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## G = C10×Dic3order 120 = 23·3·5

### Direct product of C10 and Dic3

Aliases: C10×Dic3, C6⋊C20, C305C4, C10.16D6, C30.21C22, C32(C2×C20), (C2×C6).C10, C1512(C2×C4), C22.(C5×S3), C2.2(S3×C10), (C2×C10).2S3, (C2×C30).3C2, C6.4(C2×C10), SmallGroup(120,24)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C10×Dic3
 Chief series C1 — C3 — C6 — C30 — C5×Dic3 — C10×Dic3
 Lower central C3 — C10×Dic3
 Upper central C1 — C2×C10

Generators and relations for C10×Dic3
G = < a,b,c | a10=b6=1, c2=b3, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C10×Dic3
Regular action on 120 points
Generators in S120
(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)(101 102 103 104 105 106 107 108 109 110)(111 112 113 114 115 116 117 118 119 120)
(1 75 85 105 95 70)(2 76 86 106 96 61)(3 77 87 107 97 62)(4 78 88 108 98 63)(5 79 89 109 99 64)(6 80 90 110 100 65)(7 71 81 101 91 66)(8 72 82 102 92 67)(9 73 83 103 93 68)(10 74 84 104 94 69)(11 51 41 115 40 29)(12 52 42 116 31 30)(13 53 43 117 32 21)(14 54 44 118 33 22)(15 55 45 119 34 23)(16 56 46 120 35 24)(17 57 47 111 36 25)(18 58 48 112 37 26)(19 59 49 113 38 27)(20 60 50 114 39 28)
(1 16 105 120)(2 17 106 111)(3 18 107 112)(4 19 108 113)(5 20 109 114)(6 11 110 115)(7 12 101 116)(8 13 102 117)(9 14 103 118)(10 15 104 119)(21 92 43 72)(22 93 44 73)(23 94 45 74)(24 95 46 75)(25 96 47 76)(26 97 48 77)(27 98 49 78)(28 99 50 79)(29 100 41 80)(30 91 42 71)(31 66 52 81)(32 67 53 82)(33 68 54 83)(34 69 55 84)(35 70 56 85)(36 61 57 86)(37 62 58 87)(38 63 59 88)(39 64 60 89)(40 65 51 90)

G:=sub<Sym(120)| (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)(101,102,103,104,105,106,107,108,109,110)(111,112,113,114,115,116,117,118,119,120), (1,75,85,105,95,70)(2,76,86,106,96,61)(3,77,87,107,97,62)(4,78,88,108,98,63)(5,79,89,109,99,64)(6,80,90,110,100,65)(7,71,81,101,91,66)(8,72,82,102,92,67)(9,73,83,103,93,68)(10,74,84,104,94,69)(11,51,41,115,40,29)(12,52,42,116,31,30)(13,53,43,117,32,21)(14,54,44,118,33,22)(15,55,45,119,34,23)(16,56,46,120,35,24)(17,57,47,111,36,25)(18,58,48,112,37,26)(19,59,49,113,38,27)(20,60,50,114,39,28), (1,16,105,120)(2,17,106,111)(3,18,107,112)(4,19,108,113)(5,20,109,114)(6,11,110,115)(7,12,101,116)(8,13,102,117)(9,14,103,118)(10,15,104,119)(21,92,43,72)(22,93,44,73)(23,94,45,74)(24,95,46,75)(25,96,47,76)(26,97,48,77)(27,98,49,78)(28,99,50,79)(29,100,41,80)(30,91,42,71)(31,66,52,81)(32,67,53,82)(33,68,54,83)(34,69,55,84)(35,70,56,85)(36,61,57,86)(37,62,58,87)(38,63,59,88)(39,64,60,89)(40,65,51,90)>;

