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## G = Dic6⋊D5order 240 = 24·3·5

### 1st semidirect product of Dic6 and D5 acting via D5/C5=C2

Aliases: C30.9D4, C20.6D6, C156SD16, Dic61D5, D60.4C2, C10.8D12, C12.24D10, C60.17C22, C52C83S3, C31(Q8⋊D5), C52(C24⋊C2), C4.10(S3×D5), (C5×Dic6)⋊1C2, C6.3(C5⋊D4), C2.6(C5⋊D12), (C3×C52C8)⋊3C2, SmallGroup(240,21)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — Dic6⋊D5
 Chief series C1 — C5 — C15 — C30 — C60 — C3×C5⋊2C8 — Dic6⋊D5
 Lower central C15 — C30 — C60 — Dic6⋊D5
 Upper central C1 — C2 — C4

Generators and relations for Dic6⋊D5
G = < a,b,c,d | a12=c5=d2=1, b2=a6, bab-1=dad=a-1, ac=ca, bc=cb, dbd=a3b, dcd=c-1 >

Smallest permutation representation of Dic6⋊D5
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 62 7 68)(2 61 8 67)(3 72 9 66)(4 71 10 65)(5 70 11 64)(6 69 12 63)(13 95 19 89)(14 94 20 88)(15 93 21 87)(16 92 22 86)(17 91 23 85)(18 90 24 96)(25 98 31 104)(26 97 32 103)(27 108 33 102)(28 107 34 101)(29 106 35 100)(30 105 36 99)(37 75 43 81)(38 74 44 80)(39 73 45 79)(40 84 46 78)(41 83 47 77)(42 82 48 76)(49 112 55 118)(50 111 56 117)(51 110 57 116)(52 109 58 115)(53 120 59 114)(54 119 60 113)
(1 86 102 60 46)(2 87 103 49 47)(3 88 104 50 48)(4 89 105 51 37)(5 90 106 52 38)(6 91 107 53 39)(7 92 108 54 40)(8 93 97 55 41)(9 94 98 56 42)(10 95 99 57 43)(11 96 100 58 44)(12 85 101 59 45)(13 36 110 75 71)(14 25 111 76 72)(15 26 112 77 61)(16 27 113 78 62)(17 28 114 79 63)(18 29 115 80 64)(19 30 116 81 65)(20 31 117 82 66)(21 32 118 83 67)(22 33 119 84 68)(23 34 120 73 69)(24 35 109 74 70)
(1 46)(2 45)(3 44)(4 43)(5 42)(6 41)(7 40)(8 39)(9 38)(10 37)(11 48)(12 47)(13 113)(14 112)(15 111)(16 110)(17 109)(18 120)(19 119)(20 118)(21 117)(22 116)(23 115)(24 114)(25 26)(27 36)(28 35)(29 34)(30 33)(31 32)(49 85)(50 96)(51 95)(52 94)(53 93)(54 92)(55 91)(56 90)(57 89)(58 88)(59 87)(60 86)(61 76)(62 75)(63 74)(64 73)(65 84)(66 83)(67 82)(68 81)(69 80)(70 79)(71 78)(72 77)(97 107)(98 106)(99 105)(100 104)(101 103)

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

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

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

Dic6⋊D5 is a maximal subgroup of
D5×C24⋊C2  C24⋊D10  C24.2D10  D120⋊C2  C20.60D12  C60.38D4  C20.D12  D60.C22  Dic10⋊D6  C60.19C23  Dic6⋊D10  S3×Q8⋊D5  C60.C23  C60.44C23  D20.16D6
Dic6⋊D5 is a maximal quotient of
D6015C4  C10.Dic12  C60.7Q8

33 conjugacy classes

 class 1 2A 2B 3 4A 4B 5A 5B 6 8A 8B 10A 10B 12A 12B 15A 15B 20A 20B 20C 20D 20E 20F 24A 24B 24C 24D 30A 30B 60A 60B 60C 60D order 1 2 2 3 4 4 5 5 6 8 8 10 10 12 12 15 15 20 20 20 20 20 20 24 24 24 24 30 30 60 60 60 60 size 1 1 60 2 2 12 2 2 2 10 10 2 2 2 2 4 4 4 4 12 12 12 12 10 10 10 10 4 4 4 4 4 4

33 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + image C1 C2 C2 C2 S3 D4 D5 D6 SD16 D10 D12 C5⋊D4 C24⋊C2 S3×D5 Q8⋊D5 C5⋊D12 Dic6⋊D5 kernel Dic6⋊D5 C3×C5⋊2C8 C5×Dic6 D60 C5⋊2C8 C30 Dic6 C20 C15 C12 C10 C6 C5 C4 C3 C2 C1 # reps 1 1 1 1 1 1 2 1 2 2 2 4 4 2 2 2 4

Matrix representation of Dic6⋊D5 in GL4(𝔽241) generated by

 1 0 0 0 0 1 0 0 0 0 43 142 0 0 99 142
,
 1 0 0 0 0 1 0 0 0 0 213 147 0 0 175 28
,
 0 1 0 0 240 189 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 1 0 0 0 0 0 1 0 0 0 240 240
G:=sub<GL(4,GF(241))| [1,0,0,0,0,1,0,0,0,0,43,99,0,0,142,142],[1,0,0,0,0,1,0,0,0,0,213,175,0,0,147,28],[0,240,0,0,1,189,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,240,0,0,0,240] >;

Dic6⋊D5 in GAP, Magma, Sage, TeX

{\rm Dic}_6\rtimes D_5
% in TeX

G:=Group("Dic6:D5");
// GroupNames label

G:=SmallGroup(240,21);
// by ID

G=gap.SmallGroup(240,21);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,24,169,55,116,50,490,6917]);
// Polycyclic

G:=Group<a,b,c,d|a^12=c^5=d^2=1,b^2=a^6,b*a*b^-1=d*a*d=a^-1,a*c=c*a,b*c=c*b,d*b*d=a^3*b,d*c*d=c^-1>;
// generators/relations

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