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## G = C7×C5⋊D4order 280 = 23·5·7

### Direct product of C7 and C5⋊D4

Aliases: C7×C5⋊D4, C359D4, Dic5⋊C14, D102C14, C14.17D10, C70.22C22, C52(C7×D4), C22⋊(C7×D5), (C2×C70)⋊6C2, (C2×C14)⋊1D5, (C2×C10)⋊2C14, (D5×C14)⋊5C2, C2.5(D5×C14), C10.5(C2×C14), (C7×Dic5)⋊4C2, SmallGroup(280,23)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C7×C5⋊D4
 Chief series C1 — C5 — C10 — C70 — D5×C14 — C7×C5⋊D4
 Lower central C5 — C10 — C7×C5⋊D4
 Upper central C1 — C14 — C2×C14

Generators and relations for C7×C5⋊D4
G = < a,b,c,d | a7=b5=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Smallest permutation representation of C7×C5⋊D4
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)
(1 117 109 44 76)(2 118 110 45 77)(3 119 111 46 71)(4 113 112 47 72)(5 114 106 48 73)(6 115 107 49 74)(7 116 108 43 75)(8 133 39 31 102)(9 127 40 32 103)(10 128 41 33 104)(11 129 42 34 105)(12 130 36 35 99)(13 131 37 29 100)(14 132 38 30 101)(15 57 53 124 139)(16 58 54 125 140)(17 59 55 126 134)(18 60 56 120 135)(19 61 50 121 136)(20 62 51 122 137)(21 63 52 123 138)(22 65 97 89 78)(23 66 98 90 79)(24 67 92 91 80)(25 68 93 85 81)(26 69 94 86 82)(27 70 95 87 83)(28 64 96 88 84)
(1 132 78 56)(2 133 79 50)(3 127 80 51)(4 128 81 52)(5 129 82 53)(6 130 83 54)(7 131 84 55)(8 23 61 118)(9 24 62 119)(10 25 63 113)(11 26 57 114)(12 27 58 115)(13 28 59 116)(14 22 60 117)(15 106 105 69)(16 107 99 70)(17 108 100 64)(18 109 101 65)(19 110 102 66)(20 111 103 67)(21 112 104 68)(29 96 134 43)(30 97 135 44)(31 98 136 45)(32 92 137 46)(33 93 138 47)(34 94 139 48)(35 95 140 49)(36 87 125 74)(37 88 126 75)(38 89 120 76)(39 90 121 77)(40 91 122 71)(41 85 123 72)(42 86 124 73)
(8 121)(9 122)(10 123)(11 124)(12 125)(13 126)(14 120)(15 34)(16 35)(17 29)(18 30)(19 31)(20 32)(21 33)(22 89)(23 90)(24 91)(25 85)(26 86)(27 87)(28 88)(36 58)(37 59)(38 60)(39 61)(40 62)(41 63)(42 57)(43 108)(44 109)(45 110)(46 111)(47 112)(48 106)(49 107)(50 133)(51 127)(52 128)(53 129)(54 130)(55 131)(56 132)(64 96)(65 97)(66 98)(67 92)(68 93)(69 94)(70 95)(71 119)(72 113)(73 114)(74 115)(75 116)(76 117)(77 118)(99 140)(100 134)(101 135)(102 136)(103 137)(104 138)(105 139)

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

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

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

91 conjugacy classes

 class 1 2A 2B 2C 4 5A 5B 7A ··· 7F 10A ··· 10F 14A ··· 14F 14G ··· 14L 14M ··· 14R 28A ··· 28F 35A ··· 35L 70A ··· 70AJ order 1 2 2 2 4 5 5 7 ··· 7 10 ··· 10 14 ··· 14 14 ··· 14 14 ··· 14 28 ··· 28 35 ··· 35 70 ··· 70 size 1 1 2 10 10 2 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 10 ··· 10 10 ··· 10 2 ··· 2 2 ··· 2

91 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + image C1 C2 C2 C2 C7 C14 C14 C14 D4 D5 D10 C5⋊D4 C7×D4 C7×D5 D5×C14 C7×C5⋊D4 kernel C7×C5⋊D4 C7×Dic5 D5×C14 C2×C70 C5⋊D4 Dic5 D10 C2×C10 C35 C2×C14 C14 C7 C5 C22 C2 C1 # reps 1 1 1 1 6 6 6 6 1 2 2 4 6 12 12 24

Matrix representation of C7×C5⋊D4 in GL2(𝔽281) generated by

 249 0 0 249
,
 243 280 1 0
,
 237 116 102 44
,
 1 0 243 280
G:=sub<GL(2,GF(281))| [249,0,0,249],[243,1,280,0],[237,102,116,44],[1,243,0,280] >;

C7×C5⋊D4 in GAP, Magma, Sage, TeX

C_7\times C_5\rtimes D_4
% in TeX

G:=Group("C7xC5:D4");
// GroupNames label

G:=SmallGroup(280,23);
// by ID

G=gap.SmallGroup(280,23);
# by ID

G:=PCGroup([5,-2,-2,-7,-2,-5,301,5604]);
// Polycyclic

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

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