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G = C6×Dic5order 120 = 23·3·5

Direct product of C6 and Dic5

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

Aliases: C6×Dic5, C304C4, C102C12, C6.16D10, C30.16C22, C53(C2×C12), (C2×C10).C6, C1511(C2×C4), (C2×C6).2D5, C2.2(C6×D5), C22.(C3×D5), C10.4(C2×C6), (C2×C30).2C2, SmallGroup(120,19)

Series: Derived Chief Lower central Upper central

C1C5 — C6×Dic5
C1C5C10C30C3×Dic5 — C6×Dic5
C5 — C6×Dic5
C1C2×C6

Generators and relations for C6×Dic5
 G = < a,b,c | a6=b10=1, c2=b5, ab=ba, ac=ca, cbc-1=b-1 >

5C4
5C4
5C2×C4
5C12
5C12
5C2×C12

Smallest permutation representation of C6×Dic5
Regular action on 120 points
Generators in S120
(1 49 29 39 19 59)(2 50 30 40 20 60)(3 41 21 31 11 51)(4 42 22 32 12 52)(5 43 23 33 13 53)(6 44 24 34 14 54)(7 45 25 35 15 55)(8 46 26 36 16 56)(9 47 27 37 17 57)(10 48 28 38 18 58)(61 101 81 91 71 111)(62 102 82 92 72 112)(63 103 83 93 73 113)(64 104 84 94 74 114)(65 105 85 95 75 115)(66 106 86 96 76 116)(67 107 87 97 77 117)(68 108 88 98 78 118)(69 109 89 99 79 119)(70 110 90 100 80 120)
(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 66 6 61)(2 65 7 70)(3 64 8 69)(4 63 9 68)(5 62 10 67)(11 74 16 79)(12 73 17 78)(13 72 18 77)(14 71 19 76)(15 80 20 75)(21 84 26 89)(22 83 27 88)(23 82 28 87)(24 81 29 86)(25 90 30 85)(31 94 36 99)(32 93 37 98)(33 92 38 97)(34 91 39 96)(35 100 40 95)(41 104 46 109)(42 103 47 108)(43 102 48 107)(44 101 49 106)(45 110 50 105)(51 114 56 119)(52 113 57 118)(53 112 58 117)(54 111 59 116)(55 120 60 115)

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

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

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

C6×Dic5 is a maximal subgroup of
D6⋊Dic5  D304C4  C30.Q8  Dic155C4  C6.Dic10  C158M4(2)  Dic3.D10  D5×C2×C12

48 conjugacy classes

class 1 2A2B2C3A3B4A4B4C4D5A5B6A···6F10A···10F12A···12H15A15B15C15D30A···30L
order1222334444556···610···1012···121515151530···30
size1111115555221···12···25···522222···2

48 irreducible representations

dim11111111222222
type++++-+
imageC1C2C2C3C4C6C6C12D5Dic5D10C3×D5C3×Dic5C6×D5
kernelC6×Dic5C3×Dic5C2×C30C2×Dic5C30Dic5C2×C10C10C2×C6C6C6C22C2C2
# reps12124428242484

Matrix representation of C6×Dic5 in GL4(𝔽61) generated by

1000
04700
00600
00060
,
60000
0100
00601
001644
,
11000
06000
00043
00440
G:=sub<GL(4,GF(61))| [1,0,0,0,0,47,0,0,0,0,60,0,0,0,0,60],[60,0,0,0,0,1,0,0,0,0,60,16,0,0,1,44],[11,0,0,0,0,60,0,0,0,0,0,44,0,0,43,0] >;

C6×Dic5 in GAP, Magma, Sage, TeX

C_6\times {\rm Dic}_5
% in TeX

G:=Group("C6xDic5");
// GroupNames label

G:=SmallGroup(120,19);
// by ID

G=gap.SmallGroup(120,19);
# by ID

G:=PCGroup([5,-2,-2,-3,-2,-5,60,2404]);
// Polycyclic

G:=Group<a,b,c|a^6=b^10=1,c^2=b^5,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C6×Dic5 in TeX

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