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G = D6.Dic5order 240 = 24·3·5

The non-split extension by D6 of 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 — C60 — C3×C5⋊2C8 — D6.Dic5
 Lower central C15 — C30 — D6.Dic5
 Upper central C1 — C4

Generators and relations for D6.Dic5
G = < a,b,c,d | a6=b2=1, c10=a3, d2=c5, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a3b, dcd-1=c9 >

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

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

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

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

D6.Dic5 is a maximal subgroup of
D5×C8⋊S3  C40⋊D6  C40.54D6  C40.55D6  D12.2Dic5  S3×C4.Dic5  D12.Dic5  D60.C22  C60.10C23  D2010D6  D12.9D10  D12⋊D10  Dic10.26D6  D20.28D6  C60.44C23
D6.Dic5 is a maximal quotient of
C30.22C42  C60.94D4  C60.15Q8

42 conjugacy classes

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

42 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + - + - + - image C1 C2 C2 C2 C4 C4 S3 D5 D6 M4(2) Dic5 D10 Dic5 C4×S3 C8⋊S3 C4.Dic5 S3×D5 S3×Dic5 D6.Dic5 kernel D6.Dic5 C3×C5⋊2C8 C15⋊3C8 S3×C20 C5×Dic3 S3×C10 C5⋊2C8 C4×S3 C20 C15 Dic3 C12 D6 C10 C5 C3 C4 C2 C1 # reps 1 1 1 1 2 2 1 2 1 2 2 2 2 2 4 8 2 2 4

Matrix representation of D6.Dic5 in GL4(𝔽241) generated by

 1 1 0 0 240 0 0 0 0 0 240 0 0 0 0 240
,
 1 1 0 0 0 240 0 0 0 0 240 0 0 0 208 1
,
 64 0 0 0 0 64 0 0 0 0 40 0 0 0 79 6
,
 78 156 0 0 85 163 0 0 0 0 94 228 0 0 63 147
G:=sub<GL(4,GF(241))| [1,240,0,0,1,0,0,0,0,0,240,0,0,0,0,240],[1,0,0,0,1,240,0,0,0,0,240,208,0,0,0,1],[64,0,0,0,0,64,0,0,0,0,40,79,0,0,0,6],[78,85,0,0,156,163,0,0,0,0,94,63,0,0,228,147] >;

D6.Dic5 in GAP, Magma, Sage, TeX

D_6.{\rm Dic}_5
% in TeX

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

G:=SmallGroup(240,11);
// by ID

G=gap.SmallGroup(240,11);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,24,121,50,490,6917]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^2=1,c^10=a^3,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^9>;
// generators/relations

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