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## G = C20×C7⋊C3order 420 = 22·3·5·7

### Direct product of C20 and C7⋊C3

Aliases: C20×C7⋊C3, C28⋊C15, C140⋊C3, C72C60, C358C12, C70.4C6, C14.2C30, C2.(C10×C7⋊C3), C10.2(C2×C7⋊C3), (C10×C7⋊C3).4C2, (C2×C7⋊C3).2C10, SmallGroup(420,4)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — C20×C7⋊C3
 Chief series C1 — C7 — C14 — C70 — C10×C7⋊C3 — C20×C7⋊C3
 Lower central C7 — C20×C7⋊C3
 Upper central C1 — C20

Generators and relations for C20×C7⋊C3
G = < a,b,c | a20=b7=c3=1, ab=ba, ac=ca, cbc-1=b4 >

Smallest permutation representation of C20×C7⋊C3
On 140 points
Generators in S140
(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)
(1 64 107 84 48 34 133)(2 65 108 85 49 35 134)(3 66 109 86 50 36 135)(4 67 110 87 51 37 136)(5 68 111 88 52 38 137)(6 69 112 89 53 39 138)(7 70 113 90 54 40 139)(8 71 114 91 55 21 140)(9 72 115 92 56 22 121)(10 73 116 93 57 23 122)(11 74 117 94 58 24 123)(12 75 118 95 59 25 124)(13 76 119 96 60 26 125)(14 77 120 97 41 27 126)(15 78 101 98 42 28 127)(16 79 102 99 43 29 128)(17 80 103 100 44 30 129)(18 61 104 81 45 31 130)(19 62 105 82 46 32 131)(20 63 106 83 47 33 132)
(21 91 140)(22 92 121)(23 93 122)(24 94 123)(25 95 124)(26 96 125)(27 97 126)(28 98 127)(29 99 128)(30 100 129)(31 81 130)(32 82 131)(33 83 132)(34 84 133)(35 85 134)(36 86 135)(37 87 136)(38 88 137)(39 89 138)(40 90 139)(41 77 120)(42 78 101)(43 79 102)(44 80 103)(45 61 104)(46 62 105)(47 63 106)(48 64 107)(49 65 108)(50 66 109)(51 67 110)(52 68 111)(53 69 112)(54 70 113)(55 71 114)(56 72 115)(57 73 116)(58 74 117)(59 75 118)(60 76 119)

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

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

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

100 conjugacy classes

 class 1 2 3A 3B 4A 4B 5A 5B 5C 5D 6A 6B 7A 7B 10A 10B 10C 10D 12A 12B 12C 12D 14A 14B 15A ··· 15H 20A ··· 20H 28A 28B 28C 28D 30A ··· 30H 35A ··· 35H 60A ··· 60P 70A ··· 70H 140A ··· 140P order 1 2 3 3 4 4 5 5 5 5 6 6 7 7 10 10 10 10 12 12 12 12 14 14 15 ··· 15 20 ··· 20 28 28 28 28 30 ··· 30 35 ··· 35 60 ··· 60 70 ··· 70 140 ··· 140 size 1 1 7 7 1 1 1 1 1 1 7 7 3 3 1 1 1 1 7 7 7 7 3 3 7 ··· 7 1 ··· 1 3 3 3 3 7 ··· 7 3 ··· 3 7 ··· 7 3 ··· 3 3 ··· 3

100 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 3 3 3 3 3 3 type + + image C1 C2 C3 C4 C5 C6 C10 C12 C15 C20 C30 C60 C7⋊C3 C2×C7⋊C3 C4×C7⋊C3 C5×C7⋊C3 C10×C7⋊C3 C20×C7⋊C3 kernel C20×C7⋊C3 C10×C7⋊C3 C140 C5×C7⋊C3 C4×C7⋊C3 C70 C2×C7⋊C3 C35 C28 C7⋊C3 C14 C7 C20 C10 C5 C4 C2 C1 # reps 1 1 2 2 4 2 4 4 8 8 8 16 2 2 4 8 8 16

Matrix representation of C20×C7⋊C3 in GL3(𝔽421) generated by

 408 0 0 0 408 0 0 0 408
,
 176 177 1 1 0 0 0 1 0
,
 1 0 0 244 420 420 0 1 0
G:=sub<GL(3,GF(421))| [408,0,0,0,408,0,0,0,408],[176,1,0,177,0,1,1,0,0],[1,244,0,0,420,1,0,420,0] >;

C20×C7⋊C3 in GAP, Magma, Sage, TeX

C_{20}\times C_7\rtimes C_3
% in TeX

G:=Group("C20xC7:C3");
// GroupNames label

G:=SmallGroup(420,4);
// by ID

G=gap.SmallGroup(420,4);
# by ID

G:=PCGroup([5,-2,-3,-5,-2,-7,150,1509]);
// Polycyclic

G:=Group<a,b,c|a^20=b^7=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^4>;
// generators/relations

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