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## G = C5×D28order 280 = 23·5·7

### Direct product of C5 and D28

Aliases: C5×D28, C355D4, C203D7, C281C10, C1403C2, D141C10, C10.15D14, C70.15C22, C4⋊(C5×D7), C71(C5×D4), (C10×D7)⋊4C2, C2.4(C10×D7), C14.3(C2×C10), SmallGroup(280,16)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — C5×D28
 Chief series C1 — C7 — C14 — C70 — C10×D7 — C5×D28
 Lower central C7 — C14 — C5×D28
 Upper central C1 — C10 — C20

Generators and relations for C5×D28
G = < a,b,c | a5=b28=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of C5×D28
On 140 points
Generators in S140
(1 106 44 78 119)(2 107 45 79 120)(3 108 46 80 121)(4 109 47 81 122)(5 110 48 82 123)(6 111 49 83 124)(7 112 50 84 125)(8 85 51 57 126)(9 86 52 58 127)(10 87 53 59 128)(11 88 54 60 129)(12 89 55 61 130)(13 90 56 62 131)(14 91 29 63 132)(15 92 30 64 133)(16 93 31 65 134)(17 94 32 66 135)(18 95 33 67 136)(19 96 34 68 137)(20 97 35 69 138)(21 98 36 70 139)(22 99 37 71 140)(23 100 38 72 113)(24 101 39 73 114)(25 102 40 74 115)(26 103 41 75 116)(27 104 42 76 117)(28 105 43 77 118)
(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 28)(2 27)(3 26)(4 25)(5 24)(6 23)(7 22)(8 21)(9 20)(10 19)(11 18)(12 17)(13 16)(14 15)(29 30)(31 56)(32 55)(33 54)(34 53)(35 52)(36 51)(37 50)(38 49)(39 48)(40 47)(41 46)(42 45)(43 44)(57 70)(58 69)(59 68)(60 67)(61 66)(62 65)(63 64)(71 84)(72 83)(73 82)(74 81)(75 80)(76 79)(77 78)(85 98)(86 97)(87 96)(88 95)(89 94)(90 93)(91 92)(99 112)(100 111)(101 110)(102 109)(103 108)(104 107)(105 106)(113 124)(114 123)(115 122)(116 121)(117 120)(118 119)(125 140)(126 139)(127 138)(128 137)(129 136)(130 135)(131 134)(132 133)

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

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

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

85 conjugacy classes

 class 1 2A 2B 2C 4 5A 5B 5C 5D 7A 7B 7C 10A 10B 10C 10D 10E ··· 10L 14A 14B 14C 20A 20B 20C 20D 28A ··· 28F 35A ··· 35L 70A ··· 70L 140A ··· 140X order 1 2 2 2 4 5 5 5 5 7 7 7 10 10 10 10 10 ··· 10 14 14 14 20 20 20 20 28 ··· 28 35 ··· 35 70 ··· 70 140 ··· 140 size 1 1 14 14 2 1 1 1 1 2 2 2 1 1 1 1 14 ··· 14 2 2 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

85 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + image C1 C2 C2 C5 C10 C10 D4 D7 D14 C5×D4 D28 C5×D7 C10×D7 C5×D28 kernel C5×D28 C140 C10×D7 D28 C28 D14 C35 C20 C10 C7 C5 C4 C2 C1 # reps 1 1 2 4 4 8 1 3 3 4 6 12 12 24

Matrix representation of C5×D28 in GL3(𝔽281) generated by

 90 0 0 0 1 0 0 0 1
,
 280 0 0 0 268 107 0 174 16
,
 1 0 0 0 13 156 0 107 268
G:=sub<GL(3,GF(281))| [90,0,0,0,1,0,0,0,1],[280,0,0,0,268,174,0,107,16],[1,0,0,0,13,107,0,156,268] >;

C5×D28 in GAP, Magma, Sage, TeX

C_5\times D_{28}
% in TeX

G:=Group("C5xD28");
// GroupNames label

G:=SmallGroup(280,16);
// by ID

G=gap.SmallGroup(280,16);
# by ID

G:=PCGroup([5,-2,-2,-5,-2,-7,221,106,6004]);
// Polycyclic

G:=Group<a,b,c|a^5=b^28=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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