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G = C7×D24order 336 = 24·3·7

Direct product of C7 and D24

direct product, metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C7×D24, C216D8, C565S3, C241C14, C1687C2, D121C14, C28.53D6, C42.29D4, C14.14D12, C84.69C22, C31(C7×D8), C81(S3×C7), C6.2(C7×D4), (C7×D12)⋊7C2, C4.9(S3×C14), C2.4(C7×D12), C12.9(C2×C14), SmallGroup(336,77)

Series: Derived Chief Lower central Upper central

C1C12 — C7×D24
C1C3C6C12C84C7×D12 — C7×D24
C3C6C12 — C7×D24
C1C14C28C56

Generators and relations for C7×D24
 G = < a,b,c | a7=b24=c2=1, ab=ba, ac=ca, cbc=b-1 >

12C2
12C2
6C22
6C22
4S3
4S3
12C14
12C14
3D4
3D4
2D6
2D6
6C2×C14
6C2×C14
4S3×C7
4S3×C7
3D8
3C7×D4
3C7×D4
2S3×C14
2S3×C14
3C7×D8

Smallest permutation representation of C7×D24
On 168 points
Generators in S168
(1 152 37 106 83 61 141)(2 153 38 107 84 62 142)(3 154 39 108 85 63 143)(4 155 40 109 86 64 144)(5 156 41 110 87 65 121)(6 157 42 111 88 66 122)(7 158 43 112 89 67 123)(8 159 44 113 90 68 124)(9 160 45 114 91 69 125)(10 161 46 115 92 70 126)(11 162 47 116 93 71 127)(12 163 48 117 94 72 128)(13 164 25 118 95 49 129)(14 165 26 119 96 50 130)(15 166 27 120 73 51 131)(16 167 28 97 74 52 132)(17 168 29 98 75 53 133)(18 145 30 99 76 54 134)(19 146 31 100 77 55 135)(20 147 32 101 78 56 136)(21 148 33 102 79 57 137)(22 149 34 103 80 58 138)(23 150 35 104 81 59 139)(24 151 36 105 82 60 140)
(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 24)(2 23)(3 22)(4 21)(5 20)(6 19)(7 18)(8 17)(9 16)(10 15)(11 14)(12 13)(25 48)(26 47)(27 46)(28 45)(29 44)(30 43)(31 42)(32 41)(33 40)(34 39)(35 38)(36 37)(49 72)(50 71)(51 70)(52 69)(53 68)(54 67)(55 66)(56 65)(57 64)(58 63)(59 62)(60 61)(73 92)(74 91)(75 90)(76 89)(77 88)(78 87)(79 86)(80 85)(81 84)(82 83)(93 96)(94 95)(97 114)(98 113)(99 112)(100 111)(101 110)(102 109)(103 108)(104 107)(105 106)(115 120)(116 119)(117 118)(121 136)(122 135)(123 134)(124 133)(125 132)(126 131)(127 130)(128 129)(137 144)(138 143)(139 142)(140 141)(145 158)(146 157)(147 156)(148 155)(149 154)(150 153)(151 152)(159 168)(160 167)(161 166)(162 165)(163 164)

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

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

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

105 conjugacy classes

class 1 2A2B2C 3  4  6 7A···7F8A8B12A12B14A···14F14G···14R21A···21F24A24B24C24D28A···28F42A···42F56A···56L84A···84L168A···168X
order12223467···788121214···1414···1421···212424242428···2842···4256···5684···84168···168
size1112122221···122221···112···122···222222···22···22···22···22···2

105 irreducible representations

dim111111222222222222
type+++++++++
imageC1C2C2C7C14C14S3D4D6D8D12S3×C7D24C7×D4S3×C14C7×D8C7×D12C7×D24
kernelC7×D24C168C7×D12D24C24D12C56C42C28C21C14C8C7C6C4C3C2C1
# reps1126612111226466121224

Matrix representation of C7×D24 in GL2(𝔽337) generated by

2950
0295
,
284129
208155
,
208155
284129
G:=sub<GL(2,GF(337))| [295,0,0,295],[284,208,129,155],[208,284,155,129] >;

C7×D24 in GAP, Magma, Sage, TeX

C_7\times D_{24}
% in TeX

G:=Group("C7xD24");
// GroupNames label

G:=SmallGroup(336,77);
// by ID

G=gap.SmallGroup(336,77);
# by ID

G:=PCGroup([6,-2,-2,-7,-2,-2,-3,361,511,2019,69,8069]);
// Polycyclic

G:=Group<a,b,c|a^7=b^24=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C7×D24 in TeX

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