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## G = C7×C3⋊F5order 420 = 22·3·5·7

### Direct product of C7 and C3⋊F5

Aliases: C7×C3⋊F5, C213F5, C151C28, C1053C4, C352Dic3, C3⋊(C7×F5), C5⋊(C7×Dic3), D5.(S3×C7), (C7×D5).2S3, (D5×C21).3C2, (C3×D5).1C14, SmallGroup(420,22)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C15 — C7×C3⋊F5
 Chief series C1 — C5 — C15 — C3×D5 — D5×C21 — C7×C3⋊F5
 Lower central C15 — C7×C3⋊F5
 Upper central C1 — C7

Generators and relations for C7×C3⋊F5
G = < a,b,c,d | a7=b3=c5=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c3 >

Smallest permutation representation of C7×C3⋊F5
On 105 points
Generators in S105
(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)
(1 54 94)(2 55 95)(3 56 96)(4 50 97)(5 51 98)(6 52 92)(7 53 93)(8 39 58)(9 40 59)(10 41 60)(11 42 61)(12 36 62)(13 37 63)(14 38 57)(15 85 65)(16 86 66)(17 87 67)(18 88 68)(19 89 69)(20 90 70)(21 91 64)(22 100 43)(23 101 44)(24 102 45)(25 103 46)(26 104 47)(27 105 48)(28 99 49)(29 75 79)(30 76 80)(31 77 81)(32 71 82)(33 72 83)(34 73 84)(35 74 78)
(1 70 79 102 59)(2 64 80 103 60)(3 65 81 104 61)(4 66 82 105 62)(5 67 83 99 63)(6 68 84 100 57)(7 69 78 101 58)(8 53 19 35 44)(9 54 20 29 45)(10 55 21 30 46)(11 56 15 31 47)(12 50 16 32 48)(13 51 17 33 49)(14 52 18 34 43)(22 38 92 88 73)(23 39 93 89 74)(24 40 94 90 75)(25 41 95 91 76)(26 42 96 85 77)(27 36 97 86 71)(28 37 98 87 72)
(8 23 19 74)(9 24 20 75)(10 25 21 76)(11 26 15 77)(12 27 16 71)(13 28 17 72)(14 22 18 73)(29 40 45 90)(30 41 46 91)(31 42 47 85)(32 36 48 86)(33 37 49 87)(34 38 43 88)(35 39 44 89)(50 97)(51 98)(52 92)(53 93)(54 94)(55 95)(56 96)(57 100 68 84)(58 101 69 78)(59 102 70 79)(60 103 64 80)(61 104 65 81)(62 105 66 82)(63 99 67 83)

G:=sub<Sym(105)| (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), (1,54,94)(2,55,95)(3,56,96)(4,50,97)(5,51,98)(6,52,92)(7,53,93)(8,39,58)(9,40,59)(10,41,60)(11,42,61)(12,36,62)(13,37,63)(14,38,57)(15,85,65)(16,86,66)(17,87,67)(18,88,68)(19,89,69)(20,90,70)(21,91,64)(22,100,43)(23,101,44)(24,102,45)(25,103,46)(26,104,47)(27,105,48)(28,99,49)(29,75,79)(30,76,80)(31,77,81)(32,71,82)(33,72,83)(34,73,84)(35,74,78), (1,70,79,102,59)(2,64,80,103,60)(3,65,81,104,61)(4,66,82,105,62)(5,67,83,99,63)(6,68,84,100,57)(7,69,78,101,58)(8,53,19,35,44)(9,54,20,29,45)(10,55,21,30,46)(11,56,15,31,47)(12,50,16,32,48)(13,51,17,33,49)(14,52,18,34,43)(22,38,92,88,73)(23,39,93,89,74)(24,40,94,90,75)(25,41,95,91,76)(26,42,96,85,77)(27,36,97,86,71)(28,37,98,87,72), (8,23,19,74)(9,24,20,75)(10,25,21,76)(11,26,15,77)(12,27,16,71)(13,28,17,72)(14,22,18,73)(29,40,45,90)(30,41,46,91)(31,42,47,85)(32,36,48,86)(33,37,49,87)(34,38,43,88)(35,39,44,89)(50,97)(51,98)(52,92)(53,93)(54,94)(55,95)(56,96)(57,100,68,84)(58,101,69,78)(59,102,70,79)(60,103,64,80)(61,104,65,81)(62,105,66,82)(63,99,67,83)>;

