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## G = C5⋊Dic21order 420 = 22·3·5·7

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

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

 Derived series C1 — C105 — C5⋊Dic21
 Chief series C1 — C7 — C35 — C105 — D5×C21 — C5⋊Dic21
 Lower central C105 — C5⋊Dic21
 Upper central 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 >

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

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

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

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

45 conjugacy classes

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

45 irreducible representations

 dim 1 1 1 2 2 2 2 2 2 4 4 4 4 type + + + - + - + - + image C1 C2 C4 S3 Dic3 D7 Dic7 D21 Dic21 F5 C3⋊F5 C7⋊F5 C5⋊Dic21 kernel C5⋊Dic21 D5×C21 C105 C7×D5 C35 C3×D5 C15 D5 C5 C21 C7 C3 C1 # reps 1 1 2 1 1 3 3 6 6 1 2 6 12

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

 1 0 0 287 0 1 134 340 135 236 50 263 287 108 158 368
,
 111 371 0 0 100 357 0 0 274 38 216 50 136 209 371 158
,
 340 18 0 0 384 81 0 0 406 260 370 158 313 167 234 51
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

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