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## G = Dic30order 120 = 23·3·5

### Dicyclic group

Aliases: Dic30, C4.D15, C152Q8, C52Dic6, C60.1C2, C20.1S3, C10.8D6, C12.1D5, C2.3D30, C6.8D10, C32Dic10, C30.8C22, Dic15.1C2, SmallGroup(120,26)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — Dic30
 Chief series C1 — C5 — C15 — C30 — Dic15 — Dic30
 Lower central C15 — C30 — Dic30
 Upper central C1 — C2 — C4

Generators and relations for Dic30
G = < a,b | a60=1, b2=a30, bab-1=a-1 >

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

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

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

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

Dic30 is a maximal subgroup of
C6.D20  D12.D5  C3⋊Dic20  C5⋊Dic12  C24⋊D5  Dic60  D4.D15  C157Q16  D5×Dic6  D205S3  S3×Dic10  D125D5  D6011C2  D42D15  Q8×D15  Dic90  C3⋊Dic30  C12.D15  C20.1S4  C20.2S4
Dic30 is a maximal quotient of
C30.4Q8  C605C4  Dic90  C3⋊Dic30  C12.D15  C20.1S4

33 conjugacy classes

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

33 irreducible representations

 dim 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + - + + + - + - + - image C1 C2 C2 S3 Q8 D5 D6 D10 Dic6 D15 Dic10 D30 Dic30 kernel Dic30 Dic15 C60 C20 C15 C12 C10 C6 C5 C4 C3 C2 C1 # reps 1 2 1 1 1 2 1 2 2 4 4 4 8

Matrix representation of Dic30 in GL4(𝔽61) generated by

 27 25 0 0 36 4 0 0 0 0 23 53 0 0 8 45
,
 55 52 0 0 38 6 0 0 0 0 8 45 0 0 23 53
G:=sub<GL(4,GF(61))| [27,36,0,0,25,4,0,0,0,0,23,8,0,0,53,45],[55,38,0,0,52,6,0,0,0,0,8,23,0,0,45,53] >;

Dic30 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(120,26);
// by ID

G=gap.SmallGroup(120,26);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-5,20,61,26,323,2404]);
// Polycyclic

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

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