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G = Dic6⋊D5order 240 = 24·3·5

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

metabelian, supersoluble, monomial, 2-hyperelementary

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

C1C60 — Dic6⋊D5
C1C5C15C30C60C3×C52C8 — Dic6⋊D5
C15C30C60 — Dic6⋊D5
C1C2C4

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 >

60C2
6C4
30C22
20S3
12D5
3Q8
5C8
15D4
2Dic3
10D6
6D10
6C20
4D15
15SD16
5C24
5D12
3D20
3C5×Q8
2C5×Dic3
2D30
5C24⋊C2
3Q8⋊D5

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 2A2B 3 4A4B5A5B 6 8A8B10A10B12A12B15A15B20A20B20C20D20E20F24A24B24C24D30A30B60A60B60C60D
order1223445568810101212151520202020202024242424303060606060
size116022122221010222244441212121210101010444444

33 irreducible representations

dim11112222222224444
type++++++++++++++
imageC1C2C2C2S3D4D5D6SD16D10D12C5⋊D4C24⋊C2S3×D5Q8⋊D5C5⋊D12Dic6⋊D5
kernelDic6⋊D5C3×C52C8C5×Dic6D60C52C8C30Dic6C20C15C12C10C6C5C4C3C2C1
# reps11111121222442224

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

1000
0100
0043142
0099142
,
1000
0100
00213147
0017528
,
0100
24018900
0010
0001
,
0100
1000
0010
00240240
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

Export

Subgroup lattice of Dic6⋊D5 in TeX

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