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## G = C42.C23order 336 = 24·3·7

### 17th non-split extension by C42 of C23 acting via C23/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C42 — C42.C23
 Chief series C1 — C7 — C21 — C42 — C6×D7 — Dic3×D7 — C42.C23
 Lower central C21 — C42 — C42.C23
 Upper central C1 — C2 — C22

Generators and relations for C42.C23
G = < a,b,c,d | a6=b2=1, c14=d2=a3, bab=cac-1=dad-1=a-1, cbc-1=ab, dbd-1=a4b, dcd-1=c13 >

Subgroups: 396 in 80 conjugacy classes, 32 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C6, C7, C2×C4, D4, Q8, Dic3, Dic3, C12, D6, C2×C6, C2×C6, D7, C14, C14, C4○D4, C21, Dic6, C4×S3, C2×Dic3, C3⋊D4, C3⋊D4, C3×D4, Dic7, Dic7, C28, D14, C2×C14, C2×C14, S3×C7, C3×D7, C42, C42, D42S3, Dic14, C4×D7, C2×Dic7, C7⋊D4, C7⋊D4, C7×D4, C7×Dic3, C3×Dic7, Dic21, C6×D7, S3×C14, C2×C42, D42D7, Dic3×D7, S3×Dic7, C21⋊D4, C21⋊Q8, C3×C7⋊D4, C7×C3⋊D4, C2×Dic21, C42.C23
Quotients: C1, C2, C22, S3, C23, D6, D7, C4○D4, C22×S3, D14, D42S3, C22×D7, S3×D7, D42D7, C2×S3×D7, C42.C23

