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

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

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

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

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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