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## G = Dic3.D10order 240 = 24·3·5

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — Dic3.D10
 Chief series C1 — C5 — C15 — C30 — C3×Dic5 — S3×Dic5 — Dic3.D10
 Lower central C15 — C30 — Dic3.D10
 Upper central C1 — C2 — C22

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)])

36 conjugacy classes

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

36 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + - + image C1 C2 C2 C2 C2 C2 C2 C2 S3 D5 D6 D6 C4○D4 D10 D10 D10 C4○D12 S3×D5 D4⋊2D5 C2×S3×D5 Dic3.D10 kernel Dic3.D10 S3×Dic5 D30.C2 C5⋊D12 C15⋊Q8 C6×Dic5 C5×C3⋊D4 C15⋊7D4 C2×Dic5 C3⋊D4 Dic5 C2×C10 C15 Dic3 D6 C2×C6 C5 C22 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 1 2 2 1 2 2 2 2 4 2 2 2 4

Matrix representation of Dic3.D10 in GL6(𝔽61)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 60 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 0 0 0 0 0 60 60 0 0 0 0 0 0 11 0 0 0 0 0 33 50
,
 1 43 0 0 0 0 18 43 0 0 0 0 0 0 60 0 0 0 0 0 1 1 0 0 0 0 0 0 50 48 0 0 0 0 28 11
,
 1 0 0 0 0 0 18 60 0 0 0 0 0 0 60 0 0 0 0 0 1 1 0 0 0 0 0 0 60 21 0 0 0 0 58 1

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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