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## G = Dic5.D6order 240 = 24·3·5

### 5th non-split extension by Dic5 of D6 acting via D6/S3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — Dic5.D6
 Chief series C1 — C5 — C15 — C30 — C6×D5 — D5×Dic3 — Dic5.D6
 Lower central C15 — C30 — Dic5.D6
 Upper central C1 — C2 — C22

Generators and relations for Dic5.D6
G = < a,b,c,d | a10=c6=1, b2=d2=a5, bab-1=cac-1=dad-1=a-1, cbc-1=dbd-1=a5b, dcd-1=c-1 >

Subgroups: 344 in 80 conjugacy classes, 32 normal (all characteristic)
C1, C2, C2 [×3], C3, C4 [×4], C22, C22 [×2], C5, S3, C6, C6 [×2], C2×C4 [×3], D4 [×3], Q8, D5 [×2], C10, C10, Dic3 [×2], Dic3, C12, D6, C2×C6, C2×C6, C15, C4○D4, Dic5, Dic5, C20 [×2], D10, D10, C2×C10, Dic6, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4 [×2], C3×D4, C3×D5, D15, C30, C30, Dic10, C4×D5 [×2], D20, C5⋊D4, C5⋊D4, C2×C20, D42S3, C5×Dic3 [×2], C3×Dic5, Dic15, C6×D5, D30, C2×C30, C4○D20, D5×Dic3, D30.C2, C3⋊D20, C15⋊Q8, C3×C5⋊D4, C10×Dic3, C157D4, Dic5.D6
Quotients: C1, C2 [×7], C22 [×7], S3, C23, D5, D6 [×3], C4○D4, D10 [×3], C22×S3, C22×D5, D42S3, S3×D5, C4○D20, C2×S3×D5, Dic5.D6

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

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

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

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

39 conjugacy classes

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

39 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + - + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 D5 D6 D6 D6 C4○D4 D10 D10 C4○D20 D4⋊2S3 S3×D5 C2×S3×D5 Dic5.D6 kernel Dic5.D6 D5×Dic3 D30.C2 C3⋊D20 C15⋊Q8 C3×C5⋊D4 C10×Dic3 C15⋊7D4 C5⋊D4 C2×Dic3 Dic5 D10 C2×C10 C15 Dic3 C2×C6 C3 C5 C22 C2 C1 # reps 1 1 1 1 1 1 1 1 1 2 1 1 1 2 4 2 8 1 2 2 4

Matrix representation of Dic5.D6 in GL6(𝔽61)

 60 0 0 0 0 0 0 60 0 0 0 0 0 0 44 1 0 0 0 0 16 60 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 50 0 0 0 0 0 27 11 0 0 0 0 0 0 44 43 0 0 0 0 16 17 0 0 0 0 0 0 60 0 0 0 0 0 0 60
,
 11 18 0 0 0 0 34 50 0 0 0 0 0 0 17 18 0 0 0 0 45 44 0 0 0 0 0 0 60 60 0 0 0 0 1 0
,
 60 15 0 0 0 0 8 1 0 0 0 0 0 0 44 43 0 0 0 0 16 17 0 0 0 0 0 0 1 1 0 0 0 0 0 60

G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,44,16,0,0,0,0,1,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[50,27,0,0,0,0,0,11,0,0,0,0,0,0,44,16,0,0,0,0,43,17,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[11,34,0,0,0,0,18,50,0,0,0,0,0,0,17,45,0,0,0,0,18,44,0,0,0,0,0,0,60,1,0,0,0,0,60,0],[60,8,0,0,0,0,15,1,0,0,0,0,0,0,44,16,0,0,0,0,43,17,0,0,0,0,0,0,1,0,0,0,0,0,1,60] >;

Dic5.D6 in GAP, Magma, Sage, TeX

{\rm Dic}_5.D_6
% in TeX

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

G:=SmallGroup(240,140);
// by ID

G=gap.SmallGroup(240,140);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,55,218,490,6917]);
// Polycyclic

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

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