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G = C5×C7⋊A4order 420 = 22·3·5·7

Direct product of C5 and C7⋊A4

Aliases: C5×C7⋊A4, C35⋊A4, C7⋊(C5×A4), (C2×C70)⋊2C3, (C2×C14)⋊2C15, C22⋊(C5×C7⋊C3), (C2×C10)⋊(C7⋊C3), SmallGroup(420,33)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — C5×C7⋊A4
 Chief series C1 — C7 — C2×C14 — C2×C70 — C5×C7⋊A4
 Lower central C2×C14 — C5×C7⋊A4
 Upper central C1 — C5

Generators and relations for C5×C7⋊A4
G = < a,b,c,d,e | a5=b7=c2=d2=e3=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=b4, ece-1=cd=dc, ede-1=c >

Smallest permutation representation of C5×C7⋊A4
On 140 points
Generators in S140
(1 113 85 57 29)(2 114 86 58 30)(3 115 87 59 31)(4 116 88 60 32)(5 117 89 61 33)(6 118 90 62 34)(7 119 91 63 35)(8 120 92 64 36)(9 121 93 65 37)(10 122 94 66 38)(11 123 95 67 39)(12 124 96 68 40)(13 125 97 69 41)(14 126 98 70 42)(15 127 99 71 43)(16 128 100 72 44)(17 129 101 73 45)(18 130 102 74 46)(19 131 103 75 47)(20 132 104 76 48)(21 133 105 77 49)(22 134 106 78 50)(23 135 107 79 51)(24 136 108 80 52)(25 137 109 81 53)(26 138 110 82 54)(27 139 111 83 55)(28 140 112 84 56)
(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 8)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(21 28)(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(43 50)(44 51)(45 52)(46 53)(47 54)(48 55)(49 56)(57 64)(58 65)(59 66)(60 67)(61 68)(62 69)(63 70)(71 78)(72 79)(73 80)(74 81)(75 82)(76 83)(77 84)(85 92)(86 93)(87 94)(88 95)(89 96)(90 97)(91 98)(99 106)(100 107)(101 108)(102 109)(103 110)(104 111)(105 112)(113 120)(114 121)(115 122)(116 123)(117 124)(118 125)(119 126)(127 134)(128 135)(129 136)(130 137)(131 138)(132 139)(133 140)
(1 22)(2 23)(3 24)(4 25)(5 26)(6 27)(7 28)(8 15)(9 16)(10 17)(11 18)(12 19)(13 20)(14 21)(29 50)(30 51)(31 52)(32 53)(33 54)(34 55)(35 56)(36 43)(37 44)(38 45)(39 46)(40 47)(41 48)(42 49)(57 78)(58 79)(59 80)(60 81)(61 82)(62 83)(63 84)(64 71)(65 72)(66 73)(67 74)(68 75)(69 76)(70 77)(85 106)(86 107)(87 108)(88 109)(89 110)(90 111)(91 112)(92 99)(93 100)(94 101)(95 102)(96 103)(97 104)(98 105)(113 134)(114 135)(115 136)(116 137)(117 138)(118 139)(119 140)(120 127)(121 128)(122 129)(123 130)(124 131)(125 132)(126 133)
(2 3 5)(4 7 6)(8 22 15)(9 24 19)(10 26 16)(11 28 20)(12 23 17)(13 25 21)(14 27 18)(30 31 33)(32 35 34)(36 50 43)(37 52 47)(38 54 44)(39 56 48)(40 51 45)(41 53 49)(42 55 46)(58 59 61)(60 63 62)(64 78 71)(65 80 75)(66 82 72)(67 84 76)(68 79 73)(69 81 77)(70 83 74)(86 87 89)(88 91 90)(92 106 99)(93 108 103)(94 110 100)(95 112 104)(96 107 101)(97 109 105)(98 111 102)(114 115 117)(116 119 118)(120 134 127)(121 136 131)(122 138 128)(123 140 132)(124 135 129)(125 137 133)(126 139 130)

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

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

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

60 conjugacy classes

 class 1 2 3A 3B 5A 5B 5C 5D 7A 7B 10A 10B 10C 10D 14A ··· 14F 15A ··· 15H 35A ··· 35H 70A ··· 70X order 1 2 3 3 5 5 5 5 7 7 10 10 10 10 14 ··· 14 15 ··· 15 35 ··· 35 70 ··· 70 size 1 3 28 28 1 1 1 1 3 3 3 3 3 3 3 ··· 3 28 ··· 28 3 ··· 3 3 ··· 3

60 irreducible representations

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

Matrix representation of C5×C7⋊A4 in GL4(𝔽211) generated by

 107 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 1 0 21 0 0 1 20
,
 1 0 0 0 0 84 131 187 0 99 92 49 0 131 187 34
,
 1 0 0 0 0 172 179 73 0 80 133 24 0 179 73 116
,
 14 0 0 0 0 1 0 20 0 0 0 210 0 0 1 210
G:=sub<GL(4,GF(211))| [107,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,1,0,1,21,20],[1,0,0,0,0,84,99,131,0,131,92,187,0,187,49,34],[1,0,0,0,0,172,80,179,0,179,133,73,0,73,24,116],[14,0,0,0,0,1,0,0,0,0,0,1,0,20,210,210] >;

C5×C7⋊A4 in GAP, Magma, Sage, TeX

C_5\times C_7\rtimes A_4
% in TeX

G:=Group("C5xC7:A4");
// GroupNames label

G:=SmallGroup(420,33);
// by ID

G=gap.SmallGroup(420,33);
# by ID

G:=PCGroup([5,-3,-5,-2,2,-7,452,903,3004]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^7=c^2=d^2=e^3=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e^-1=b^4,e*c*e^-1=c*d=d*c,e*d*e^-1=c>;
// generators/relations

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