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## G = D6⋊Dic5order 240 = 24·3·5

### The semidirect product of D6 and Dic5 acting via Dic5/C10=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — D6⋊Dic5
 Chief series C1 — C5 — C15 — C30 — C2×C30 — C6×Dic5 — D6⋊Dic5
 Lower central C15 — C30 — D6⋊Dic5
 Upper central C1 — C22

Generators and relations for D6⋊Dic5
G = < a,b,c,d | a6=b2=c10=1, d2=c5, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a3b, dcd-1=c-1 >

Subgroups: 240 in 68 conjugacy classes, 30 normal (26 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C22, C22 [×4], C5, S3 [×2], C6 [×3], C2×C4 [×2], C23, C10 [×3], C10 [×2], Dic3, C12, D6 [×2], D6 [×2], C2×C6, C15, C22⋊C4, Dic5 [×2], C2×C10, C2×C10 [×4], C2×Dic3, C2×C12, C22×S3, C5×S3 [×2], C30 [×3], C2×Dic5, C2×Dic5, C22×C10, D6⋊C4, C3×Dic5, Dic15, S3×C10 [×2], S3×C10 [×2], C2×C30, C23.D5, C6×Dic5, C2×Dic15, S3×C2×C10, D6⋊Dic5
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D5, D6, C22⋊C4, Dic5 [×2], D10, C4×S3, D12, C3⋊D4, C2×Dic5, C5⋊D4 [×2], D6⋊C4, S3×D5, C23.D5, S3×Dic5, C15⋊D4, C5⋊D12, D6⋊Dic5

Smallest permutation representation of D6⋊Dic5
On 120 points
Generators in S120
(1 14 41 59 28 65)(2 15 42 60 29 66)(3 16 43 51 30 67)(4 17 44 52 21 68)(5 18 45 53 22 69)(6 19 46 54 23 70)(7 20 47 55 24 61)(8 11 48 56 25 62)(9 12 49 57 26 63)(10 13 50 58 27 64)(31 80 99 104 116 82)(32 71 100 105 117 83)(33 72 91 106 118 84)(34 73 92 107 119 85)(35 74 93 108 120 86)(36 75 94 109 111 87)(37 76 95 110 112 88)(38 77 96 101 113 89)(39 78 97 102 114 90)(40 79 98 103 115 81)
(1 70)(2 61)(3 62)(4 63)(5 64)(6 65)(7 66)(8 67)(9 68)(10 69)(11 30)(12 21)(13 22)(14 23)(15 24)(16 25)(17 26)(18 27)(19 28)(20 29)(31 111)(32 112)(33 113)(34 114)(35 115)(36 116)(37 117)(38 118)(39 119)(40 120)(41 54)(42 55)(43 56)(44 57)(45 58)(46 59)(47 60)(48 51)(49 52)(50 53)(71 110)(72 101)(73 102)(74 103)(75 104)(76 105)(77 106)(78 107)(79 108)(80 109)(81 86)(82 87)(83 88)(84 89)(85 90)(91 96)(92 97)(93 98)(94 99)(95 100)
(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 80 6 75)(2 79 7 74)(3 78 8 73)(4 77 9 72)(5 76 10 71)(11 92 16 97)(12 91 17 96)(13 100 18 95)(14 99 19 94)(15 98 20 93)(21 89 26 84)(22 88 27 83)(23 87 28 82)(24 86 29 81)(25 85 30 90)(31 70 36 65)(32 69 37 64)(33 68 38 63)(34 67 39 62)(35 66 40 61)(41 104 46 109)(42 103 47 108)(43 102 48 107)(44 101 49 106)(45 110 50 105)(51 114 56 119)(52 113 57 118)(53 112 58 117)(54 111 59 116)(55 120 60 115)

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 5A 5B 6A 6B 6C 10A ··· 10F 10G ··· 10N 12A 12B 12C 12D 15A 15B 30A ··· 30F order 1 2 2 2 2 2 3 4 4 4 4 5 5 6 6 6 10 ··· 10 10 ··· 10 12 12 12 12 15 15 30 ··· 30 size 1 1 1 1 6 6 2 10 10 30 30 2 2 2 2 2 2 ··· 2 6 ··· 6 10 10 10 10 4 4 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 D6 Dic5 D10 C4×S3 D12 C3⋊D4 C5⋊D4 S3×D5 S3×Dic5 C15⋊D4 C5⋊D12 kernel D6⋊Dic5 C6×Dic5 C2×Dic15 S3×C2×C10 S3×C10 C2×Dic5 C30 C22×S3 C2×C10 D6 C2×C6 C10 C10 C10 C6 C22 C2 C2 C2 # reps 1 1 1 1 4 1 2 2 1 4 2 2 2 2 8 2 2 2 2

Matrix representation of D6⋊Dic5 in GL4(𝔽61) generated by

 60 60 0 0 1 0 0 0 0 0 60 0 0 0 0 60
,
 1 1 0 0 0 60 0 0 0 0 1 0 0 0 14 60
,
 60 0 0 0 0 60 0 0 0 0 41 0 0 0 22 3
,
 50 0 0 0 0 50 0 0 0 0 50 19 0 0 0 11
G:=sub<GL(4,GF(61))| [60,1,0,0,60,0,0,0,0,0,60,0,0,0,0,60],[1,0,0,0,1,60,0,0,0,0,1,14,0,0,0,60],[60,0,0,0,0,60,0,0,0,0,41,22,0,0,0,3],[50,0,0,0,0,50,0,0,0,0,50,0,0,0,19,11] >;

D6⋊Dic5 in GAP, Magma, Sage, TeX

D_6\rtimes {\rm Dic}_5
% in TeX

G:=Group("D6:Dic5");
// GroupNames label

G:=SmallGroup(240,27);
// by ID

G=gap.SmallGroup(240,27);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,24,121,490,6917]);
// Polycyclic

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

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