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G = C5⋊Dic21order 420 = 22·3·5·7

The semidirect product of C5 and Dic21 acting via Dic21/C21=C4

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

Aliases: C5⋊Dic21, C211F5, C1051C4, D5.D21, C151Dic7, C351Dic3, C3⋊(C7⋊F5), C7⋊(C3⋊F5), (C3×D5).1D7, (C7×D5).1S3, (D5×C21).1C2, SmallGroup(420,23)

Series: Derived Chief Lower central Upper central

C1C105 — C5⋊Dic21
C1C7C35C105D5×C21 — C5⋊Dic21
C105 — C5⋊Dic21
C1

Generators and relations for C5⋊Dic21
 G = < a,b,c | a5=b42=1, c2=b21, bab-1=a-1, cac-1=a3, cbc-1=b-1 >

5C2
105C4
5C6
5C14
35Dic3
21F5
15Dic7
5C42
7C3⋊F5
5Dic21
3C7⋊F5

Smallest permutation representation of C5⋊Dic21
On 105 points
Generators in S105
(1 100 23 44 79)(2 80 45 24 101)(3 102 25 46 81)(4 82 47 26 103)(5 104 27 48 83)(6 84 49 28 105)(7 64 29 50 85)(8 86 51 30 65)(9 66 31 52 87)(10 88 53 32 67)(11 68 33 54 89)(12 90 55 34 69)(13 70 35 56 91)(14 92 57 36 71)(15 72 37 58 93)(16 94 59 38 73)(17 74 39 60 95)(18 96 61 40 75)(19 76 41 62 97)(20 98 63 42 77)(21 78 43 22 99)
(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 14)(2 13)(3 12)(4 11)(5 10)(6 9)(7 8)(15 21)(16 20)(17 19)(22 72 43 93)(23 71 44 92)(24 70 45 91)(25 69 46 90)(26 68 47 89)(27 67 48 88)(28 66 49 87)(29 65 50 86)(30 64 51 85)(31 105 52 84)(32 104 53 83)(33 103 54 82)(34 102 55 81)(35 101 56 80)(36 100 57 79)(37 99 58 78)(38 98 59 77)(39 97 60 76)(40 96 61 75)(41 95 62 74)(42 94 63 73)

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

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

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

45 conjugacy classes

class 1  2  3 4A4B 5  6 7A7B7C14A14B14C15A15B21A···21F35A···35F42A···42F105A···105L
order1234456777141414151521···2135···3542···42105···105
size152105105410222101010442···24···410···104···4

45 irreducible representations

dim1112222224444
type+++-+-+-+
imageC1C2C4S3Dic3D7Dic7D21Dic21F5C3⋊F5C7⋊F5C5⋊Dic21
kernelC5⋊Dic21D5×C21C105C7×D5C35C3×D5C15D5C5C21C7C3C1
# reps11211336612612

Matrix representation of C5⋊Dic21 in GL4(𝔽421) generated by

100287
01134340
13523650263
287108158368
,
11137100
10035700
2743821650
136209371158
,
3401800
3848100
406260370158
31316723451
G:=sub<GL(4,GF(421))| [1,0,135,287,0,1,236,108,0,134,50,158,287,340,263,368],[111,100,274,136,371,357,38,209,0,0,216,371,0,0,50,158],[340,384,406,313,18,81,260,167,0,0,370,234,0,0,158,51] >;

C5⋊Dic21 in GAP, Magma, Sage, TeX

C_5\rtimes {\rm Dic}_{21}
% in TeX

G:=Group("C5:Dic21");
// GroupNames label

G:=SmallGroup(420,23);
// by ID

G=gap.SmallGroup(420,23);
# by ID

G:=PCGroup([5,-2,-2,-3,-5,-7,10,122,483,488,9004]);
// Polycyclic

G:=Group<a,b,c|a^5=b^42=1,c^2=b^21,b*a*b^-1=a^-1,c*a*c^-1=a^3,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C5⋊Dic21 in TeX

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