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## G = D10⋊Dic3order 240 = 24·3·5

### 1st semidirect product of D10 and Dic3 acting via Dic3/C6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — D10⋊Dic3
 Chief series C1 — C5 — C15 — C30 — C2×C30 — D5×C2×C6 — D10⋊Dic3
 Lower central C15 — C30 — D10⋊Dic3
 Upper central C1 — C22

Generators and relations for D10⋊Dic3
G = < a,b,c,d | a10=b2=c6=1, d2=c3, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a5b, dcd-1=c-1 >

Subgroups: 272 in 68 conjugacy classes, 30 normal (26 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C22, C22 [×4], C5, C6 [×3], C6 [×2], C2×C4 [×2], C23, D5 [×2], C10 [×3], Dic3 [×2], C2×C6, C2×C6 [×4], C15, C22⋊C4, Dic5, C20, D10 [×2], D10 [×2], C2×C10, C2×Dic3, C2×Dic3, C22×C6, C3×D5 [×2], C30 [×3], C2×Dic5, C2×C20, C22×D5, C6.D4, C5×Dic3, Dic15, C6×D5 [×2], C6×D5 [×2], C2×C30, D10⋊C4, C10×Dic3, C2×Dic15, D5×C2×C6, D10⋊Dic3
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D5, Dic3 [×2], D6, C22⋊C4, D10, C2×Dic3, C3⋊D4 [×2], C4×D5, D20, C5⋊D4, C6.D4, S3×D5, D10⋊C4, D5×Dic3, C15⋊D4, C3⋊D20, D10⋊Dic3

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

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

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

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

42 conjugacy classes

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

42 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + - + + + + - - + image C1 C2 C2 C2 C4 S3 D4 D5 Dic3 D6 D10 C3⋊D4 C4×D5 D20 C5⋊D4 S3×D5 D5×Dic3 C15⋊D4 C3⋊D20 kernel D10⋊Dic3 C10×Dic3 C2×Dic15 D5×C2×C6 C6×D5 C22×D5 C30 C2×Dic3 D10 C2×C10 C2×C6 C10 C6 C6 C6 C22 C2 C2 C2 # reps 1 1 1 1 4 1 2 2 2 1 2 4 4 4 4 2 2 2 2

Matrix representation of D10⋊Dic3 in GL4(𝔽61) generated by

 17 17 0 0 44 1 0 0 0 0 60 0 0 0 0 60
,
 44 44 0 0 60 17 0 0 0 0 60 0 0 0 23 1
,
 60 0 0 0 0 60 0 0 0 0 13 0 0 0 25 47
,
 29 54 0 0 7 32 0 0 0 0 27 5 0 0 13 34
G:=sub<GL(4,GF(61))| [17,44,0,0,17,1,0,0,0,0,60,0,0,0,0,60],[44,60,0,0,44,17,0,0,0,0,60,23,0,0,0,1],[60,0,0,0,0,60,0,0,0,0,13,25,0,0,0,47],[29,7,0,0,54,32,0,0,0,0,27,13,0,0,5,34] >;

D10⋊Dic3 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(240,26);
// by ID

G=gap.SmallGroup(240,26);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,121,31,490,6917]);
// Polycyclic

G:=Group<a,b,c,d|a^10=b^2=c^6=1,d^2=c^3,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^5*b,d*c*d^-1=c^-1>;
// generators/relations

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