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G = C7×C24⋊C2order 336 = 24·3·7

Direct product of C7 and C24⋊C2

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

Aliases: C7×C24⋊C2, C566S3, C242C14, C16810C2, C219SD16, C42.28D4, C28.52D6, Dic61C14, D12.1C14, C14.13D12, C84.68C22, C82(S3×C7), C6.1(C7×D4), C31(C7×SD16), C4.8(S3×C14), C2.3(C7×D12), C12.8(C2×C14), (C7×Dic6)⋊7C2, (C7×D12).3C2, SmallGroup(336,76)

Series: Derived Chief Lower central Upper central

C1C12 — C7×C24⋊C2
C1C3C6C12C84C7×D12 — C7×C24⋊C2
C3C6C12 — C7×C24⋊C2
C1C14C28C56

Generators and relations for C7×C24⋊C2
 G = < a,b,c | a7=b24=c2=1, ab=ba, ac=ca, cbc=b11 >

12C2
6C22
6C4
4S3
12C14
3Q8
3D4
2Dic3
2D6
6C28
6C2×C14
4S3×C7
3SD16
3C7×Q8
3C7×D4
2S3×C14
2C7×Dic3
3C7×SD16

Smallest permutation representation of C7×C24⋊C2
On 168 points
Generators in S168
(1 167 111 53 42 90 131)(2 168 112 54 43 91 132)(3 145 113 55 44 92 133)(4 146 114 56 45 93 134)(5 147 115 57 46 94 135)(6 148 116 58 47 95 136)(7 149 117 59 48 96 137)(8 150 118 60 25 73 138)(9 151 119 61 26 74 139)(10 152 120 62 27 75 140)(11 153 97 63 28 76 141)(12 154 98 64 29 77 142)(13 155 99 65 30 78 143)(14 156 100 66 31 79 144)(15 157 101 67 32 80 121)(16 158 102 68 33 81 122)(17 159 103 69 34 82 123)(18 160 104 70 35 83 124)(19 161 105 71 36 84 125)(20 162 106 72 37 85 126)(21 163 107 49 38 86 127)(22 164 108 50 39 87 128)(23 165 109 51 40 88 129)(24 166 110 52 41 89 130)
(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)
(2 12)(3 23)(4 10)(5 21)(6 8)(7 19)(9 17)(11 15)(14 24)(16 22)(18 20)(25 47)(26 34)(27 45)(28 32)(29 43)(31 41)(33 39)(35 37)(36 48)(38 46)(40 44)(49 57)(50 68)(51 55)(52 66)(54 64)(56 62)(58 60)(59 71)(61 69)(63 67)(70 72)(73 95)(74 82)(75 93)(76 80)(77 91)(79 89)(81 87)(83 85)(84 96)(86 94)(88 92)(97 101)(98 112)(100 110)(102 108)(103 119)(104 106)(105 117)(107 115)(109 113)(114 120)(116 118)(121 141)(122 128)(123 139)(124 126)(125 137)(127 135)(129 133)(130 144)(132 142)(134 140)(136 138)(145 165)(146 152)(147 163)(148 150)(149 161)(151 159)(153 157)(154 168)(156 166)(158 164)(160 162)

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

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

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

105 conjugacy classes

class 1 2A2B 3 4A4B 6 7A···7F8A8B12A12B14A···14F14G···14L21A···21F24A24B24C24D28A···28F28G···28L42A···42F56A···56L84A···84L168A···168X
order12234467···788121214···1414···1421···212424242428···2828···2842···4256···5684···84168···168
size1112221221···122221···112···122···222222···212···122···22···22···22···2

105 irreducible representations

dim11111111222222222222
type++++++++
imageC1C2C2C2C7C14C14C14S3D4D6SD16D12S3×C7C24⋊C2C7×D4S3×C14C7×SD16C7×D12C7×C24⋊C2
kernelC7×C24⋊C2C168C7×Dic6C7×D12C24⋊C2C24Dic6D12C56C42C28C21C14C8C7C6C4C3C2C1
# reps11116666111226466121224

Matrix representation of C7×C24⋊C2 in GL4(𝔽337) generated by

175000
017500
0080
0008
,
133600
1000
000294
0047141
,
133600
033600
0010
00224336
G:=sub<GL(4,GF(337))| [175,0,0,0,0,175,0,0,0,0,8,0,0,0,0,8],[1,1,0,0,336,0,0,0,0,0,0,47,0,0,294,141],[1,0,0,0,336,336,0,0,0,0,1,224,0,0,0,336] >;

C7×C24⋊C2 in GAP, Magma, Sage, TeX

C_7\times C_{24}\rtimes C_2
% in TeX

G:=Group("C7xC24:C2");
// GroupNames label

G:=SmallGroup(336,76);
// by ID

G=gap.SmallGroup(336,76);
# by ID

G:=PCGroup([6,-2,-2,-7,-2,-2,-3,361,175,2019,69,8069]);
// 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^11>;
// generators/relations

Export

Subgroup lattice of C7×C24⋊C2 in TeX

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