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## G = C7⋊D20order 280 = 23·5·7

### The semidirect product of C7 and D20 acting via D20/D10=C2

Aliases: C353D4, C72D20, Dic7⋊D5, D704C2, D102D7, C14.6D10, C10.6D14, C70.6C22, C51(C7⋊D4), C2.6(D5×D7), (D5×C14)⋊2C2, (C5×Dic7)⋊3C2, SmallGroup(280,12)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C70 — C7⋊D20
 Chief series C1 — C7 — C35 — C70 — C5×Dic7 — C7⋊D20
 Lower central C35 — C70 — C7⋊D20
 Upper central C1 — C2

Generators and relations for C7⋊D20
G = < a,b,c | a7=b20=c2=1, bab-1=cac=a-1, cbc=b-1 >

Smallest permutation representation of C7⋊D20
On 140 points
Generators in S140
(1 90 32 63 129 103 53)(2 54 104 130 64 33 91)(3 92 34 65 131 105 55)(4 56 106 132 66 35 93)(5 94 36 67 133 107 57)(6 58 108 134 68 37 95)(7 96 38 69 135 109 59)(8 60 110 136 70 39 97)(9 98 40 71 137 111 41)(10 42 112 138 72 21 99)(11 100 22 73 139 113 43)(12 44 114 140 74 23 81)(13 82 24 75 121 115 45)(14 46 116 122 76 25 83)(15 84 26 77 123 117 47)(16 48 118 124 78 27 85)(17 86 28 79 125 119 49)(18 50 120 126 80 29 87)(19 88 30 61 127 101 51)(20 52 102 128 62 31 89)
(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 5)(2 4)(6 20)(7 19)(8 18)(9 17)(10 16)(11 15)(12 14)(21 118)(22 117)(23 116)(24 115)(25 114)(26 113)(27 112)(28 111)(29 110)(30 109)(31 108)(32 107)(33 106)(34 105)(35 104)(36 103)(37 102)(38 101)(39 120)(40 119)(41 86)(42 85)(43 84)(44 83)(45 82)(46 81)(47 100)(48 99)(49 98)(50 97)(51 96)(52 95)(53 94)(54 93)(55 92)(56 91)(57 90)(58 89)(59 88)(60 87)(61 135)(62 134)(63 133)(64 132)(65 131)(66 130)(67 129)(68 128)(69 127)(70 126)(71 125)(72 124)(73 123)(74 122)(75 121)(76 140)(77 139)(78 138)(79 137)(80 136)

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

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

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

37 conjugacy classes

 class 1 2A 2B 2C 4 5A 5B 7A 7B 7C 10A 10B 14A 14B 14C 14D ··· 14I 20A 20B 20C 20D 35A ··· 35F 70A ··· 70F order 1 2 2 2 4 5 5 7 7 7 10 10 14 14 14 14 ··· 14 20 20 20 20 35 ··· 35 70 ··· 70 size 1 1 10 70 14 2 2 2 2 2 2 2 2 2 2 10 ··· 10 14 14 14 14 4 ··· 4 4 ··· 4

37 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + image C1 C2 C2 C2 D4 D5 D7 D10 D14 D20 C7⋊D4 D5×D7 C7⋊D20 kernel C7⋊D20 C5×Dic7 D5×C14 D70 C35 Dic7 D10 C14 C10 C7 C5 C2 C1 # reps 1 1 1 1 1 2 3 2 3 4 6 6 6

Matrix representation of C7⋊D20 in GL6(𝔽281)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 6 1 0 0 0 0 279 234 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 242 236 0 0 0 0 222 76 0 0 0 0 0 0 42 6 0 0 0 0 34 239 0 0 0 0 0 0 0 280 0 0 0 0 1 0
,
 280 0 0 0 0 0 65 1 0 0 0 0 0 0 239 275 0 0 0 0 247 42 0 0 0 0 0 0 1 0 0 0 0 0 0 280

G:=sub<GL(6,GF(281))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,6,279,0,0,0,0,1,234,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[242,222,0,0,0,0,236,76,0,0,0,0,0,0,42,34,0,0,0,0,6,239,0,0,0,0,0,0,0,1,0,0,0,0,280,0],[280,65,0,0,0,0,0,1,0,0,0,0,0,0,239,247,0,0,0,0,275,42,0,0,0,0,0,0,1,0,0,0,0,0,0,280] >;

C7⋊D20 in GAP, Magma, Sage, TeX

C_7\rtimes D_{20}
% in TeX

G:=Group("C7:D20");
// GroupNames label

G:=SmallGroup(280,12);
// by ID

G=gap.SmallGroup(280,12);
# by ID

G:=PCGroup([5,-2,-2,-2,-5,-7,20,61,328,6004]);
// Polycyclic

G:=Group<a,b,c|a^7=b^20=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations

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