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## G = C3⋊Dic30order 360 = 23·32·5

### The semidirect product of C3 and Dic30 acting via Dic30/Dic15=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C30 — C3⋊Dic30
 Chief series C1 — C5 — C15 — C3×C15 — C3×C30 — C3×Dic15 — C3⋊Dic30
 Lower central C3×C15 — C3×C30 — C3⋊Dic30
 Upper central C1 — C2

Generators and relations for C3⋊Dic30
G = < a,b,c | a3=b60=1, c2=b30, bab-1=a-1, ac=ca, cbc-1=b-1 >

Subgroups: 316 in 54 conjugacy classes, 24 normal (all characteristic)
C1, C2, C3, C3, C4, C5, C6, C6, Q8, C32, C10, Dic3, Dic3, C12, C15, C15, C3×C6, Dic5, C20, Dic6, C30, C30, C3×Dic3, C3×Dic3, C3⋊Dic3, Dic10, C3×C15, C5×Dic3, C3×Dic5, Dic15, Dic15, C60, C322Q8, C3×C30, C15⋊Q8, Dic30, Dic3×C15, C3×Dic15, C3⋊Dic15, C3⋊Dic30
Quotients: C1, C2, C22, S3, Q8, D5, D6, D10, Dic6, D15, S32, Dic10, S3×D5, D30, C322Q8, C15⋊Q8, Dic30, S3×D15, C3⋊Dic30

Smallest permutation representation of C3⋊Dic30
On 120 points
Generators in S120
(1 21 41)(2 42 22)(3 23 43)(4 44 24)(5 25 45)(6 46 26)(7 27 47)(8 48 28)(9 29 49)(10 50 30)(11 31 51)(12 52 32)(13 33 53)(14 54 34)(15 35 55)(16 56 36)(17 37 57)(18 58 38)(19 39 59)(20 60 40)(61 101 81)(62 82 102)(63 103 83)(64 84 104)(65 105 85)(66 86 106)(67 107 87)(68 88 108)(69 109 89)(70 90 110)(71 111 91)(72 92 112)(73 113 93)(74 94 114)(75 115 95)(76 96 116)(77 117 97)(78 98 118)(79 119 99)(80 100 120)
(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 91 31 61)(2 90 32 120)(3 89 33 119)(4 88 34 118)(5 87 35 117)(6 86 36 116)(7 85 37 115)(8 84 38 114)(9 83 39 113)(10 82 40 112)(11 81 41 111)(12 80 42 110)(13 79 43 109)(14 78 44 108)(15 77 45 107)(16 76 46 106)(17 75 47 105)(18 74 48 104)(19 73 49 103)(20 72 50 102)(21 71 51 101)(22 70 52 100)(23 69 53 99)(24 68 54 98)(25 67 55 97)(26 66 56 96)(27 65 57 95)(28 64 58 94)(29 63 59 93)(30 62 60 92)

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

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

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

51 conjugacy classes

 class 1 2 3A 3B 3C 4A 4B 4C 5A 5B 6A 6B 6C 10A 10B 12A 12B 12C 12D 15A 15B 15C 15D 15E ··· 15J 20A 20B 20C 20D 30A 30B 30C 30D 30E ··· 30J 60A ··· 60H order 1 2 3 3 3 4 4 4 5 5 6 6 6 10 10 12 12 12 12 15 15 15 15 15 ··· 15 20 20 20 20 30 30 30 30 30 ··· 30 60 ··· 60 size 1 1 2 2 4 6 30 90 2 2 2 2 4 2 2 6 6 30 30 2 2 2 2 4 ··· 4 6 6 6 6 2 2 2 2 4 ··· 4 6 ··· 6

51 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + - + + + - + - + - + + - - + - image C1 C2 C2 C2 S3 S3 Q8 D5 D6 D10 Dic6 D15 Dic10 D30 Dic30 S32 S3×D5 C32⋊2Q8 C15⋊Q8 S3×D15 C3⋊Dic30 kernel C3⋊Dic30 Dic3×C15 C3×Dic15 C3⋊Dic15 C5×Dic3 Dic15 C3×C15 C3×Dic3 C30 C3×C6 C15 Dic3 C32 C6 C3 C10 C6 C5 C3 C2 C1 # reps 1 1 1 1 1 1 1 2 2 2 4 4 4 4 8 1 2 1 2 4 4

Matrix representation of C3⋊Dic30 in GL4(𝔽61) generated by

 0 60 0 0 1 60 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 1 0 0 0 0 0 36 8 0 0 53 44
,
 60 0 0 0 0 60 0 0 0 0 40 23 0 0 2 21
G:=sub<GL(4,GF(61))| [0,1,0,0,60,60,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,36,53,0,0,8,44],[60,0,0,0,0,60,0,0,0,0,40,2,0,0,23,21] >;

C3⋊Dic30 in GAP, Magma, Sage, TeX

C_3\rtimes {\rm Dic}_{30}
% in TeX

G:=Group("C3:Dic30");
// GroupNames label

G:=SmallGroup(360,83);
// by ID

G=gap.SmallGroup(360,83);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-5,24,73,31,201,1444,10373]);
// Polycyclic

G:=Group<a,b,c|a^3=b^60=1,c^2=b^30,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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