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## G = C2×C10×C7⋊C3order 420 = 22·3·5·7

### Direct product of C2×C10 and C7⋊C3

Aliases: C2×C10×C7⋊C3, C704C6, C142C30, (C2×C70)⋊3C3, C356(C2×C6), C72(C2×C30), (C2×C14)⋊3C15, SmallGroup(420,31)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — C2×C10×C7⋊C3
 Chief series C1 — C7 — C35 — C5×C7⋊C3 — C10×C7⋊C3 — C2×C10×C7⋊C3
 Lower central C7 — C2×C10×C7⋊C3
 Upper central C1 — C2×C10

Generators and relations for C2×C10×C7⋊C3
G = < a,b,c,d | a2=b10=c7=d3=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c4 >

Smallest permutation representation of C2×C10×C7⋊C3
On 140 points
Generators in S140
(1 35)(2 36)(3 37)(4 38)(5 39)(6 40)(7 31)(8 32)(9 33)(10 34)(11 101)(12 102)(13 103)(14 104)(15 105)(16 106)(17 107)(18 108)(19 109)(20 110)(21 79)(22 80)(23 71)(24 72)(25 73)(26 74)(27 75)(28 76)(29 77)(30 78)(41 85)(42 86)(43 87)(44 88)(45 89)(46 90)(47 81)(48 82)(49 83)(50 84)(51 129)(52 130)(53 121)(54 122)(55 123)(56 124)(57 125)(58 126)(59 127)(60 128)(61 116)(62 117)(63 118)(64 119)(65 120)(66 111)(67 112)(68 113)(69 114)(70 115)(91 137)(92 138)(93 139)(94 140)(95 131)(96 132)(97 133)(98 134)(99 135)(100 136)
(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)
(1 117 96 78 110 129 87)(2 118 97 79 101 130 88)(3 119 98 80 102 121 89)(4 120 99 71 103 122 90)(5 111 100 72 104 123 81)(6 112 91 73 105 124 82)(7 113 92 74 106 125 83)(8 114 93 75 107 126 84)(9 115 94 76 108 127 85)(10 116 95 77 109 128 86)(11 52 44 36 63 133 21)(12 53 45 37 64 134 22)(13 54 46 38 65 135 23)(14 55 47 39 66 136 24)(15 56 48 40 67 137 25)(16 57 49 31 68 138 26)(17 58 50 32 69 139 27)(18 59 41 33 70 140 28)(19 60 42 34 61 131 29)(20 51 43 35 62 132 30)
(11 63 133)(12 64 134)(13 65 135)(14 66 136)(15 67 137)(16 68 138)(17 69 139)(18 70 140)(19 61 131)(20 62 132)(21 44 52)(22 45 53)(23 46 54)(24 47 55)(25 48 56)(26 49 57)(27 50 58)(28 41 59)(29 42 60)(30 43 51)(71 90 122)(72 81 123)(73 82 124)(74 83 125)(75 84 126)(76 85 127)(77 86 128)(78 87 129)(79 88 130)(80 89 121)(91 105 112)(92 106 113)(93 107 114)(94 108 115)(95 109 116)(96 110 117)(97 101 118)(98 102 119)(99 103 120)(100 104 111)

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

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

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

100 conjugacy classes

 class 1 2A 2B 2C 3A 3B 5A 5B 5C 5D 6A ··· 6F 7A 7B 10A ··· 10L 14A ··· 14F 15A ··· 15H 30A ··· 30X 35A ··· 35H 70A ··· 70X order 1 2 2 2 3 3 5 5 5 5 6 ··· 6 7 7 10 ··· 10 14 ··· 14 15 ··· 15 30 ··· 30 35 ··· 35 70 ··· 70 size 1 1 1 1 7 7 1 1 1 1 7 ··· 7 3 3 1 ··· 1 3 ··· 3 7 ··· 7 7 ··· 7 3 ··· 3 3 ··· 3

100 irreducible representations

 dim 1 1 1 1 1 1 1 1 3 3 3 3 type + + image C1 C2 C3 C5 C6 C10 C15 C30 C7⋊C3 C2×C7⋊C3 C5×C7⋊C3 C10×C7⋊C3 kernel C2×C10×C7⋊C3 C10×C7⋊C3 C2×C70 C22×C7⋊C3 C70 C2×C7⋊C3 C2×C14 C14 C2×C10 C10 C22 C2 # reps 1 3 2 4 6 12 8 24 2 6 8 24

Matrix representation of C2×C10×C7⋊C3 in GL4(𝔽211) generated by

 1 0 0 0 0 210 0 0 0 0 210 0 0 0 0 210
,
 210 0 0 0 0 23 0 0 0 0 23 0 0 0 0 23
,
 1 0 0 0 0 210 190 1 0 0 190 1 0 210 191 1
,
 196 0 0 0 0 191 1 21 0 1 0 0 0 1 1 20
G:=sub<GL(4,GF(211))| [1,0,0,0,0,210,0,0,0,0,210,0,0,0,0,210],[210,0,0,0,0,23,0,0,0,0,23,0,0,0,0,23],[1,0,0,0,0,210,0,210,0,190,190,191,0,1,1,1],[196,0,0,0,0,191,1,1,0,1,0,1,0,21,0,20] >;

C2×C10×C7⋊C3 in GAP, Magma, Sage, TeX

C_2\times C_{10}\times C_7\rtimes C_3
% in TeX

G:=Group("C2xC10xC7:C3");
// GroupNames label

G:=SmallGroup(420,31);
// by ID

G=gap.SmallGroup(420,31);
# by ID

G:=PCGroup([5,-2,-2,-3,-5,-7,764]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^10=c^7=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^4>;
// generators/relations

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