Copied to
clipboard

G = C2.D84order 336 = 24·3·7

2nd central extension by C2 of D84

Series: Derived Chief Lower central Upper central

 Derived series C1 — C42 — C2.D84
 Chief series C1 — C7 — C21 — C42 — C2×C42 — C22×D21 — C2.D84
 Lower central C21 — C42 — C2.D84
 Upper central C1 — C22 — C2×C4

Generators and relations for C2.D84
G = < a,b,c | a2=b84=1, c2=a, ab=ba, ac=ca, cbc-1=ab-1 >

Subgroups: 512 in 68 conjugacy classes, 29 normal (27 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C7, C2×C4, C2×C4, C23, Dic3, C12, D6, C2×C6, D7, C14, C22⋊C4, C21, C2×Dic3, C2×C12, C22×S3, Dic7, C28, D14, C2×C14, D21, C42, D6⋊C4, C2×Dic7, C2×C28, C22×D7, Dic21, C84, D42, D42, C2×C42, D14⋊C4, C2×Dic21, C2×C84, C22×D21, C2.D84
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, D7, C22⋊C4, C4×S3, D12, C3⋊D4, D14, D21, D6⋊C4, C4×D7, D28, C7⋊D4, D42, D14⋊C4, C4×D21, D84, C217D4, C2.D84

Smallest permutation representation of C2.D84
On 168 points
Generators in S168
```(1 123)(2 124)(3 125)(4 126)(5 127)(6 128)(7 129)(8 130)(9 131)(10 132)(11 133)(12 134)(13 135)(14 136)(15 137)(16 138)(17 139)(18 140)(19 141)(20 142)(21 143)(22 144)(23 145)(24 146)(25 147)(26 148)(27 149)(28 150)(29 151)(30 152)(31 153)(32 154)(33 155)(34 156)(35 157)(36 158)(37 159)(38 160)(39 161)(40 162)(41 163)(42 164)(43 165)(44 166)(45 167)(46 168)(47 85)(48 86)(49 87)(50 88)(51 89)(52 90)(53 91)(54 92)(55 93)(56 94)(57 95)(58 96)(59 97)(60 98)(61 99)(62 100)(63 101)(64 102)(65 103)(66 104)(67 105)(68 106)(69 107)(70 108)(71 109)(72 110)(73 111)(74 112)(75 113)(76 114)(77 115)(78 116)(79 117)(80 118)(81 119)(82 120)(83 121)(84 122)
(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)
(1 122 123 84)(2 83 124 121)(3 120 125 82)(4 81 126 119)(5 118 127 80)(6 79 128 117)(7 116 129 78)(8 77 130 115)(9 114 131 76)(10 75 132 113)(11 112 133 74)(12 73 134 111)(13 110 135 72)(14 71 136 109)(15 108 137 70)(16 69 138 107)(17 106 139 68)(18 67 140 105)(19 104 141 66)(20 65 142 103)(21 102 143 64)(22 63 144 101)(23 100 145 62)(24 61 146 99)(25 98 147 60)(26 59 148 97)(27 96 149 58)(28 57 150 95)(29 94 151 56)(30 55 152 93)(31 92 153 54)(32 53 154 91)(33 90 155 52)(34 51 156 89)(35 88 157 50)(36 49 158 87)(37 86 159 48)(38 47 160 85)(39 168 161 46)(40 45 162 167)(41 166 163 44)(42 43 164 165)```

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

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

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

90 conjugacy classes

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

90 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C4 S3 D4 D6 D7 C4×S3 D12 C3⋊D4 D14 D21 C4×D7 D28 C7⋊D4 D42 C4×D21 D84 C21⋊7D4 kernel C2.D84 C2×Dic21 C2×C84 C22×D21 D42 C2×C28 C42 C2×C14 C2×C12 C14 C14 C14 C2×C6 C2×C4 C6 C6 C6 C22 C2 C2 C2 # reps 1 1 1 1 4 1 2 1 3 2 2 2 3 6 6 6 6 6 12 12 12

Matrix representation of C2.D84 in GL5(𝔽337)

 336 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 336 0 0 0 0 0 336
,
 148 0 0 0 0 0 177 16 0 0 0 321 307 0 0 0 0 0 203 286 0 0 0 89 54
,
 189 0 0 0 0 0 177 16 0 0 0 64 160 0 0 0 0 0 203 286 0 0 0 101 134

`G:=sub<GL(5,GF(337))| [336,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,336,0,0,0,0,0,336],[148,0,0,0,0,0,177,321,0,0,0,16,307,0,0,0,0,0,203,89,0,0,0,286,54],[189,0,0,0,0,0,177,64,0,0,0,16,160,0,0,0,0,0,203,101,0,0,0,286,134] >;`

C2.D84 in GAP, Magma, Sage, TeX

`C_2.D_{84}`
`% in TeX`

`G:=Group("C2.D84");`
`// GroupNames label`

`G:=SmallGroup(336,100);`
`// by ID`

`G=gap.SmallGroup(336,100);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-3,-7,121,31,964,10373]);`
`// Polycyclic`

`G:=Group<a,b,c|a^2=b^84=1,c^2=a,a*b=b*a,a*c=c*a,c*b*c^-1=a*b^-1>;`
`// generators/relations`

׿
×
𝔽