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G = Dic30order 120 = 23·3·5

Dicyclic group

metacyclic, supersoluble, monomial, 2-hyperelementary

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

C1C30 — Dic30
C1C5C15C30Dic15 — Dic30
C15C30 — Dic30
C1C2C4

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

15C4
15C4
15Q8
5Dic3
5Dic3
3Dic5
3Dic5
5Dic6
3Dic10

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 4A4B4C5A5B 6 10A10B12A12B15A15B15C15D20A20B20C20D30A30B30C30D60A···60H
order1234445561010121215151515202020203030303060···60
size1122303022222222222222222222···2

33 irreducible representations

dim1112222222222
type++++-+++-+-+-
imageC1C2C2S3Q8D5D6D10Dic6D15Dic10D30Dic30
kernelDic30Dic15C60C20C15C12C10C6C5C4C3C2C1
# reps1211121224448

Matrix representation of Dic30 in GL4(𝔽61) generated by

272500
36400
002353
00845
,
555200
38600
00845
002353
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

Export

Subgroup lattice of Dic30 in TeX

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