177)(37,161,156,53)(38,168,145,60)(39,163,146,55)(40,158,147,50)(41,165,148,57)(42,160,149,52)(43,167,150,59)(44,162,151,54)(45,157,152,49)(46,164,153,56)(47,159,154,51)(48,166,155,58)(85,143,101,125)(86,138,102,132)(87,133,103,127)(88,140,104,122)(89,135,105,129)(90,142,106,124)(91,137,107,131)(92,144,108,126)(93,139,97,121)(94,134,98,128)(95,141,99,123)(96,136,100,130), (1,127,7,121)(2,122,8,128)(3,129,9,123)(4,124,10,130)(5,131,11,125)(6,126,12,132)(13,90,19,96)(14,85,20,91)(15,92,21,86)(16,87,22,93)(17,94,23,88)(18,89,24,95)(25,157,31,163)(26,164,32,158)(27,159,33,165)(28,166,34,160)(29,161,35,167)(30,168,36,162)(37,170,43,176)(38,177,44,171)(39,172,45,178)(40,179,46,173)(41,174,47,180)(42,169,48,175)(49,188,55,182)(50,183,56,189)(51,190,57,184)(52,185,58,191)(53,192,59,186)(54,187,60,181)(61,108,67,102)(62,103,68,97)(63,98,69,104)(64,105,70,99)(65,100,71,106)(66,107,72,101)(73,149,79,155)(74,156,80,150)(75,151,81,145)(76,146,82,152)(77,153,83,147)(78,148,84,154)(109,143,115,137)(110,138,116,144)(111,133,117,139)(112,140,118,134)(113,135,119,141)(114,142,120,136), (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), (1,47,121,174,7,41,127,180)(2,173,128,46,8,179,122,40)(3,45,123,172,9,39,129,178)(4,171,130,44,10,177,124,38)(5,43,125,170,11,37,131,176)(6,169,132,42,12,175,126,48)(13,36,96,168,19,30,90,162)(14,167,91,35,20,161,85,29)(15,34,86,166,21,28,92,160)(16,165,93,33,22,159,87,27)(17,32,88,164,23,26,94,158)(18,163,95,31,24,157,89,25)(49,105,182,64,55,99,188,70)(50,63,189,104,56,69,183,98)(51,103,184,62,57,97,190,68)(52,61,191,102,58,67,185,108)(53,101,186,72,59,107,192,66)(54,71,181,100,60,65,187,106)(73,144,155,116,79,138,149,110)(74,115,150,143,80,109,156,137)(75,142,145,114,81,136,151,120)(76,113,152,141,82,119,146,135)(77,140,147,112,83,134,153,118)(78,111,154,139,84,117,148,133)>;`

