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G = C7×Dic6order 168 = 23·3·7

Direct product of C7 and Dic6

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

Aliases: C7×Dic6, C214Q8, C84.5C2, C28.3S3, C12.1C14, C14.13D6, Dic3.C14, C42.18C22, C3⋊(C7×Q8), C4.(S3×C7), C2.3(S3×C14), C6.1(C2×C14), (C7×Dic3).2C2, SmallGroup(168,29)

Series: Derived Chief Lower central Upper central

C1C6 — C7×Dic6
C1C3C6C42C7×Dic3 — C7×Dic6
C3C6 — C7×Dic6
C1C14C28

Generators and relations for C7×Dic6
 G = < a,b,c | a7=b12=1, c2=b6, ab=ba, ac=ca, cbc-1=b-1 >

3C4
3C4
3Q8
3C28
3C28
3C7×Q8

Smallest permutation representation of C7×Dic6
Regular action on 168 points
Generators in S168
(1 161 68 87 81 19 151)(2 162 69 88 82 20 152)(3 163 70 89 83 21 153)(4 164 71 90 84 22 154)(5 165 72 91 73 23 155)(6 166 61 92 74 24 156)(7 167 62 93 75 13 145)(8 168 63 94 76 14 146)(9 157 64 95 77 15 147)(10 158 65 96 78 16 148)(11 159 66 85 79 17 149)(12 160 67 86 80 18 150)(25 99 136 45 54 126 113)(26 100 137 46 55 127 114)(27 101 138 47 56 128 115)(28 102 139 48 57 129 116)(29 103 140 37 58 130 117)(30 104 141 38 59 131 118)(31 105 142 39 60 132 119)(32 106 143 40 49 121 120)(33 107 144 41 50 122 109)(34 108 133 42 51 123 110)(35 97 134 43 52 124 111)(36 98 135 44 53 125 112)
(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 46 7 40)(2 45 8 39)(3 44 9 38)(4 43 10 37)(5 42 11 48)(6 41 12 47)(13 106 19 100)(14 105 20 99)(15 104 21 98)(16 103 22 97)(17 102 23 108)(18 101 24 107)(25 76 31 82)(26 75 32 81)(27 74 33 80)(28 73 34 79)(29 84 35 78)(30 83 36 77)(49 161 55 167)(50 160 56 166)(51 159 57 165)(52 158 58 164)(53 157 59 163)(54 168 60 162)(61 122 67 128)(62 121 68 127)(63 132 69 126)(64 131 70 125)(65 130 71 124)(66 129 72 123)(85 116 91 110)(86 115 92 109)(87 114 93 120)(88 113 94 119)(89 112 95 118)(90 111 96 117)(133 149 139 155)(134 148 140 154)(135 147 141 153)(136 146 142 152)(137 145 143 151)(138 156 144 150)

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

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

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

C7×Dic6 is a maximal subgroup of
C28.D6  Dic6⋊D7  C21⋊Q16  C7⋊Dic12  D28⋊S3  D21⋊Q8  D14.D6  S3×C7×Q8

63 conjugacy classes

class 1  2  3 4A4B4C 6 7A···7F12A12B14A···14F21A···21F28A···28F28G···28R42A···42F84A···84L
order12344467···7121214···1421···2128···2828···2842···4284···84
size11226621···1221···12···22···26···62···22···2

63 irreducible representations

dim11111122222222
type++++-+-
imageC1C2C2C7C14C14S3Q8D6Dic6S3×C7C7×Q8S3×C14C7×Dic6
kernelC7×Dic6C7×Dic3C84Dic6Dic3C12C28C21C14C7C4C3C2C1
# reps1216126111266612

Matrix representation of C7×Dic6 in GL4(𝔽337) generated by

79000
07900
00790
00079
,
033600
1100
001224
003336
,
29614600
1874100
00302240
0030135
G:=sub<GL(4,GF(337))| [79,0,0,0,0,79,0,0,0,0,79,0,0,0,0,79],[0,1,0,0,336,1,0,0,0,0,1,3,0,0,224,336],[296,187,0,0,146,41,0,0,0,0,302,301,0,0,240,35] >;

C7×Dic6 in GAP, Magma, Sage, TeX

C_7\times {\rm Dic}_6
% in TeX

G:=Group("C7xDic6");
// GroupNames label

G:=SmallGroup(168,29);
// by ID

G=gap.SmallGroup(168,29);
# by ID

G:=PCGroup([5,-2,-2,-7,-2,-3,140,301,146,2804]);
// Polycyclic

G:=Group<a,b,c|a^7=b^12=1,c^2=b^6,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C7×Dic6 in TeX

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