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G = C10×Dic3order 120 = 23·3·5

Direct product of C10 and Dic3

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

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

C1C3 — C10×Dic3
C1C3C6C30C5×Dic3 — C10×Dic3
C3 — C10×Dic3
C1C2×C10

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

3C4
3C4
3C2×C4
3C20
3C20
3C2×C20

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 2A2B2C 3 4A4B4C4D5A5B5C5D6A6B6C10A···10L15A15B15C15D20A···20P30A···30L
order122234444555566610···101515151520···2030···30
size11112333311112221···122223···32···2

60 irreducible representations

dim11111111222222
type++++-+
imageC1C2C2C4C5C10C10C20S3Dic3D6C5×S3C5×Dic3S3×C10
kernelC10×Dic3C5×Dic3C2×C30C30C2×Dic3Dic3C2×C6C6C2×C10C10C10C22C2C2
# reps121448416121484

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

1000
06000
00200
00020
,
60000
06000
00601
00600
,
50000
05000
00600
00601
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

Export

Subgroup lattice of C10×Dic3 in TeX

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