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)(101,102,103,104,105,106,107,108,109,110)(111,112,113,114,115,116,117,118,119,120), (1,75,85,105,95,70)(2,76,86,106,96,61)(3,77,87,107,97,62)(4,78,88,108,98,63)(5,79,89,109,99,64)(6,80,90,110,100,65)(7,71,81,101,91,66)(8,72,82,102,92,67)(9,73,83,103,93,68)(10,74,84,104,94,69)(11,51,41,115,40,29)(12,52,42,116,31,30)(13,53,43,117,32,21)(14,54,44,118,33,22)(15,55,45,119,34,23)(16,56,46,120,35,24)(17,57,47,111,36,25)(18,58,48,112,37,26)(19,59,49,113,38,27)(20,60,50,114,39,28), (1,16,105,120)(2,17,106,111)(3,18,107,112)(4,19,108,113)(5,20,109,114)(6,11,110,115)(7,12,101,116)(8,13,102,117)(9,14,103,118)(10,15,104,119)(21,92,43,72)(22,93,44,73)(23,94,45,74)(24,95,46,75)(25,96,47,76)(26,97,48,77)(27,98,49,78)(28,99,50,79)(29,100,41,80)(30,91,42,71)(31,66,52,81)(32,67,53,82)(33,68,54,83)(34,69,55,84)(35,70,56,85)(36,61,57,86)(37,62,58,87)(38,63,59,88)(39,64,60,89)(40,65,51,90) );

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),(101,102,103,104,105,106,107,108,109,110),(111,112,113,114,115,116,117,118,119,120)], [(1,75,85,105,95,70),(2,76,86,106,96,61),(3,77,87,107,97,62),(4,78,88,108,98,63),(5,79,89,109,99,64),(6,80,90,110,100,65),(7,71,81,101,91,66),(8,72,82,102,92,67),(9,73,83,103,93,68),(10,74,84,104,94,69),(11,51,41,115,40,29),(12,52,42,116,31,30),(13,53,43,117,32,21),(14,54,44,118,33,22),(15,55,45,119,34,23),(16,56,46,120,35,24),(17,57,47,111,36,25),(18,58,48,112,37,26),(19,59,49,113,38,27),(20,60,50,114,39,28)], [(1,16,105,120),(2,17,106,111),(3,18,107,112),(4,19,108,113),(5,20,109,114),(6,11,110,115),(7,12,101,116),(8,13,102,117),(9,14,103,118),(10,15,104,119),(21,92,43,72),(22,93,44,73),(23,94,45,74),(24,95,46,75),(25,96,47,76),(26,97,48,77),(27,98,49,78),(28,99,50,79),(29,100,41,80),(30,91,42,71),(31,66,52,81),(32,67,53,82),(33,68,54,83),(34,69,55,84),(35,70,56,85),(36,61,57,86),(37,62,58,87),(38,63,59,88),(39,64,60,89),(40,65,51,90)]])

C10×Dic3 is a maximal subgroup of   D10⋊Dic3  D304C4  C30.Q8  Dic155C4  C6.Dic10  Dic5.D6  S3×C2×C20

60 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 5A 5B 5C 5D 6A 6B 6C 10A ··· 10L 15A 15B 15C 15D 20A ··· 20P 30A ··· 30L order 1 2 2 2 3 4 4 4 4 5 5 5 5 6 6 6 10 ··· 10 15 15 15 15 20 ··· 20 30 ··· 30 size 1 1 1 1 2 3 3 3 3 1 1 1 1 2 2 2 1 ··· 1 2 2 2 2 3 ··· 3 2 ··· 2

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + - + image C1 C2 C2 C4 C5 C10 C10 C20 S3 Dic3 D6 C5×S3 C5×Dic3 S3×C10 kernel C10×Dic3 C5×Dic3 C2×C30 C30 C2×Dic3 Dic3 C2×C6 C6 C2×C10 C10 C10 C22 C2 C2 # reps 1 2 1 4 4 8 4 16 1 2 1 4 8 4

Matrix representation of C10×Dic3 in GL4(𝔽61) generated by

 1 0 0 0 0 60 0 0 0 0 20 0 0 0 0 20
,
 60 0 0 0 0 60 0 0 0 0 60 1 0 0 60 0
,
 50 0 0 0 0 50 0 0 0 0 60 0 0 0 60 1
G:=sub<GL(4,GF(61))| [1,0,0,0,0,60,0,0,0,0,20,0,0,0,0,20],[60,0,0,0,0,60,0,0,0,0,60,60,0,0,1,0],[50,0,0,0,0,50,0,0,0,0,60,60,0,0,0,1] >;

C10×Dic3 in GAP, Magma, Sage, TeX

C_{10}\times {\rm Dic}_3
% in TeX

G:=Group("C10xDic3");
// GroupNames label

G:=SmallGroup(120,24);
// by ID

G=gap.SmallGroup(120,24);
# by ID

G:=PCGroup([5,-2,-2,-5,-2,-3,100,2004]);
// Polycyclic

G:=Group<a,b,c|a^10=b^6=1,c^2=b^3,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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