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## G = C5×C7⋊C12order 420 = 22·3·5·7

### Direct product of C5 and C7⋊C12

Aliases: C5×C7⋊C12, C7⋊C60, C354C12, C14.C30, Dic7⋊C15, C70.2C6, C10.2F7, C7⋊C3⋊C20, C2.(C5×F7), (C5×Dic7)⋊C3, (C5×C7⋊C3)⋊4C4, (C2×C7⋊C3).C10, (C10×C7⋊C3).2C2, SmallGroup(420,1)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — C5×C7⋊C12
 Chief series C1 — C7 — C14 — C70 — C10×C7⋊C3 — C5×C7⋊C12
 Lower central C7 — C5×C7⋊C12
 Upper central C1 — C10

Generators and relations for C5×C7⋊C12
G = < a,b,c | a5=b7=c12=1, ab=ba, ac=ca, cbc-1=b5 >

Smallest permutation representation of C5×C7⋊C12
On 140 points
Generators in S140
(1 8 10 19 16)(2 5 11 20 13)(3 6 12 17 14)(4 7 9 18 15)(21 126 60 56 35)(22 127 61 45 36)(23 128 62 46 37)(24 117 63 47 38)(25 118 64 48 39)(26 119 65 49 40)(27 120 66 50 41)(28 121 67 51 42)(29 122 68 52 43)(30 123 57 53 44)(31 124 58 54 33)(32 125 59 55 34)(69 93 137 112 86)(70 94 138 113 87)(71 95 139 114 88)(72 96 140 115 89)(73 97 129 116 90)(74 98 130 105 91)(75 99 131 106 92)(76 100 132 107 81)(77 101 133 108 82)(78 102 134 109 83)(79 103 135 110 84)(80 104 136 111 85)
(1 138 134 28 130 32 24)(2 21 29 139 25 131 135)(3 132 140 22 136 26 30)(4 27 23 133 31 137 129)(5 126 122 114 118 106 110)(6 107 115 127 111 119 123)(7 120 128 108 124 112 116)(8 113 109 121 105 125 117)(9 66 62 82 58 86 90)(10 87 83 67 91 59 63)(11 60 68 88 64 92 84)(12 81 89 61 85 65 57)(13 35 43 95 39 99 103)(14 100 96 36 104 40 44)(15 41 37 101 33 93 97)(16 94 102 42 98 34 38)(17 76 72 45 80 49 53)(18 50 46 77 54 69 73)(19 70 78 51 74 55 47)(20 56 52 71 48 75 79)
(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)

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

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

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

70 conjugacy classes

 class 1 2 3A 3B 4A 4B 5A 5B 5C 5D 6A 6B 7 10A 10B 10C 10D 12A 12B 12C 12D 14 15A ··· 15H 20A ··· 20H 30A ··· 30H 35A 35B 35C 35D 60A ··· 60P 70A 70B 70C 70D order 1 2 3 3 4 4 5 5 5 5 6 6 7 10 10 10 10 12 12 12 12 14 15 ··· 15 20 ··· 20 30 ··· 30 35 35 35 35 60 ··· 60 70 70 70 70 size 1 1 7 7 7 7 1 1 1 1 7 7 6 1 1 1 1 7 7 7 7 6 7 ··· 7 7 ··· 7 7 ··· 7 6 6 6 6 7 ··· 7 6 6 6 6

70 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 6 6 6 6 type + + + - image C1 C2 C3 C4 C5 C6 C10 C12 C15 C20 C30 C60 F7 C7⋊C12 C5×F7 C5×C7⋊C12 kernel C5×C7⋊C12 C10×C7⋊C3 C5×Dic7 C5×C7⋊C3 C7⋊C12 C70 C2×C7⋊C3 C35 Dic7 C7⋊C3 C14 C7 C10 C5 C2 C1 # reps 1 1 2 2 4 2 4 4 8 8 8 16 1 1 4 4

Matrix representation of C5×C7⋊C12 in GL7(𝔽421)

 279 0 0 0 0 0 0 0 377 0 0 0 0 0 0 0 377 0 0 0 0 0 0 0 377 0 0 0 0 0 0 0 377 0 0 0 0 0 0 0 377 0 0 0 0 0 0 0 377
,
 1 0 0 0 0 0 0 0 0 0 0 0 0 420 0 1 0 0 0 0 420 0 0 1 0 0 0 420 0 0 0 1 0 0 420 0 0 0 0 1 0 420 0 0 0 0 0 1 420
,
 262 0 0 0 0 0 0 0 201 0 0 201 199 220 0 201 201 199 0 220 0 0 400 0 220 201 220 0 0 0 201 220 201 0 199 0 0 201 0 400 220 220 0 201 400 220 0 0 220

G:=sub<GL(7,GF(421))| [279,0,0,0,0,0,0,0,377,0,0,0,0,0,0,0,377,0,0,0,0,0,0,0,377,0,0,0,0,0,0,0,377,0,0,0,0,0,0,0,377,0,0,0,0,0,0,0,377],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,420,420,420,420,420,420],[262,0,0,0,0,0,0,0,201,201,400,0,0,201,0,0,201,0,201,201,400,0,0,199,220,220,0,220,0,201,0,201,201,400,0,0,199,220,220,0,220,0,0,220,0,0,199,220,220] >;

C5×C7⋊C12 in GAP, Magma, Sage, TeX

C_5\times C_7\rtimes C_{12}
% in TeX

G:=Group("C5xC7:C12");
// GroupNames label

G:=SmallGroup(420,1);
// by ID

G=gap.SmallGroup(420,1);
# by ID

G:=PCGroup([5,-2,-3,-5,-2,-7,150,9004,3009]);
// Polycyclic

G:=Group<a,b,c|a^5=b^7=c^12=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^5>;
// generators/relations

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