`G:=Group( (1,16,111,62)(2,23,112,69)(3,18,113,64)(4,13,114,71)(5,20,115,66)(6,15,116,61)(7,22,117,68)(8,17,118,63)(9,24,119,70)(10,19,120,65)(11,14,109,72)(12,21,110,67)(25,82,182,172)(26,77,183,179)(27,84,184,174)(28,79,185,169)(29,74,186,176)(30,81,187,171)(31,76,188,178)(32,83,189,173)(33,78,190,180)(34,73,191,175)(35,80,192,170)(36,75,181,177)(37,161,156,53)(38,168,145,60)(39,163,146,55)(40,158,147,50)(41,165,148,57)(42,160,149,52)(43,167,150,59)(44,162,151,54)(45,157,152,49)(46,164,153,56)(47,159,154,51)(48,166,155,58)(85,143,101,125)(86,138,102,132)(87,133,103,127)(88,140,104,122)(89,135,105,129)(90,142,106,124)(91,137,107,131)(92,144,108,126)(93,139,97,121)(94,134,98,128)(95,141,99,123)(96,136,100,130), (1,127,7,121)(2,122,8,128)(3,129,9,123)(4,124,10,130)(5,131,11,125)(6,126,12,132)(13,90,19,96)(14,85,20,91)(15,92,21,86)(16,87,22,93)(17,94,23,88)(18,89,24,95)(25,157,31,163)(26,164,32,158)(27,159,33,165)(28,166,34,160)(29,161,35,167)(30,168,36,162)(37,170,43,176)(38,177,44,171)(39,172,45,178)(40,179,46,173)(41,174,47,180)(42,169,48,175)(49,188,55,182)(50,183,56,189)(51,190,57,184)(52,185,58,191)(53,192,59,186)(54,187,60,181)(61,108,67,102)(62,103,68,97)(63,98,69,104)(64,105,70,99)(65,100,71,106)(66,107,72,101)(73,149,79,155)(74,156,80,150)(75,151,81,145)(76,146,82,152)(77,153,83,147)(78,148,84,154)(109,143,115,137)(110,138,116,144)(111,133,117,139)(112,140,118,134)(113,135,119,141)(114,142,120,136), (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), (1,47,121,174,7,41,127,180)(2,173,128,46,8,179,122,40)(3,45,123,172,9,39,129,178)(4,171,130,44,10,177,124,38)(5,43,125,170,11,37,131,176)(6,169,132,42,12,175,126,48)(13,36,96,168,19,30,90,162)(14,167,91,35,20,161,85,29)(15,34,86,166,21,28,92,160)(16,165,93,33,22,159,87,27)(17,32,88,164,23,26,94,158)(18,163,95,31,24,157,89,25)(49,105,182,64,55,99,188,70)(50,63,189,104,56,69,183,98)(51,103,184,62,57,97,190,68)(52,61,191,102,58,67,185,108)(53,101,186,72,59,107,192,66)(54,71,181,100,60,65,187,106)(73,144,155,116,79,138,149,110)(74,115,150,143,80,109,156,137)(75,142,145,114,81,136,151,120)(76,113,152,141,82,119,146,135)(77,140,147,112,83,134,153,118)(78,111,154,139,84,117,148,133) );`

`G=PermutationGroup([(1,16,111,62),(2,23,112,69),(3,18,113,64),(4,13,114,71),(5,20,115,66),(6,15,116,61),(7,22,117,68),(8,17,118,63),(9,24,119,70),(10,19,120,65),(11,14,109,72),(12,21,110,67),(25,82,182,172),(26,77,183,179),(27,84,184,174),(28,79,185,169),(29,74,186,176),(30,81,187,171),(31,76,188,178),(32,83,189,173),(33,78,190,180),(34,73,191,175),(35,80,192,170),(36,75,181,177),(37,161,156,53),(38,168,145,60),(39,163,146,55),(40,158,147,50),(41,165,148,57),(42,160,149,52),(43,167,150,59),(44,162,151,54),(45,157,152,49),(46,164,153,56),(47,159,154,51),(48,166,155,58),(85,143,101,125),(86,138,102,132),(87,133,103,127),(88,140,104,122),(89,135,105,129),(90,142,106,124),(91,137,107,131),(92,144,108,126),(93,139,97,121),(94,134,98,128),(95,141,99,123),(96,136,100,130)], [(1,127,7,121),(2,122,8,128),(3,129,9,123),(4,124,10,130),(5,131,11,125),(6,126,12,132),(13,90,19,96),(14,85,20,91),(15,92,21,86),(16,87,22,93),(17,94,23,88),(18,89,24,95),(25,157,31,163),(26,164,32,158),(27,159,33,165),(28,166,34,160),(29,161,35,167),(30,168,36,162),(37,170,43,176),(38,177,44,171),(39,172,45,178),(40,179,46,173),(41,174,47,180),(42,169,48,175),(49,188,55,182),(50,183,56,189),(51,190,57,184),(52,185,58,191),(53,192,59,186),(54,187,60,181),(61,108,67,102),(62,103,68,97),(63,98,69,104),(64,105,70,99),(65,100,71,106),(66,107,72,101),(73,149,79,155),(74,156,80,150),(75,151,81,145),(76,146,82,152),(77,153,83,147),(78,148,84,154),(109,143,115,137),(110,138,116,144),(111,133,117,139),(112,140,118,134),(113,135,119,141),(114,142,120,136)], [(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)], [(1,47,121,174,7,41,127,180),(2,173,128,46,8,179,122,40),(3,45,123,172,9,39,129,178),(4,171,130,44,10,177,124,38),(5,43,125,170,11,37,131,176),(6,169,132,42,12,175,126,48),(13,36,96,168,19,30,90,162),(14,167,91,35,20,161,85,29),(15,34,86,166,21,28,92,160),(16,165,93,33,22,159,87,27),(17,32,88,164,23,26,94,158),(18,163,95,31,24,157,89,25),(49,105,182,64,55,99,188,70),(50,63,189,104,56,69,183,98),(51,103,184,62,57,97,190,68),(52,61,191,102,58,67,185,108),(53,101,186,72,59,107,192,66),(54,71,181,100,60,65,187,106),(73,144,155,116,79,138,149,110),(74,115,150,143,80,109,156,137),(75,142,145,114,81,136,151,120),(76,113,152,141,82,119,146,135),(77,140,147,112,83,134,153,118),(78,111,154,139,84,117,148,133)])`