Smallest permutation representation of C42.C23
On 168 points
Generators in S168
```(1 72 37 15 58 51)(2 52 59 16 38 73)(3 74 39 17 60 53)(4 54 61 18 40 75)(5 76 41 19 62 55)(6 56 63 20 42 77)(7 78 43 21 64 29)(8 30 65 22 44 79)(9 80 45 23 66 31)(10 32 67 24 46 81)(11 82 47 25 68 33)(12 34 69 26 48 83)(13 84 49 27 70 35)(14 36 71 28 50 57)(85 115 143 99 129 157)(86 158 130 100 144 116)(87 117 145 101 131 159)(88 160 132 102 146 118)(89 119 147 103 133 161)(90 162 134 104 148 120)(91 121 149 105 135 163)(92 164 136 106 150 122)(93 123 151 107 137 165)(94 166 138 108 152 124)(95 125 153 109 139 167)(96 168 140 110 154 126)(97 127 155 111 113 141)(98 142 114 112 156 128)
(1 163)(2 136)(3 165)(4 138)(5 167)(6 140)(7 141)(8 114)(9 143)(10 116)(11 145)(12 118)(13 147)(14 120)(15 149)(16 122)(17 151)(18 124)(19 153)(20 126)(21 155)(22 128)(23 157)(24 130)(25 159)(26 132)(27 161)(28 134)(29 97)(30 142)(31 99)(32 144)(33 101)(34 146)(35 103)(36 148)(37 105)(38 150)(39 107)(40 152)(41 109)(42 154)(43 111)(44 156)(45 85)(46 158)(47 87)(48 160)(49 89)(50 162)(51 91)(52 164)(53 93)(54 166)(55 95)(56 168)(57 90)(58 121)(59 92)(60 123)(61 94)(62 125)(63 96)(64 127)(65 98)(66 129)(67 100)(68 131)(69 102)(70 133)(71 104)(72 135)(73 106)(74 137)(75 108)(76 139)(77 110)(78 113)(79 112)(80 115)(81 86)(82 117)(83 88)(84 119)
(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)
(1 14 15 28)(2 27 16 13)(3 12 17 26)(4 25 18 11)(5 10 19 24)(6 23 20 9)(7 8 21 22)(29 30 43 44)(31 56 45 42)(32 41 46 55)(33 54 47 40)(34 39 48 53)(35 52 49 38)(36 37 50 51)(57 58 71 72)(59 84 73 70)(60 69 74 83)(61 82 75 68)(62 67 76 81)(63 80 77 66)(64 65 78 79)(85 96 99 110)(86 109 100 95)(87 94 101 108)(88 107 102 93)(89 92 103 106)(90 105 104 91)(97 112 111 98)(113 128 127 114)(115 126 129 140)(116 139 130 125)(117 124 131 138)(118 137 132 123)(119 122 133 136)(120 135 134 121)(141 156 155 142)(143 154 157 168)(144 167 158 153)(145 152 159 166)(146 165 160 151)(147 150 161 164)(148 163 162 149)```

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 4E 6A 6B 6C 7A 7B 7C 12 14A 14B 14C 14D 14E 14F 14G 14H 14I 21A 21B 21C 28A 28B 28C 42A ··· 42I order 1 2 2 2 2 3 4 4 4 4 4 6 6 6 7 7 7 12 14 14 14 14 14 14 14 14 14 21 21 21 28 28 28 42 ··· 42 size 1 1 2 6 14 2 6 14 21 21 42 2 4 28 2 2 2 28 2 2 2 4 4 4 12 12 12 4 4 4 12 12 12 4 ··· 4

42 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + + + + + - + - + - image C1 C2 C2 C2 C2 C2 C2 C2 S3 D6 D6 D6 D7 C4○D4 D14 D14 D14 D4⋊2S3 S3×D7 D4⋊2D7 C2×S3×D7 C42.C23 kernel C42.C23 Dic3×D7 S3×Dic7 C21⋊D4 C21⋊Q8 C3×C7⋊D4 C7×C3⋊D4 C2×Dic21 C7⋊D4 Dic7 D14 C2×C14 C3⋊D4 C21 Dic3 D6 C2×C6 C7 C22 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 1 1 3 2 3 3 3 1 3 3 3 6

Matrix representation of C42.C23 in GL6(𝔽337)

 1 290 0 0 0 0 208 335 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 336 0 0 0 0 0 0 336
,
 198 39 0 0 0 0 49 139 0 0 0 0 0 0 336 0 0 0 0 0 0 336 0 0 0 0 0 0 189 75 0 0 0 0 18 148
,
 254 246 0 0 0 0 309 83 0 0 0 0 0 0 35 193 0 0 0 0 288 192 0 0 0 0 0 0 189 0 0 0 0 0 18 148
,
 254 246 0 0 0 0 309 83 0 0 0 0 0 0 144 336 0 0 0 0 178 193 0 0 0 0 0 0 189 0 0 0 0 0 0 189

`G:=sub<GL(6,GF(337))| [1,208,0,0,0,0,290,335,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,336,0,0,0,0,0,0,336],[198,49,0,0,0,0,39,139,0,0,0,0,0,0,336,0,0,0,0,0,0,336,0,0,0,0,0,0,189,18,0,0,0,0,75,148],[254,309,0,0,0,0,246,83,0,0,0,0,0,0,35,288,0,0,0,0,193,192,0,0,0,0,0,0,189,18,0,0,0,0,0,148],[254,309,0,0,0,0,246,83,0,0,0,0,0,0,144,178,0,0,0,0,336,193,0,0,0,0,0,0,189,0,0,0,0,0,0,189] >;`

C42.C23 in GAP, Magma, Sage, TeX

`C_{42}.C_2^3`
`% in TeX`

`G:=Group("C42.C2^3");`
`// GroupNames label`

`G:=SmallGroup(336,153);`
`// by ID`

`G=gap.SmallGroup(336,153);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-3,-7,55,218,116,490,10373]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^6=b^2=1,c^14=d^2=a^3,b*a*b=c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=a*b,d*b*d^-1=a^4*b,d*c*d^-1=c^13>;`
`// generators/relations`

׿
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