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G = Dic3.D10order 240 = 24·3·5

6th non-split extension by Dic3 of D10 acting via D10/D5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D6.4D10, C30.19C23, Dic5.13D6, Dic3.6D10, D30.4C22, Dic15.6C22, C15⋊Q86C2, C3⋊D43D5, C54(C4○D12), C5⋊D124C2, C157D44C2, (C2×C10).2D6, C1510(C4○D4), C33(D42D5), D30.C24C2, (C6×Dic5)⋊4C2, (S3×Dic5)⋊4C2, (C2×Dic5)⋊3S3, (C2×C6).14D10, C22.2(S3×D5), C6.19(C22×D5), (S3×C10).4C22, C10.19(C22×S3), (C2×C30).13C22, (C5×Dic3).6C22, (C3×Dic5).15C22, C2.20(C2×S3×D5), (C5×C3⋊D4)⋊2C2, SmallGroup(240,143)

Series: Derived Chief Lower central Upper central

C1C30 — Dic3.D10
C1C5C15C30C3×Dic5S3×Dic5 — Dic3.D10
C15C30 — Dic3.D10
C1C2C22

Generators and relations for Dic3.D10
 G = < a,b,c,d | a6=c10=1, b2=d2=a3, bab-1=cac-1=dad-1=a-1, cbc-1=dbd-1=a3b, dcd-1=c-1 >

Subgroups: 336 in 80 conjugacy classes, 32 normal (all characteristic)
C1, C2, C2 [×3], C3, C4 [×4], C22, C22 [×2], C5, S3 [×2], C6, C6, C2×C4 [×3], D4 [×3], Q8, D5, C10, C10 [×2], Dic3, Dic3, C12 [×2], D6, D6, C2×C6, C15, C4○D4, Dic5 [×2], Dic5, C20, D10, C2×C10, C2×C10, Dic6, C4×S3 [×2], D12, C3⋊D4, C3⋊D4, C2×C12, C5×S3, D15, C30, C30, Dic10, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4 [×2], C5×D4, C4○D12, C5×Dic3, C3×Dic5 [×2], Dic15, S3×C10, D30, C2×C30, D42D5, S3×Dic5, D30.C2, C5⋊D12, C15⋊Q8, C6×Dic5, C5×C3⋊D4, C157D4, Dic3.D10
Quotients: C1, C2 [×7], C22 [×7], S3, C23, D5, D6 [×3], C4○D4, D10 [×3], C22×S3, C22×D5, C4○D12, S3×D5, D42D5, C2×S3×D5, Dic3.D10

Smallest permutation representation of Dic3.D10
On 120 points
Generators in S120
(1 103 93 50 75 22)(2 23 76 41 94 104)(3 105 95 42 77 24)(4 25 78 43 96 106)(5 107 97 44 79 26)(6 27 80 45 98 108)(7 109 99 46 71 28)(8 29 72 47 100 110)(9 101 91 48 73 30)(10 21 74 49 92 102)(11 31 112 59 65 84)(12 85 66 60 113 32)(13 33 114 51 67 86)(14 87 68 52 115 34)(15 35 116 53 69 88)(16 89 70 54 117 36)(17 37 118 55 61 90)(18 81 62 56 119 38)(19 39 120 57 63 82)(20 83 64 58 111 40)
(1 118 50 90)(2 81 41 119)(3 120 42 82)(4 83 43 111)(5 112 44 84)(6 85 45 113)(7 114 46 86)(8 87 47 115)(9 116 48 88)(10 89 49 117)(11 26 59 97)(12 98 60 27)(13 28 51 99)(14 100 52 29)(15 30 53 91)(16 92 54 21)(17 22 55 93)(18 94 56 23)(19 24 57 95)(20 96 58 25)(31 79 65 107)(32 108 66 80)(33 71 67 109)(34 110 68 72)(35 73 69 101)(36 102 70 74)(37 75 61 103)(38 104 62 76)(39 77 63 105)(40 106 64 78)
(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 89 50 117)(2 88 41 116)(3 87 42 115)(4 86 43 114)(5 85 44 113)(6 84 45 112)(7 83 46 111)(8 82 47 120)(9 81 48 119)(10 90 49 118)(11 80 59 108)(12 79 60 107)(13 78 51 106)(14 77 52 105)(15 76 53 104)(16 75 54 103)(17 74 55 102)(18 73 56 101)(19 72 57 110)(20 71 58 109)(21 61 92 37)(22 70 93 36)(23 69 94 35)(24 68 95 34)(25 67 96 33)(26 66 97 32)(27 65 98 31)(28 64 99 40)(29 63 100 39)(30 62 91 38)

