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## G = D5×C2×C14order 280 = 23·5·7

### Direct product of C2×C14 and D5

Aliases: D5×C2×C14, C354C23, C704C22, C10⋊(C2×C14), C5⋊(C22×C14), (C2×C70)⋊7C2, (C2×C10)⋊3C14, SmallGroup(280,38)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — D5×C2×C14
 Chief series C1 — C5 — C35 — C7×D5 — D5×C14 — D5×C2×C14
 Lower central C5 — D5×C2×C14
 Upper central C1 — C2×C14

Generators and relations for D5×C2×C14
G = < a,b,c,d | a2=b14=c5=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 152 in 64 conjugacy classes, 42 normal (10 characteristic)
C1, C2, C2, C22, C22, C5, C7, C23, D5, C10, C14, C14, D10, C2×C10, C2×C14, C2×C14, C35, C22×D5, C22×C14, C7×D5, C70, D5×C14, C2×C70, D5×C2×C14
Quotients: C1, C2, C22, C7, C23, D5, C14, D10, C2×C14, C22×D5, C22×C14, C7×D5, D5×C14, D5×C2×C14

Smallest permutation representation of D5×C2×C14
On 140 points
Generators in S140
(1 133)(2 134)(3 135)(4 136)(5 137)(6 138)(7 139)(8 140)(9 127)(10 128)(11 129)(12 130)(13 131)(14 132)(15 68)(16 69)(17 70)(18 57)(19 58)(20 59)(21 60)(22 61)(23 62)(24 63)(25 64)(26 65)(27 66)(28 67)(29 88)(30 89)(31 90)(32 91)(33 92)(34 93)(35 94)(36 95)(37 96)(38 97)(39 98)(40 85)(41 86)(42 87)(43 74)(44 75)(45 76)(46 77)(47 78)(48 79)(49 80)(50 81)(51 82)(52 83)(53 84)(54 71)(55 72)(56 73)(99 125)(100 126)(101 113)(102 114)(103 115)(104 116)(105 117)(106 118)(107 119)(108 120)(109 121)(110 122)(111 123)(112 124)
(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 39 70 79 101)(2 40 57 80 102)(3 41 58 81 103)(4 42 59 82 104)(5 29 60 83 105)(6 30 61 84 106)(7 31 62 71 107)(8 32 63 72 108)(9 33 64 73 109)(10 34 65 74 110)(11 35 66 75 111)(12 36 67 76 112)(13 37 68 77 99)(14 38 69 78 100)(15 46 125 131 96)(16 47 126 132 97)(17 48 113 133 98)(18 49 114 134 85)(19 50 115 135 86)(20 51 116 136 87)(21 52 117 137 88)(22 53 118 138 89)(23 54 119 139 90)(24 55 120 140 91)(25 56 121 127 92)(26 43 122 128 93)(27 44 123 129 94)(28 45 124 130 95)
(1 113)(2 114)(3 115)(4 116)(5 117)(6 118)(7 119)(8 120)(9 121)(10 122)(11 123)(12 124)(13 125)(14 126)(15 68)(16 69)(17 70)(18 57)(19 58)(20 59)(21 60)(22 61)(23 62)(24 63)(25 64)(26 65)(27 66)(28 67)(29 52)(30 53)(31 54)(32 55)(33 56)(34 43)(35 44)(36 45)(37 46)(38 47)(39 48)(40 49)(41 50)(42 51)(71 90)(72 91)(73 92)(74 93)(75 94)(76 95)(77 96)(78 97)(79 98)(80 85)(81 86)(82 87)(83 88)(84 89)(99 131)(100 132)(101 133)(102 134)(103 135)(104 136)(105 137)(106 138)(107 139)(108 140)(109 127)(110 128)(111 129)(112 130)

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

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

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

112 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 5A 5B 7A ··· 7F 10A ··· 10F 14A ··· 14R 14S ··· 14AP 35A ··· 35L 70A ··· 70AJ order 1 2 2 2 2 2 2 2 5 5 7 ··· 7 10 ··· 10 14 ··· 14 14 ··· 14 35 ··· 35 70 ··· 70 size 1 1 1 1 5 5 5 5 2 2 1 ··· 1 2 ··· 2 1 ··· 1 5 ··· 5 2 ··· 2 2 ··· 2

112 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + + + image C1 C2 C2 C7 C14 C14 D5 D10 C7×D5 D5×C14 kernel D5×C2×C14 D5×C14 C2×C70 C22×D5 D10 C2×C10 C2×C14 C14 C22 C2 # reps 1 6 1 6 36 6 2 6 12 36

Matrix representation of D5×C2×C14 in GL3(𝔽71) generated by

 70 0 0 0 1 0 0 0 1
,
 70 0 0 0 26 0 0 0 26
,
 1 0 0 0 8 1 0 70 0
,
 1 0 0 0 1 8 0 0 70
G:=sub<GL(3,GF(71))| [70,0,0,0,1,0,0,0,1],[70,0,0,0,26,0,0,0,26],[1,0,0,0,8,70,0,1,0],[1,0,0,0,1,0,0,8,70] >;

D5×C2×C14 in GAP, Magma, Sage, TeX

D_5\times C_2\times C_{14}
% in TeX

G:=Group("D5xC2xC14");
// GroupNames label

G:=SmallGroup(280,38);
// by ID

G=gap.SmallGroup(280,38);
# by ID

G:=PCGroup([5,-2,-2,-2,-7,-5,5604]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^14=c^5=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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