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), (1,54,94)(2,55,95)(3,56,96)(4,50,97)(5,51,98)(6,52,92)(7,53,93)(8,39,58)(9,40,59)(10,41,60)(11,42,61)(12,36,62)(13,37,63)(14,38,57)(15,85,65)(16,86,66)(17,87,67)(18,88,68)(19,89,69)(20,90,70)(21,91,64)(22,100,43)(23,101,44)(24,102,45)(25,103,46)(26,104,47)(27,105,48)(28,99,49)(29,75,79)(30,76,80)(31,77,81)(32,71,82)(33,72,83)(34,73,84)(35,74,78), (1,70,79,102,59)(2,64,80,103,60)(3,65,81,104,61)(4,66,82,105,62)(5,67,83,99,63)(6,68,84,100,57)(7,69,78,101,58)(8,53,19,35,44)(9,54,20,29,45)(10,55,21,30,46)(11,56,15,31,47)(12,50,16,32,48)(13,51,17,33,49)(14,52,18,34,43)(22,38,92,88,73)(23,39,93,89,74)(24,40,94,90,75)(25,41,95,91,76)(26,42,96,85,77)(27,36,97,86,71)(28,37,98,87,72), (8,23,19,74)(9,24,20,75)(10,25,21,76)(11,26,15,77)(12,27,16,71)(13,28,17,72)(14,22,18,73)(29,40,45,90)(30,41,46,91)(31,42,47,85)(32,36,48,86)(33,37,49,87)(34,38,43,88)(35,39,44,89)(50,97)(51,98)(52,92)(53,93)(54,94)(55,95)(56,96)(57,100,68,84)(58,101,69,78)(59,102,70,79)(60,103,64,80)(61,104,65,81)(62,105,66,82)(63,99,67,83) );

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)], [(1,54,94),(2,55,95),(3,56,96),(4,50,97),(5,51,98),(6,52,92),(7,53,93),(8,39,58),(9,40,59),(10,41,60),(11,42,61),(12,36,62),(13,37,63),(14,38,57),(15,85,65),(16,86,66),(17,87,67),(18,88,68),(19,89,69),(20,90,70),(21,91,64),(22,100,43),(23,101,44),(24,102,45),(25,103,46),(26,104,47),(27,105,48),(28,99,49),(29,75,79),(30,76,80),(31,77,81),(32,71,82),(33,72,83),(34,73,84),(35,74,78)], [(1,70,79,102,59),(2,64,80,103,60),(3,65,81,104,61),(4,66,82,105,62),(5,67,83,99,63),(6,68,84,100,57),(7,69,78,101,58),(8,53,19,35,44),(9,54,20,29,45),(10,55,21,30,46),(11,56,15,31,47),(12,50,16,32,48),(13,51,17,33,49),(14,52,18,34,43),(22,38,92,88,73),(23,39,93,89,74),(24,40,94,90,75),(25,41,95,91,76),(26,42,96,85,77),(27,36,97,86,71),(28,37,98,87,72)], [(8,23,19,74),(9,24,20,75),(10,25,21,76),(11,26,15,77),(12,27,16,71),(13,28,17,72),(14,22,18,73),(29,40,45,90),(30,41,46,91),(31,42,47,85),(32,36,48,86),(33,37,49,87),(34,38,43,88),(35,39,44,89),(50,97),(51,98),(52,92),(53,93),(54,94),(55,95),(56,96),(57,100,68,84),(58,101,69,78),(59,102,70,79),(60,103,64,80),(61,104,65,81),(62,105,66,82),(63,99,67,83)]])

63 conjugacy classes

 class 1 2 3 4A 4B 5 6 7A ··· 7F 14A ··· 14F 15A 15B 21A ··· 21F 28A ··· 28L 35A ··· 35F 42A ··· 42F 105A ··· 105L order 1 2 3 4 4 5 6 7 ··· 7 14 ··· 14 15 15 21 ··· 21 28 ··· 28 35 ··· 35 42 ··· 42 105 ··· 105 size 1 5 2 15 15 4 10 1 ··· 1 5 ··· 5 4 4 2 ··· 2 15 ··· 15 4 ··· 4 10 ··· 10 4 ··· 4

63 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 4 4 4 4 type + + + - + image C1 C2 C4 C7 C14 C28 S3 Dic3 S3×C7 C7×Dic3 F5 C3⋊F5 C7×F5 C7×C3⋊F5 kernel C7×C3⋊F5 D5×C21 C105 C3⋊F5 C3×D5 C15 C7×D5 C35 D5 C5 C21 C7 C3 C1 # reps 1 1 2 6 6 12 1 1 6 6 1 2 6 12

Matrix representation of C7×C3⋊F5 in GL6(𝔽421)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 75 0 0 0 0 0 0 75 0 0 0 0 0 0 75 0 0 0 0 0 0 75
,
 100 169 0 0 0 0 5 320 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 420 420 420 420
,
 18 220 0 0 0 0 276 403 0 0 0 0 0 0 189 0 377 377 0 0 377 377 0 189 0 0 44 233 44 0 0 0 232 188 188 232

G:=sub<GL(6,GF(421))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,75,0,0,0,0,0,0,75,0,0,0,0,0,0,75,0,0,0,0,0,0,75],[100,5,0,0,0,0,169,320,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,420,0,0,1,0,0,420,0,0,0,1,0,420,0,0,0,0,1,420],[18,276,0,0,0,0,220,403,0,0,0,0,0,0,189,377,44,232,0,0,0,377,233,188,0,0,377,0,44,188,0,0,377,189,0,232] >;

C7×C3⋊F5 in GAP, Magma, Sage, TeX

C_7\times C_3\rtimes F_5
% in TeX

G:=Group("C7xC3:F5");
// GroupNames label

G:=SmallGroup(420,22);
// by ID

G=gap.SmallGroup(420,22);
# by ID

G:=PCGroup([5,-2,-7,-2,-3,-5,70,1123,6304,614]);
// Polycyclic

G:=Group<a,b,c,d|a^7=b^3=c^5=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^3>;
// generators/relations

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