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G = C6×Dic15order 360 = 23·32·5

Direct product of C6 and Dic15

Series: Derived Chief Lower central Upper central

 Derived series C1 — C15 — C6×Dic15
 Chief series C1 — C5 — C15 — C30 — C3×C30 — C3×Dic15 — C6×Dic15
 Lower central C15 — C6×Dic15
 Upper central C1 — C2×C6

Generators and relations for C6×Dic15
G = < a,b,c | a6=b30=1, c2=b15, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 204 in 74 conjugacy classes, 46 normal (26 characteristic)
C1, C2, C2, C3, C3, C4, C22, C5, C6, C6, C6, C2×C4, C32, C10, C10, Dic3, C12, C2×C6, C2×C6, C15, C15, C3×C6, C3×C6, Dic5, C2×C10, C2×Dic3, C2×C12, C30, C30, C30, C3×Dic3, C62, C2×Dic5, C3×C15, C3×Dic5, Dic15, C2×C30, C2×C30, C6×Dic3, C3×C30, C3×C30, C6×Dic5, C2×Dic15, C3×Dic15, C6×C30, C6×Dic15
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, D5, Dic3, C12, D6, C2×C6, C3×S3, Dic5, D10, C2×Dic3, C2×C12, C3×D5, D15, C3×Dic3, S3×C6, C2×Dic5, C3×Dic5, Dic15, C6×D5, D30, C6×Dic3, C3×D15, C6×Dic5, C2×Dic15, C3×Dic15, C6×D15, C6×Dic15

Smallest permutation representation of C6×Dic15
On 120 points
Generators in S120
(1 48 21 38 11 58)(2 49 22 39 12 59)(3 50 23 40 13 60)(4 51 24 41 14 31)(5 52 25 42 15 32)(6 53 26 43 16 33)(7 54 27 44 17 34)(8 55 28 45 18 35)(9 56 29 46 19 36)(10 57 30 47 20 37)(61 101 71 111 81 91)(62 102 72 112 82 92)(63 103 73 113 83 93)(64 104 74 114 84 94)(65 105 75 115 85 95)(66 106 76 116 86 96)(67 107 77 117 87 97)(68 108 78 118 88 98)(69 109 79 119 89 99)(70 110 80 120 90 100)
(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 86 16 71)(2 85 17 70)(3 84 18 69)(4 83 19 68)(5 82 20 67)(6 81 21 66)(7 80 22 65)(8 79 23 64)(9 78 24 63)(10 77 25 62)(11 76 26 61)(12 75 27 90)(13 74 28 89)(14 73 29 88)(15 72 30 87)(31 113 46 98)(32 112 47 97)(33 111 48 96)(34 110 49 95)(35 109 50 94)(36 108 51 93)(37 107 52 92)(38 106 53 91)(39 105 54 120)(40 104 55 119)(41 103 56 118)(42 102 57 117)(43 101 58 116)(44 100 59 115)(45 99 60 114)

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

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

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

108 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 4A 4B 4C 4D 5A 5B 6A ··· 6F 6G ··· 6O 10A ··· 10F 12A ··· 12H 15A ··· 15P 30A ··· 30AV order 1 2 2 2 3 3 3 3 3 4 4 4 4 5 5 6 ··· 6 6 ··· 6 10 ··· 10 12 ··· 12 15 ··· 15 30 ··· 30 size 1 1 1 1 1 1 2 2 2 15 15 15 15 2 2 1 ··· 1 2 ··· 2 2 ··· 2 15 ··· 15 2 ··· 2 2 ··· 2

108 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + - + - + + - + image C1 C2 C2 C3 C4 C6 C6 C12 S3 D5 Dic3 D6 C3×S3 Dic5 D10 C3×D5 D15 C3×Dic3 S3×C6 C3×Dic5 Dic15 C6×D5 D30 C3×D15 C3×Dic15 C6×D15 kernel C6×Dic15 C3×Dic15 C6×C30 C2×Dic15 C3×C30 Dic15 C2×C30 C30 C2×C30 C62 C30 C30 C2×C10 C3×C6 C3×C6 C2×C6 C2×C6 C10 C10 C6 C6 C6 C6 C22 C2 C2 # reps 1 2 1 2 4 4 2 8 1 2 2 1 2 4 2 4 4 4 2 8 8 4 4 8 16 8

Matrix representation of C6×Dic15 in GL3(𝔽61) generated by

 48 0 0 0 13 0 0 0 13
,
 1 0 0 0 45 0 0 0 19
,
 60 0 0 0 0 1 0 60 0
G:=sub<GL(3,GF(61))| [48,0,0,0,13,0,0,0,13],[1,0,0,0,45,0,0,0,19],[60,0,0,0,0,60,0,1,0] >;

C6×Dic15 in GAP, Magma, Sage, TeX

C_6\times {\rm Dic}_{15}
% in TeX

G:=Group("C6xDic15");
// GroupNames label

G:=SmallGroup(360,103);
// by ID

G=gap.SmallGroup(360,103);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-3,-5,72,1444,10373]);
// Polycyclic

G:=Group<a,b,c|a^6=b^30=1,c^2=b^15,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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