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

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

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

Dic3.D10 is a maximal subgroup of
C5⋊C8.D6  D15⋊C8⋊C2  C30.C24  D5×C4○D12  C15⋊2- 1+4  S3×D42D5  D30.C23  D1214D10  C15⋊2+ 1+4
Dic3.D10 is a maximal quotient of
Dic55Dic6  Dic15.2Q8  D6⋊C4.D5  C4⋊Dic5⋊S3  Dic3.2Dic10  Dic5.8D12  C5⋊(C423S3)  Dic5.7Dic6  (C4×Dic5)⋊S3  D6.(C4×D5)  D30.C2⋊C4  Dic54D12  D6.D20  D304Q8  D64Dic10  D30.7D4  C23.D5⋊S3  Dic15.19D4  (C6×Dic5)⋊7C4  C23.13(S3×D5)  C23.14(S3×D5)  C30.(C2×D4)  C6.D4⋊D5  Dic5×C3⋊D4  C1526(C4×D4)  (S3×C10).D4  D307D4  Dic154D4  D30.16D4  (C2×C10)⋊4D12  (C2×C30)⋊Q8

36 conjugacy classes

class 1 2A2B2C2D 3 4A4B4C4D4E5A5B6A6B6C10A10B10C10D10E10F12A12B12C12D15A15B20A20B30A···30F
order1222234444455666101010101010121212121515202030···30
size112630255610302222222441212101010104412124···4

36 irreducible representations

dim111111112222222224444
type++++++++++++++++-+
imageC1C2C2C2C2C2C2C2S3D5D6D6C4○D4D10D10D10C4○D12S3×D5D42D5C2×S3×D5Dic3.D10
kernelDic3.D10S3×Dic5D30.C2C5⋊D12C15⋊Q8C6×Dic5C5×C3⋊D4C157D4C2×Dic5C3⋊D4Dic5C2×C10C15Dic3D6C2×C6C5C22C3C2C1
# reps111111111221222242224

Matrix representation of Dic3.D10 in GL6(𝔽61)

100000
010000
000100
00606000
0000600
0000060
,
100000
010000
001000
00606000
0000110
00003350
,
1430000
18430000
0060000
001100
00005048
00002811
,
100000
18600000
0060000
001100
00006021
0000581

G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,60,0,0,0,0,1,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,60,0,0,0,0,0,60,0,0,0,0,0,0,11,33,0,0,0,0,0,50],[1,18,0,0,0,0,43,43,0,0,0,0,0,0,60,1,0,0,0,0,0,1,0,0,0,0,0,0,50,28,0,0,0,0,48,11],[1,18,0,0,0,0,0,60,0,0,0,0,0,0,60,1,0,0,0,0,0,1,0,0,0,0,0,0,60,58,0,0,0,0,21,1] >;

Dic3.D10 in GAP, Magma, Sage, TeX

{\rm Dic}_3.D_{10}
% in TeX

G:=Group("Dic3.D10");
// GroupNames label

G:=SmallGroup(240,143);
// by ID

G=gap.SmallGroup(240,143);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,48,116,490,6917]);
// Polycyclic

G:=Group<a,b,c,d|a^6=c^10=1,b^2=d^2=a^3,b*a*b^-1=c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=a^3*b,d*c*d^-1=c^-1>;
// generators/relations

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