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G = Dic5.D6order 240 = 24·3·5

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D10.3D6, Dic5.5D6, C30.16C23, D30.3C22, Dic3.12D10, Dic15.5C22, C15⋊Q84C2, C5⋊D43S3, C34(C4○D20), C158(C4○D4), C3⋊D204C2, C157D43C2, (C2×C6).1D10, C53(D42S3), D30.C23C2, (C2×Dic3)⋊3D5, (D5×Dic3)⋊3C2, (C2×C10).13D6, C22.1(S3×D5), (C10×Dic3)⋊4C2, (C6×D5).3C22, C6.16(C22×D5), C10.16(C22×S3), (C2×C30).10C22, (C3×Dic5).5C22, (C5×Dic3).11C22, C2.18(C2×S3×D5), (C3×C5⋊D4)⋊1C2, SmallGroup(240,140)

Series: Derived Chief Lower central Upper central

C1C30 — Dic5.D6
C1C5C15C30C6×D5D5×Dic3 — Dic5.D6
C15C30 — Dic5.D6
C1C2C22

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)])

Dic5.D6 is a maximal subgroup of
D20.38D6  S3×C4○D20  C15⋊2- 1+4  D5×D42S3  D30.C23  D2014D6  C15⋊2+ 1+4
Dic5.D6 is a maximal quotient of
Dic35Dic10  Dic5.2Dic6  Dic15.Q8  C4⋊Dic3⋊D5  Dic3.D20  D30.D4  (C2×C12).D10  (C4×Dic3)⋊D5  Dic3.3Dic10  C10.D4⋊S3  (D5×Dic3)⋊C4  Dic34D20  D30.23(C2×C4)  D10.16D12  D302Q8  Dic15.D4  Dic15.19D4  C23.26(S3×D5)  C23.13(S3×D5)  C23.14(S3×D5)  D306D4  C6.(D4×D5)  (C2×C30).D4  C10.(C2×D12)  C23.17(S3×D5)  Dic153D4  Dic3×C5⋊D4  C1528(C4×D4)  D30.16D4  (C2×C6)⋊D20  (C2×C10)⋊8Dic6

39 conjugacy classes

class 1 2A2B2C2D 3 4A4B4C4D4E5A5B6A6B6C10A···10F 12 15A15B20A···20H30A···30F
order122223444445566610···1012151520···2030···30
size1121030233610302224202···220446···64···4

39 irreducible representations

dim111111112222222224444
type+++++++++++++++-++
imageC1C2C2C2C2C2C2C2S3D5D6D6D6C4○D4D10D10C4○D20D42S3S3×D5C2×S3×D5Dic5.D6
kernelDic5.D6D5×Dic3D30.C2C3⋊D20C15⋊Q8C3×C5⋊D4C10×Dic3C157D4C5⋊D4C2×Dic3Dic5D10C2×C10C15Dic3C2×C6C3C5C22C2C1
# reps111111111211124281224

Matrix representation of Dic5.D6 in GL6(𝔽61)

6000000
0600000
0044100
00166000
000010
000001
,
5000000
27110000
00444300
00161700
0000600
0000060
,
11180000
34500000
00171800
00454400
00006060
000010
,
60150000
810000
00444300
00161700
000011
0000060

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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