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G = C3×Dic14order 168 = 23·3·7

Direct product of C3 and Dic14

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

Aliases: C3×Dic14, C213Q8, C28.5C6, C84.3C2, C12.3D7, C6.13D14, Dic7.2C6, C42.13C22, C4.(C3×D7), C73(C3×Q8), C2.3(C6×D7), C14.9(C2×C6), (C3×Dic7).2C2, SmallGroup(168,24)

Series: Derived Chief Lower central Upper central

C1C14 — C3×Dic14
C1C7C14C42C3×Dic7 — C3×Dic14
C7C14 — C3×Dic14
C1C6C12

Generators and relations for C3×Dic14
 G = < a,b,c | a3=b28=1, c2=b14, ab=ba, ac=ca, cbc-1=b-1 >

7C4
7C4
7Q8
7C12
7C12
7C3×Q8

Smallest permutation representation of C3×Dic14
Regular action on 168 points
Generators in S168
(1 131 105)(2 132 106)(3 133 107)(4 134 108)(5 135 109)(6 136 110)(7 137 111)(8 138 112)(9 139 85)(10 140 86)(11 113 87)(12 114 88)(13 115 89)(14 116 90)(15 117 91)(16 118 92)(17 119 93)(18 120 94)(19 121 95)(20 122 96)(21 123 97)(22 124 98)(23 125 99)(24 126 100)(25 127 101)(26 128 102)(27 129 103)(28 130 104)(29 160 79)(30 161 80)(31 162 81)(32 163 82)(33 164 83)(34 165 84)(35 166 57)(36 167 58)(37 168 59)(38 141 60)(39 142 61)(40 143 62)(41 144 63)(42 145 64)(43 146 65)(44 147 66)(45 148 67)(46 149 68)(47 150 69)(48 151 70)(49 152 71)(50 153 72)(51 154 73)(52 155 74)(53 156 75)(54 157 76)(55 158 77)(56 159 78)
(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)
(1 151 15 165)(2 150 16 164)(3 149 17 163)(4 148 18 162)(5 147 19 161)(6 146 20 160)(7 145 21 159)(8 144 22 158)(9 143 23 157)(10 142 24 156)(11 141 25 155)(12 168 26 154)(13 167 27 153)(14 166 28 152)(29 110 43 96)(30 109 44 95)(31 108 45 94)(32 107 46 93)(33 106 47 92)(34 105 48 91)(35 104 49 90)(36 103 50 89)(37 102 51 88)(38 101 52 87)(39 100 53 86)(40 99 54 85)(41 98 55 112)(42 97 56 111)(57 130 71 116)(58 129 72 115)(59 128 73 114)(60 127 74 113)(61 126 75 140)(62 125 76 139)(63 124 77 138)(64 123 78 137)(65 122 79 136)(66 121 80 135)(67 120 81 134)(68 119 82 133)(69 118 83 132)(70 117 84 131)

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

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

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

C3×Dic14 is a maximal subgroup of
C42.D4  C21⋊SD16  C21⋊Q16  C3⋊Dic28  D12⋊D7  D84⋊C2  D21⋊Q8  C3×Q8×D7

51 conjugacy classes

class 1  2 3A3B4A4B4C6A6B7A7B7C12A12B12C12D12E12F14A14B14C21A···21F28A···28F42A···42F84A···84L
order12334446677712121212121214141421···2128···2842···4284···84
size1111214141122222141414142222···22···22···22···2

51 irreducible representations

dim11111122222222
type+++-++-
imageC1C2C2C3C6C6Q8D7C3×Q8D14C3×D7Dic14C6×D7C3×Dic14
kernelC3×Dic14C3×Dic7C84Dic14Dic7C28C21C12C7C6C4C3C2C1
# reps121242132366612

Matrix representation of C3×Dic14 in GL2(𝔽337) generated by

1280
0128
,
225243
94118
,
16149
200176
G:=sub<GL(2,GF(337))| [128,0,0,128],[225,94,243,118],[161,200,49,176] >;

C3×Dic14 in GAP, Magma, Sage, TeX

C_3\times {\rm Dic}_{14}
% in TeX

G:=Group("C3xDic14");
// GroupNames label

G:=SmallGroup(168,24);
// by ID

G=gap.SmallGroup(168,24);
# by ID

G:=PCGroup([5,-2,-2,-3,-2,-7,60,141,66,3604]);
// Polycyclic

G:=Group<a,b,c|a^3=b^28=1,c^2=b^14,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C3×Dic14 in TeX

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