Copied to
clipboard

## G = D14.D6order 336 = 24·3·7

### 3rd non-split extension by D14 of D6 acting via D6/C6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C42 — D14.D6
 Chief series C1 — C7 — C21 — C42 — C6×D7 — C3⋊D28 — D14.D6
 Lower central C21 — C42 — D14.D6
 Upper central C1 — C2 — C4

Generators and relations for D14.D6
G = < a,b,c | a12=c2=1, b14=a6, bab-1=cac=a-1, cbc=a6b13 >

Subgroups: 500 in 80 conjugacy classes, 32 normal (22 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C6, C7, C2×C4, D4, Q8, Dic3, C12, C12, D6, C2×C6, D7, C14, C4○D4, C21, Dic6, C4×S3, D12, C3⋊D4, C2×C12, Dic7, C28, C28, D14, D14, C3×D7, D21, C42, C4○D12, C4×D7, C4×D7, D28, C7×Q8, C7×Dic3, C3×Dic7, C84, C6×D7, D42, Q82D7, D21⋊C4, C3⋊D28, C12×D7, C7×Dic6, D84, D14.D6
Quotients: C1, C2, C22, S3, C23, D6, D7, C4○D4, C22×S3, D14, C4○D12, C22×D7, S3×D7, Q82D7, C2×S3×D7, D14.D6

Smallest permutation representation of D14.D6
On 168 points
Generators in S168
```(1 158 100 134 80 30 15 144 86 120 66 44)(2 45 67 121 87 145 16 31 81 135 101 159)(3 160 102 136 82 32 17 146 88 122 68 46)(4 47 69 123 89 147 18 33 83 137 103 161)(5 162 104 138 84 34 19 148 90 124 70 48)(6 49 71 125 91 149 20 35 57 139 105 163)(7 164 106 140 58 36 21 150 92 126 72 50)(8 51 73 127 93 151 22 37 59 113 107 165)(9 166 108 114 60 38 23 152 94 128 74 52)(10 53 75 129 95 153 24 39 61 115 109 167)(11 168 110 116 62 40 25 154 96 130 76 54)(12 55 77 131 97 155 26 41 63 117 111 141)(13 142 112 118 64 42 27 156 98 132 78 56)(14 29 79 133 99 157 28 43 65 119 85 143)
(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 21)(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 12)(22 28)(23 27)(24 26)(29 165)(30 164)(31 163)(32 162)(33 161)(34 160)(35 159)(36 158)(37 157)(38 156)(39 155)(40 154)(41 153)(42 152)(43 151)(44 150)(45 149)(46 148)(47 147)(48 146)(49 145)(50 144)(51 143)(52 142)(53 141)(54 168)(55 167)(56 166)(57 101)(58 100)(59 99)(60 98)(61 97)(62 96)(63 95)(64 94)(65 93)(66 92)(67 91)(68 90)(69 89)(70 88)(71 87)(72 86)(73 85)(74 112)(75 111)(76 110)(77 109)(78 108)(79 107)(80 106)(81 105)(82 104)(83 103)(84 102)(113 133)(114 132)(115 131)(116 130)(117 129)(118 128)(119 127)(120 126)(121 125)(122 124)(134 140)(135 139)(136 138)```

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

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

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

45 conjugacy classes

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 4E 6A 6B 6C 7A 7B 7C 12A 12B 12C 12D 14A 14B 14C 21A 21B 21C 28A 28B 28C 28D ··· 28I 42A 42B 42C 84A ··· 84F order 1 2 2 2 2 3 4 4 4 4 4 6 6 6 7 7 7 12 12 12 12 14 14 14 21 21 21 28 28 28 28 ··· 28 42 42 42 84 ··· 84 size 1 1 14 42 42 2 2 6 6 7 7 2 14 14 2 2 2 2 2 14 14 2 2 2 4 4 4 4 4 4 12 ··· 12 4 4 4 4 ··· 4

45 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 S3 D6 D6 D6 D7 C4○D4 D14 D14 C4○D12 S3×D7 Q8⋊2D7 C2×S3×D7 D14.D6 kernel D14.D6 D21⋊C4 C3⋊D28 C12×D7 C7×Dic6 D84 C4×D7 Dic7 C28 D14 Dic6 C21 Dic3 C12 C7 C4 C3 C2 C1 # reps 1 2 2 1 1 1 1 1 1 1 3 2 6 3 4 3 3 3 6

Matrix representation of D14.D6 in GL4(𝔽337) generated by

 292 315 0 0 46 0 0 0 0 0 1 0 0 0 0 1
,
 257 109 0 0 269 80 0 0 0 0 0 109 0 0 34 303
,
 1 0 0 0 289 336 0 0 0 0 34 109 0 0 178 303
`G:=sub<GL(4,GF(337))| [292,46,0,0,315,0,0,0,0,0,1,0,0,0,0,1],[257,269,0,0,109,80,0,0,0,0,0,34,0,0,109,303],[1,289,0,0,0,336,0,0,0,0,34,178,0,0,109,303] >;`

D14.D6 in GAP, Magma, Sage, TeX

`D_{14}.D_6`
`% in TeX`

`G:=Group("D14.D6");`
`// GroupNames label`

`G:=SmallGroup(336,146);`
`// by ID`

`G=gap.SmallGroup(336,146);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-3,-7,121,55,116,50,490,10373]);`
`// Polycyclic`

`G:=Group<a,b,c|a^12=c^2=1,b^14=a^6,b*a*b^-1=c*a*c=a^-1,c*b*c=a^6*b^13>;`
`// generators/relations`

׿
×
𝔽