Copied to
clipboard

## G = C14×D12order 336 = 24·3·7

### Direct product of C14 and D12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C14×D12
 Chief series C1 — C3 — C6 — C42 — S3×C14 — S3×C2×C14 — C14×D12
 Lower central C3 — C6 — C14×D12
 Upper central C1 — C2×C14 — C2×C28

Generators and relations for C14×D12
G = < a,b,c | a14=b12=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 248 in 108 conjugacy classes, 54 normal (18 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, S3, C6, C6, C7, C2×C4, D4, C23, C12, D6, D6, C2×C6, C14, C14, C14, C2×D4, C21, D12, C2×C12, C22×S3, C28, C2×C14, C2×C14, S3×C7, C42, C42, C2×D12, C2×C28, C7×D4, C22×C14, C84, S3×C14, S3×C14, C2×C42, D4×C14, C7×D12, C2×C84, S3×C2×C14, C14×D12
Quotients: C1, C2, C22, S3, C7, D4, C23, D6, C14, C2×D4, D12, C22×S3, C2×C14, S3×C7, C2×D12, C7×D4, C22×C14, S3×C14, D4×C14, C7×D12, S3×C2×C14, C14×D12

Smallest permutation representation of C14×D12
On 168 points
Generators in S168
(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 50 92 114 31 78 142 160 112 62 130 20)(2 51 93 115 32 79 143 161 99 63 131 21)(3 52 94 116 33 80 144 162 100 64 132 22)(4 53 95 117 34 81 145 163 101 65 133 23)(5 54 96 118 35 82 146 164 102 66 134 24)(6 55 97 119 36 83 147 165 103 67 135 25)(7 56 98 120 37 84 148 166 104 68 136 26)(8 43 85 121 38 71 149 167 105 69 137 27)(9 44 86 122 39 72 150 168 106 70 138 28)(10 45 87 123 40 73 151 155 107 57 139 15)(11 46 88 124 41 74 152 156 108 58 140 16)(12 47 89 125 42 75 153 157 109 59 127 17)(13 48 90 126 29 76 154 158 110 60 128 18)(14 49 91 113 30 77 141 159 111 61 129 19)
(1 20)(2 21)(3 22)(4 23)(5 24)(6 25)(7 26)(8 27)(9 28)(10 15)(11 16)(12 17)(13 18)(14 19)(29 158)(30 159)(31 160)(32 161)(33 162)(34 163)(35 164)(36 165)(37 166)(38 167)(39 168)(40 155)(41 156)(42 157)(43 137)(44 138)(45 139)(46 140)(47 127)(48 128)(49 129)(50 130)(51 131)(52 132)(53 133)(54 134)(55 135)(56 136)(57 87)(58 88)(59 89)(60 90)(61 91)(62 92)(63 93)(64 94)(65 95)(66 96)(67 97)(68 98)(69 85)(70 86)(71 149)(72 150)(73 151)(74 152)(75 153)(76 154)(77 141)(78 142)(79 143)(80 144)(81 145)(82 146)(83 147)(84 148)(99 115)(100 116)(101 117)(102 118)(103 119)(104 120)(105 121)(106 122)(107 123)(108 124)(109 125)(110 126)(111 113)(112 114)

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

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

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

126 conjugacy classes

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

126 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C2 C7 C14 C14 C14 S3 D4 D6 D6 D12 S3×C7 C7×D4 S3×C14 S3×C14 C7×D12 kernel C14×D12 C7×D12 C2×C84 S3×C2×C14 C2×D12 D12 C2×C12 C22×S3 C2×C28 C42 C28 C2×C14 C14 C2×C4 C6 C4 C22 C2 # reps 1 4 1 2 6 24 6 12 1 2 2 1 4 6 12 12 6 24

Matrix representation of C14×D12 in GL5(𝔽337)

 336 0 0 0 0 0 42 0 0 0 0 0 42 0 0 0 0 0 52 0 0 0 0 0 52
,
 1 0 0 0 0 0 327 264 0 0 0 6 10 0 0 0 0 0 1 1 0 0 0 336 0
,
 336 0 0 0 0 0 10 185 0 0 0 331 327 0 0 0 0 0 1 1 0 0 0 0 336

G:=sub<GL(5,GF(337))| [336,0,0,0,0,0,42,0,0,0,0,0,42,0,0,0,0,0,52,0,0,0,0,0,52],[1,0,0,0,0,0,327,6,0,0,0,264,10,0,0,0,0,0,1,336,0,0,0,1,0],[336,0,0,0,0,0,10,331,0,0,0,185,327,0,0,0,0,0,1,0,0,0,0,1,336] >;

C14×D12 in GAP, Magma, Sage, TeX

C_{14}\times D_{12}
% in TeX

G:=Group("C14xD12");
// GroupNames label

G:=SmallGroup(336,186);
// by ID

G=gap.SmallGroup(336,186);
# by ID

G:=PCGroup([6,-2,-2,-2,-7,-2,-3,1082,266,8069]);
// Polycyclic

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

׿
×
𝔽