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## G = F5×C21order 420 = 22·3·5·7

### Direct product of C21 and F5

Aliases: F5×C21, C5⋊C84, C357C12, C152C28, C1054C4, D5.C42, (C7×D5).4C6, (C3×D5).2C14, (D5×C21).4C2, SmallGroup(420,20)

Series: Derived Chief Lower central Upper central

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

Generators and relations for F5×C21
G = < a,b,c | a21=b5=c4=1, ab=ba, ac=ca, cbc-1=b3 >

Smallest permutation representation of F5×C21
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 51 23 83 103)(2 52 24 84 104)(3 53 25 64 105)(4 54 26 65 85)(5 55 27 66 86)(6 56 28 67 87)(7 57 29 68 88)(8 58 30 69 89)(9 59 31 70 90)(10 60 32 71 91)(11 61 33 72 92)(12 62 34 73 93)(13 63 35 74 94)(14 43 36 75 95)(15 44 37 76 96)(16 45 38 77 97)(17 46 39 78 98)(18 47 40 79 99)(19 48 41 80 100)(20 49 42 81 101)(21 50 22 82 102)
(22 102 82 50)(23 103 83 51)(24 104 84 52)(25 105 64 53)(26 85 65 54)(27 86 66 55)(28 87 67 56)(29 88 68 57)(30 89 69 58)(31 90 70 59)(32 91 71 60)(33 92 72 61)(34 93 73 62)(35 94 74 63)(36 95 75 43)(37 96 76 44)(38 97 77 45)(39 98 78 46)(40 99 79 47)(41 100 80 48)(42 101 81 49)

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

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

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

105 conjugacy classes

 class 1 2 3A 3B 4A 4B 5 6A 6B 7A ··· 7F 12A 12B 12C 12D 14A ··· 14F 15A 15B 21A ··· 21L 28A ··· 28L 35A ··· 35F 42A ··· 42L 84A ··· 84X 105A ··· 105L order 1 2 3 3 4 4 5 6 6 7 ··· 7 12 12 12 12 14 ··· 14 15 15 21 ··· 21 28 ··· 28 35 ··· 35 42 ··· 42 84 ··· 84 105 ··· 105 size 1 5 1 1 5 5 4 5 5 1 ··· 1 5 5 5 5 5 ··· 5 4 4 1 ··· 1 5 ··· 5 4 ··· 4 5 ··· 5 5 ··· 5 4 ··· 4

105 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 4 4 4 4 type + + + image C1 C2 C3 C4 C6 C7 C12 C14 C21 C28 C42 C84 F5 C3×F5 C7×F5 F5×C21 kernel F5×C21 D5×C21 C7×F5 C105 C7×D5 C3×F5 C35 C3×D5 F5 C15 D5 C5 C21 C7 C3 C1 # reps 1 1 2 2 2 6 4 6 12 12 12 24 1 2 6 12

Matrix representation of F5×C21 in GL4(𝔽421) generated by

 229 0 0 0 0 229 0 0 0 0 229 0 0 0 0 229
,
 420 420 420 420 1 0 0 0 0 1 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 1 0 1 0 0 420 420 420 420
G:=sub<GL(4,GF(421))| [229,0,0,0,0,229,0,0,0,0,229,0,0,0,0,229],[420,1,0,0,420,0,1,0,420,0,0,1,420,0,0,0],[1,0,0,420,0,0,1,420,0,0,0,420,0,1,0,420] >;

F5×C21 in GAP, Magma, Sage, TeX

F_5\times C_{21}
% in TeX

G:=Group("F5xC21");
// GroupNames label

G:=SmallGroup(420,20);
// by ID

G=gap.SmallGroup(420,20);
# by ID

G:=PCGroup([5,-2,-3,-7,-2,-5,210,4204,219]);
// Polycyclic

G:=Group<a,b,c|a^21=b^5=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^3>;
// generators/relations

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