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## G = A4×C35order 420 = 22·3·5·7

### Direct product of C35 and A4

Aliases: A4×C35, C22⋊C105, (C2×C10)⋊C21, (C2×C70)⋊1C3, (C2×C14)⋊1C15, SmallGroup(420,32)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — A4×C35
 Chief series C1 — C22 — C2×C14 — C2×C70 — A4×C35
 Lower central C22 — A4×C35
 Upper central C1 — C35

Generators and relations for A4×C35
G = < a,b,c,d | a35=b2=c2=d3=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=b >

Smallest permutation representation of A4×C35
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)
(1 103)(2 104)(3 105)(4 71)(5 72)(6 73)(7 74)(8 75)(9 76)(10 77)(11 78)(12 79)(13 80)(14 81)(15 82)(16 83)(17 84)(18 85)(19 86)(20 87)(21 88)(22 89)(23 90)(24 91)(25 92)(26 93)(27 94)(28 95)(29 96)(30 97)(31 98)(32 99)(33 100)(34 101)(35 102)(36 107)(37 108)(38 109)(39 110)(40 111)(41 112)(42 113)(43 114)(44 115)(45 116)(46 117)(47 118)(48 119)(49 120)(50 121)(51 122)(52 123)(53 124)(54 125)(55 126)(56 127)(57 128)(58 129)(59 130)(60 131)(61 132)(62 133)(63 134)(64 135)(65 136)(66 137)(67 138)(68 139)(69 140)(70 106)
(1 139)(2 140)(3 106)(4 107)(5 108)(6 109)(7 110)(8 111)(9 112)(10 113)(11 114)(12 115)(13 116)(14 117)(15 118)(16 119)(17 120)(18 121)(19 122)(20 123)(21 124)(22 125)(23 126)(24 127)(25 128)(26 129)(27 130)(28 131)(29 132)(30 133)(31 134)(32 135)(33 136)(34 137)(35 138)(36 71)(37 72)(38 73)(39 74)(40 75)(41 76)(42 77)(43 78)(44 79)(45 80)(46 81)(47 82)(48 83)(49 84)(50 85)(51 86)(52 87)(53 88)(54 89)(55 90)(56 91)(57 92)(58 93)(59 94)(60 95)(61 96)(62 97)(63 98)(64 99)(65 100)(66 101)(67 102)(68 103)(69 104)(70 105)
(36 71 107)(37 72 108)(38 73 109)(39 74 110)(40 75 111)(41 76 112)(42 77 113)(43 78 114)(44 79 115)(45 80 116)(46 81 117)(47 82 118)(48 83 119)(49 84 120)(50 85 121)(51 86 122)(52 87 123)(53 88 124)(54 89 125)(55 90 126)(56 91 127)(57 92 128)(58 93 129)(59 94 130)(60 95 131)(61 96 132)(62 97 133)(63 98 134)(64 99 135)(65 100 136)(66 101 137)(67 102 138)(68 103 139)(69 104 140)(70 105 106)

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

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

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

140 conjugacy classes

 class 1 2 3A 3B 5A 5B 5C 5D 7A ··· 7F 10A 10B 10C 10D 14A ··· 14F 15A ··· 15H 21A ··· 21L 35A ··· 35X 70A ··· 70X 105A ··· 105AV order 1 2 3 3 5 5 5 5 7 ··· 7 10 10 10 10 14 ··· 14 15 ··· 15 21 ··· 21 35 ··· 35 70 ··· 70 105 ··· 105 size 1 3 4 4 1 1 1 1 1 ··· 1 3 3 3 3 3 ··· 3 4 ··· 4 4 ··· 4 1 ··· 1 3 ··· 3 4 ··· 4

140 irreducible representations

 dim 1 1 1 1 1 1 1 1 3 3 3 3 type + + image C1 C3 C5 C7 C15 C21 C35 C105 A4 C5×A4 C7×A4 A4×C35 kernel A4×C35 C2×C70 C7×A4 C5×A4 C2×C14 C2×C10 A4 C22 C35 C7 C5 C1 # reps 1 2 4 6 8 12 24 48 1 4 6 24

Matrix representation of A4×C35 in GL3(𝔽211) generated by

 184 0 0 0 184 0 0 0 184
,
 0 0 1 210 210 210 1 0 0
,
 0 1 0 1 0 0 210 210 210
,
 1 0 0 0 0 1 210 210 210
G:=sub<GL(3,GF(211))| [184,0,0,0,184,0,0,0,184],[0,210,1,0,210,0,1,210,0],[0,1,210,1,0,210,0,0,210],[1,0,210,0,0,210,0,1,210] >;

A4×C35 in GAP, Magma, Sage, TeX

A_4\times C_{35}
% in TeX

G:=Group("A4xC35");
// GroupNames label

G:=SmallGroup(420,32);
// by ID

G=gap.SmallGroup(420,32);
# by ID

G:=PCGroup([5,-3,-5,-7,-2,2,4203,7879]);
// Polycyclic

G:=Group<a,b,c,d|a^35=b^2=c^2=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,d*c*d^-1=b>;
// generators/relations

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