Copied to
clipboard

## G = C7×C3⋊C8order 168 = 23·3·7

### Direct product of C7 and C3⋊C8

Aliases: C7×C3⋊C8, C3⋊C56, C213C8, C6.C28, C84.6C2, C42.3C4, C28.4S3, C12.2C14, C14.2Dic3, C4.2(S3×C7), C2.(C7×Dic3), SmallGroup(168,3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C7×C3⋊C8
 Chief series C1 — C3 — C6 — C12 — C84 — C7×C3⋊C8
 Lower central C3 — C7×C3⋊C8
 Upper central C1 — C28

Generators and relations for C7×C3⋊C8
G = < a,b,c | a7=b3=c8=1, ab=ba, ac=ca, cbc-1=b-1 >

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

C7×C3⋊C8 is a maximal subgroup of
D21⋊C8  C28.32D6  D42.C4  C3⋊D56  C6.D28  C21⋊SD16  C3⋊Dic28  S3×C56

84 conjugacy classes

 class 1 2 3 4A 4B 6 7A ··· 7F 8A 8B 8C 8D 12A 12B 14A ··· 14F 21A ··· 21F 28A ··· 28L 42A ··· 42F 56A ··· 56X 84A ··· 84L order 1 2 3 4 4 6 7 ··· 7 8 8 8 8 12 12 14 ··· 14 21 ··· 21 28 ··· 28 42 ··· 42 56 ··· 56 84 ··· 84 size 1 1 2 1 1 2 1 ··· 1 3 3 3 3 2 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 3 ··· 3 2 ··· 2

84 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + - image C1 C2 C4 C7 C8 C14 C28 C56 S3 Dic3 C3⋊C8 S3×C7 C7×Dic3 C7×C3⋊C8 kernel C7×C3⋊C8 C84 C42 C3⋊C8 C21 C12 C6 C3 C28 C14 C7 C4 C2 C1 # reps 1 1 2 6 4 6 12 24 1 1 2 6 6 12

Matrix representation of C7×C3⋊C8 in GL2(𝔽29) generated by

 16 0 0 16
,
 14 2 25 14
,
 0 8 16 0
G:=sub<GL(2,GF(29))| [16,0,0,16],[14,25,2,14],[0,16,8,0] >;

C7×C3⋊C8 in GAP, Magma, Sage, TeX

C_7\times C_3\rtimes C_8
% in TeX

G:=Group("C7xC3:C8");
// GroupNames label

G:=SmallGroup(168,3);
// by ID

G=gap.SmallGroup(168,3);
# by ID

G:=PCGroup([5,-2,-7,-2,-2,-3,70,42,2804]);
// Polycyclic

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

Export

׿
×
𝔽