42 conjugacy classes

 class 1 2A 2B 2C 3 4A ··· 4F 4G 4H 4I 4J 4K 4L 4M 4N 6A 6B 6C 8A 8B 8C 8D 12A 12B 12C 12D 12E ··· 12P order 1 2 2 2 3 4 ··· 4 4 4 4 4 4 4 4 4 6 6 6 8 8 8 8 12 12 12 12 12 ··· 12 size 1 1 1 1 2 2 ··· 2 4 4 4 4 12 12 12 12 2 2 2 12 12 12 12 2 2 2 2 4 ··· 4

42 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + - - image C1 C2 C2 C2 C2 C2 C2 C2 C4 S3 D4 D6 D6 D6 C4○D4 C3⋊D4 C4×S3 C4○D12 C8.C22 Q8.11D6 Q8.14D6 kernel C42.59D6 C42.S3 C12.Q8 C6.SD16 Q8⋊2Dic3 C4×Dic6 C2×C3⋊Q16 Q8×C12 C3⋊Q16 C4×Q8 C2×C12 C42 C4⋊C4 C2×Q8 C12 C2×C4 Q8 C4 C6 C2 C2 # reps 1 1 1 1 1 1 1 1 8 1 2 1 1 1 2 4 4 4 2 2 2

Matrix representation of C42.59D6 in GL6(𝔽73)

 27 0 0 0 0 0 0 27 0 0 0 0 0 0 43 13 60 13 0 0 60 30 0 60 0 0 17 17 43 60 0 0 0 17 13 30
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 1 0 25 25 0 0 0 1 23 25 0 0 72 1 72 0 0 0 71 72 0 72
,
 30 43 0 0 0 0 30 60 0 0 0 0 0 0 14 13 1 0 0 0 60 1 72 1 0 0 46 23 72 60 0 0 23 46 13 59
,
 63 51 0 0 0 0 61 10 0 0 0 0 0 0 51 46 56 36 0 0 68 22 36 54 0 0 8 19 7 69 0 0 19 11 3 66

`G:=sub<GL(6,GF(73))| [27,0,0,0,0,0,0,27,0,0,0,0,0,0,43,60,17,0,0,0,13,30,17,17,0,0,60,0,43,13,0,0,13,60,60,30],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,72,71,0,0,0,1,1,72,0,0,25,23,72,0,0,0,25,25,0,72],[30,30,0,0,0,0,43,60,0,0,0,0,0,0,14,60,46,23,0,0,13,1,23,46,0,0,1,72,72,13,0,0,0,1,60,59],[63,61,0,0,0,0,51,10,0,0,0,0,0,0,51,68,8,19,0,0,46,22,19,11,0,0,56,36,7,3,0,0,36,54,69,66] >;`

C42.59D6 in GAP, Magma, Sage, TeX

`C_4^2._{59}D_6`
`% in TeX`

`G:=Group("C4^2.59D6");`
`// GroupNames label`

`G:=SmallGroup(192,589);`
`// by ID`

`G=gap.SmallGroup(192,589);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,232,387,58,1684,851,102,6278]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^4=1,c^6=b^2,d^2=c*b*c^-1=b^-1,a*b=b*a,c*a*c^-1=d*a*d^-1=a*b^2,b*d=d*b,d*c*d^-1=b^-1*c^5>;`
`// generators/relations`

׿
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