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G = F5xC21order 420 = 22·3·5·7

Direct product of C21 and F5

direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: F5xC21, C5:C84, C35:7C12, C15:2C28, C105:4C4, D5.C42, (C7xD5).4C6, (C3xD5).2C14, (D5xC21).4C2, SmallGroup(420,20)

Series: Derived Chief Lower central Upper central

C1C5 — F5xC21
C1C5D5C7xD5D5xC21 — F5xC21
C5 — F5xC21
C1C21

Generators and relations for F5xC21
 G = < a,b,c | a21=b5=c4=1, ab=ba, ac=ca, cbc-1=b3 >

Subgroups: 56 in 24 conjugacy classes, 16 normal (all characteristic)
Quotients: C1, C2, C3, C4, C6, C7, C12, C14, F5, C21, C28, C42, C3xF5, C84, C7xF5, F5xC21
5C2
5C4
5C6
5C14
5C12
5C28
5C42
5C84

Smallest permutation representation of F5xC21
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 3A3B4A4B 5 6A6B7A···7F12A12B12C12D14A···14F15A15B21A···21L28A···28L35A···35F42A···42L84A···84X105A···105L
order1233445667···71212121214···14151521···2128···2835···3542···4284···84105···105
size1511554551···155555···5441···15···54···45···55···54···4

105 irreducible representations

dim1111111111114444
type+++
imageC1C2C3C4C6C7C12C14C21C28C42C84F5C3xF5C7xF5F5xC21
kernelF5xC21D5xC21C7xF5C105C7xD5C3xF5C35C3xD5F5C15D5C5C21C7C3C1
# reps112226461212122412612

Matrix representation of F5xC21 in GL4(F421) generated by

229000
022900
002290
000229
,
420420420420
1000
0100
0010
,
1000
0001
0100
420420420420
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] >;

F5xC21 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

Export

Subgroup lattice of F5xC21 in TeX

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