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G = D6⋊Dic5order 240 = 24·3·5

The semidirect product of D6 and Dic5 acting via Dic5/C10=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D6⋊Dic5, C30.14D4, C10.10D12, C54(D6⋊C4), (S3×C10)⋊3C4, (C2×C6).7D10, (C2×C10).7D6, (C22×S3).D5, C10.20(C4×S3), C155(C22⋊C4), C30.30(C2×C4), (C2×Dic5)⋊1S3, (C6×Dic5)⋊1C2, C2.4(S3×Dic5), C6.4(C2×Dic5), C31(C23.D5), C22.6(S3×D5), (C2×Dic15)⋊6C2, C6.12(C5⋊D4), C2.2(C15⋊D4), C2.1(C5⋊D12), (C2×C30).4C22, C10.12(C3⋊D4), (S3×C2×C10).1C2, SmallGroup(240,27)

Series: Derived Chief Lower central Upper central

C1C30 — D6⋊Dic5
C1C5C15C30C2×C30C6×Dic5 — D6⋊Dic5
C15C30 — D6⋊Dic5
C1C22

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

Subgroups: 240 in 68 conjugacy classes, 30 normal (26 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C22, C22 [×4], C5, S3 [×2], C6 [×3], C2×C4 [×2], C23, C10 [×3], C10 [×2], Dic3, C12, D6 [×2], D6 [×2], C2×C6, C15, C22⋊C4, Dic5 [×2], C2×C10, C2×C10 [×4], C2×Dic3, C2×C12, C22×S3, C5×S3 [×2], C30 [×3], C2×Dic5, C2×Dic5, C22×C10, D6⋊C4, C3×Dic5, Dic15, S3×C10 [×2], S3×C10 [×2], C2×C30, C23.D5, C6×Dic5, C2×Dic15, S3×C2×C10, D6⋊Dic5
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D5, D6, C22⋊C4, Dic5 [×2], D10, C4×S3, D12, C3⋊D4, C2×Dic5, C5⋊D4 [×2], D6⋊C4, S3×D5, C23.D5, S3×Dic5, C15⋊D4, C5⋊D12, D6⋊Dic5

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

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

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

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

D6⋊Dic5 is a maximal subgroup of
(C2×C20).D6  (S3×C20)⋊5C4  D6⋊C4.D5  C605C4⋊C2  Dic5.8D12  D6⋊Dic5⋊C2  D6⋊Dic10  D30.35D4  C60.45D4  D6⋊Dic5.C2  C60.46D4  C60.89D4  (S3×C20)⋊7C4  C5⋊(C423S3)  C60.69D4  D6.(C4×D5)  (S3×Dic5)⋊C4  D61Dic10  D10.17D12  Dic5×D12  Dic5⋊D12  D62Dic10  (C2×D12).D5  D63Dic10  Dic158D4  D64Dic10  D30.7D4  C4×C15⋊D4  C1517(C4×D4)  C4×C5⋊D12  C1522(C4×D4)  D6⋊C4⋊D5  C60⋊D4  D10⋊C4⋊S3  Dic152D4  Dic15.10D4  Dic15.31D4  C202D12  D5×D6⋊C4  D30.27D4  D304D4  C23.D5⋊S3  C30.(C2×D4)  (C2×C10).D12  (C6×D5)⋊D4  Dic5×C3⋊D4  S3×C23.D5  (S3×C10).D4  D307D4  Dic1517D4  (C2×C10)⋊4D12  C15⋊C22≀C2  (C2×C10)⋊11D12  Dic1518D4  D308D4
D6⋊Dic5 is a maximal quotient of
C60.94D4  C20.5D12  C60.54D4  D12⋊Dic5  C10.D24  Dic6⋊Dic5  C10.Dic12  C60.98D4  C60.99D4  C30.24C42  C158(C23⋊C4)

42 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D5A5B6A6B6C10A···10F10G···10N12A12B12C12D15A15B30A···30F
order122222344445566610···1010···1012121212151530···30
size111166210103030222222···26···610101010444···4

42 irreducible representations

dim1111122222222224444
type++++++++-+++--+
imageC1C2C2C2C4S3D4D5D6Dic5D10C4×S3D12C3⋊D4C5⋊D4S3×D5S3×Dic5C15⋊D4C5⋊D12
kernelD6⋊Dic5C6×Dic5C2×Dic15S3×C2×C10S3×C10C2×Dic5C30C22×S3C2×C10D6C2×C6C10C10C10C6C22C2C2C2
# reps1111412214222282222

Matrix representation of D6⋊Dic5 in GL4(𝔽61) generated by

606000
1000
00600
00060
,
1100
06000
0010
001460
,
60000
06000
00410
00223
,
50000
05000
005019
00011
G:=sub<GL(4,GF(61))| [60,1,0,0,60,0,0,0,0,0,60,0,0,0,0,60],[1,0,0,0,1,60,0,0,0,0,1,14,0,0,0,60],[60,0,0,0,0,60,0,0,0,0,41,22,0,0,0,3],[50,0,0,0,0,50,0,0,0,0,50,0,0,0,19,11] >;

D6⋊Dic5 in GAP, Magma, Sage, TeX

D_6\rtimes {\rm Dic}_5
% in TeX

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

G:=SmallGroup(240,27);
// by ID

G=gap.SmallGroup(240,27);
# by ID

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

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

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