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G = C5×D28order 280 = 23·5·7

Direct product of C5 and D28

direct product, metacyclic, supersoluble, monomial, 2-hyperelementary

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

C1C14 — C5×D28
C1C7C14C70C10×D7 — C5×D28
C7C14 — C5×D28
C1C10C20

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

14C2
14C2
7C22
7C22
14C10
14C10
2D7
2D7
7D4
7C2×C10
7C2×C10
2C5×D7
2C5×D7
7C5×D4

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

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

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

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

85 conjugacy classes

class 1 2A2B2C 4 5A5B5C5D7A7B7C10A10B10C10D10E···10L14A14B14C20A20B20C20D28A···28F35A···35L70A···70L140A···140X
order1222455557771010101010···101414142020202028···2835···3570···70140···140
size11141421111222111114···1422222222···22···22···22···2

85 irreducible representations

dim11111122222222
type+++++++
imageC1C2C2C5C10C10D4D7D14C5×D4D28C5×D7C10×D7C5×D28
kernelC5×D28C140C10×D7D28C28D14C35C20C10C7C5C4C2C1
# reps11244813346121224

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

9000
010
001
,
28000
0268107
017416
,
100
013156
0107268
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

Export

Subgroup lattice of C5×D28 in TeX

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