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G = C3×C7⋊C8order 168 = 23·3·7

Direct product of C3 and C7⋊C8

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

Aliases: C3×C7⋊C8, C212C8, C73C24, C84.4C2, C42.2C4, C28.6C6, C12.4D7, C14.3C12, C6.2Dic7, C4.2(C3×D7), C2.(C3×Dic7), SmallGroup(168,4)

Series: Derived Chief Lower central Upper central

C1C7 — C3×C7⋊C8
C1C7C14C28C84 — C3×C7⋊C8
C7 — C3×C7⋊C8
C1C12

Generators and relations for C3×C7⋊C8
 G = < a,b,c | a3=b7=c8=1, ab=ba, ac=ca, cbc-1=b-1 >

7C8
7C24

Smallest permutation representation of C3×C7⋊C8
Regular action on 168 points
Generators in S168
(1 132 165)(2 133 166)(3 134 167)(4 135 168)(5 136 161)(6 129 162)(7 130 163)(8 131 164)(9 116 72)(10 117 65)(11 118 66)(12 119 67)(13 120 68)(14 113 69)(15 114 70)(16 115 71)(17 124 150)(18 125 151)(19 126 152)(20 127 145)(21 128 146)(22 121 147)(23 122 148)(24 123 149)(25 138 88)(26 139 81)(27 140 82)(28 141 83)(29 142 84)(30 143 85)(31 144 86)(32 137 87)(33 58 96)(34 59 89)(35 60 90)(36 61 91)(37 62 92)(38 63 93)(39 64 94)(40 57 95)(41 159 104)(42 160 97)(43 153 98)(44 154 99)(45 155 100)(46 156 101)(47 157 102)(48 158 103)(49 75 112)(50 76 105)(51 77 106)(52 78 107)(53 79 108)(54 80 109)(55 73 110)(56 74 111)
(1 25 36 71 152 157 76)(2 77 158 145 72 37 26)(3 27 38 65 146 159 78)(4 79 160 147 66 39 28)(5 29 40 67 148 153 80)(6 73 154 149 68 33 30)(7 31 34 69 150 155 74)(8 75 156 151 70 35 32)(9 62 139 133 106 103 20)(10 21 104 107 134 140 63)(11 64 141 135 108 97 22)(12 23 98 109 136 142 57)(13 58 143 129 110 99 24)(14 17 100 111 130 144 59)(15 60 137 131 112 101 18)(16 19 102 105 132 138 61)(41 52 167 82 93 117 128)(42 121 118 94 83 168 53)(43 54 161 84 95 119 122)(44 123 120 96 85 162 55)(45 56 163 86 89 113 124)(46 125 114 90 87 164 49)(47 50 165 88 91 115 126)(48 127 116 92 81 166 51)
(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)

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

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

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

C3×C7⋊C8 is a maximal subgroup of
D21⋊C8  D6.Dic7  D42.C4  C7⋊D24  D12.D7  Dic6⋊D7  C7⋊Dic12  D7×C24

60 conjugacy classes

class 1  2 3A3B4A4B6A6B7A7B7C8A8B8C8D12A12B12C12D14A14B14C21A···21F24A···24H28A···28F42A···42F84A···84L
order1233446677788881212121214141421···2124···2428···2842···4284···84
size11111111222777711112222···27···72···22···22···2

60 irreducible representations

dim11111111222222
type+++-
imageC1C2C3C4C6C8C12C24D7Dic7C3×D7C7⋊C8C3×Dic7C3×C7⋊C8
kernelC3×C7⋊C8C84C7⋊C8C42C28C21C14C7C12C6C4C3C2C1
# reps112224483366612

Matrix representation of C3×C7⋊C8 in GL2(𝔽13) generated by

30
03
,
47
93
,
05
10
G:=sub<GL(2,GF(13))| [3,0,0,3],[4,9,7,3],[0,1,5,0] >;

C3×C7⋊C8 in GAP, Magma, Sage, TeX

C_3\times C_7\rtimes C_8
% in TeX

G:=Group("C3xC7:C8");
// GroupNames label

G:=SmallGroup(168,4);
// by ID

G=gap.SmallGroup(168,4);
# by ID

G:=PCGroup([5,-2,-3,-2,-2,-7,30,42,3604]);
// Polycyclic

G:=Group<a,b,c|a^3=b^7=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C3×C7⋊C8 in TeX

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