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G = D6.D10order 240 = 24·3·5

3rd non-split extension by D6 of D10 acting via D10/C10=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D10.9D6, D6.7D10, C20.35D6, C12.35D10, C30.8C23, C60.35C22, Dic5.12D6, Dic3.8D10, D30.11C22, Dic15.13C22, C15⋊Q87C2, (C4×S3)⋊4D5, (C4×D5)⋊4S3, (S3×C20)⋊4C2, (D5×C12)⋊4C2, (C4×D15)⋊8C2, C52(C4○D12), C155(C4○D4), C32(C4○D20), C3⋊D207C2, C5⋊D127C2, C15⋊D47C2, C4.28(S3×D5), C6.8(C22×D5), C10.8(C22×S3), (S3×C10).8C22, (C6×D5).10C22, (C5×Dic3).10C22, (C3×Dic5).12C22, C2.12(C2×S3×D5), SmallGroup(240,132)

Series: Derived Chief Lower central Upper central

C1C30 — D6.D10
C1C5C15C30C6×D5C15⋊D4 — D6.D10
C15C30 — D6.D10
C1C4

Generators and relations for D6.D10
 G = < a,b,c,d | a6=b2=1, c10=d2=a3, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a3b, dcd-1=c9 >

Subgroups: 352 in 80 conjugacy classes, 32 normal (all characteristic)
C1, C2, C2 [×3], C3, C4, C4 [×3], C22 [×3], C5, S3 [×2], C6, C6, C2×C4 [×3], D4 [×3], Q8, D5 [×2], C10, C10, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C15, C4○D4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, Dic6, C4×S3, C4×S3, D12, C3⋊D4 [×2], C2×C12, C5×S3, C3×D5, D15, C30, Dic10, C4×D5, C4×D5, D20, C5⋊D4 [×2], C2×C20, C4○D12, C5×Dic3, C3×Dic5, Dic15, C60, C6×D5, S3×C10, D30, C4○D20, C15⋊D4, C3⋊D20, C5⋊D12, C15⋊Q8, D5×C12, S3×C20, C4×D15, D6.D10
Quotients: C1, C2 [×7], C22 [×7], S3, C23, D5, D6 [×3], C4○D4, D10 [×3], C22×S3, C22×D5, C4○D12, S3×D5, C4○D20, C2×S3×D5, D6.D10

Smallest permutation representation of D6.D10
On 120 points
Generators in S120
(1 29 71 11 39 61)(2 30 72 12 40 62)(3 31 73 13 21 63)(4 32 74 14 22 64)(5 33 75 15 23 65)(6 34 76 16 24 66)(7 35 77 17 25 67)(8 36 78 18 26 68)(9 37 79 19 27 69)(10 38 80 20 28 70)(41 85 117 51 95 107)(42 86 118 52 96 108)(43 87 119 53 97 109)(44 88 120 54 98 110)(45 89 101 55 99 111)(46 90 102 56 100 112)(47 91 103 57 81 113)(48 92 104 58 82 114)(49 93 105 59 83 115)(50 94 106 60 84 116)
(1 115)(2 116)(3 117)(4 118)(5 119)(6 120)(7 101)(8 102)(9 103)(10 104)(11 105)(12 106)(13 107)(14 108)(15 109)(16 110)(17 111)(18 112)(19 113)(20 114)(21 95)(22 96)(23 97)(24 98)(25 99)(26 100)(27 81)(28 82)(29 83)(30 84)(31 85)(32 86)(33 87)(34 88)(35 89)(36 90)(37 91)(38 92)(39 93)(40 94)(41 73)(42 74)(43 75)(44 76)(45 77)(46 78)(47 79)(48 80)(49 61)(50 62)(51 63)(52 64)(53 65)(54 66)(55 67)(56 68)(57 69)(58 70)(59 71)(60 72)
(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 53 11 43)(2 42 12 52)(3 51 13 41)(4 60 14 50)(5 49 15 59)(6 58 16 48)(7 47 17 57)(8 56 18 46)(9 45 19 55)(10 54 20 44)(21 85 31 95)(22 94 32 84)(23 83 33 93)(24 92 34 82)(25 81 35 91)(26 90 36 100)(27 99 37 89)(28 88 38 98)(29 97 39 87)(30 86 40 96)(61 119 71 109)(62 108 72 118)(63 117 73 107)(64 106 74 116)(65 115 75 105)(66 104 76 114)(67 113 77 103)(68 102 78 112)(69 111 79 101)(70 120 80 110)

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

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

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

D6.D10 is a maximal subgroup of
C40.54D6  C40.34D6  C40.55D6  C40.35D6  D5×C4○D12  S3×C4○D20  D2024D6  C15⋊2- 1+4  D2013D6  D2014D6  D1214D10  D20.29D6  C30.33C24  D12.29D10  D2017D6
D6.D10 is a maximal quotient of
Dic3⋊C4⋊D5  D10⋊Dic6  Dic5.8D12  D6⋊Dic5⋊C2  D6⋊Dic10  Dic3.D20  D30.34D4  D30.35D4  (D5×C12)⋊C4  (C4×Dic3)⋊D5  (C4×Dic15)⋊C2  D6⋊Dic5.C2  (S3×C20)⋊7C4  C5⋊(C423S3)  D308Q8  Dic5.7Dic6  Dic3.3Dic10  C10.D4⋊S3  Dic15.4Q8  (C4×D15)⋊10C4  (C4×Dic5)⋊S3  C4×C15⋊D4  C4×C3⋊D20  C4×C5⋊D12  D6⋊C4⋊D5  D10⋊D12  D10⋊C4⋊S3  D6⋊D20  D3012D4  Dic15.31D4  C4×C15⋊Q8

42 conjugacy classes

class 1 2A2B2C2D 3 4A4B4C4D4E5A5B6A6B6C10A10B10C10D10E10F12A12B12C12D15A15B20A20B20C20D20E20F20G20H30A30B60A60B60C60D
order12222344444556661010101010101212121215152020202020202020303060606060
size11610302116103022210102266662210104422226666444444

42 irreducible representations

dim1111111122222222222444
type++++++++++++++++++
imageC1C2C2C2C2C2C2C2S3D5D6D6D6C4○D4D10D10D10C4○D12C4○D20S3×D5C2×S3×D5D6.D10
kernelD6.D10C15⋊D4C3⋊D20C5⋊D12C15⋊Q8D5×C12S3×C20C4×D15C4×D5C4×S3Dic5C20D10C15Dic3C12D6C5C3C4C2C1
# reps1111111112111222248224

Matrix representation of D6.D10 in GL4(𝔽61) generated by

1000
0100
00246
004960
,
60000
06000
00021
00320
,
434400
18000
00110
00011
,
436000
181800
005319
0038
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,2,49,0,0,46,60],[60,0,0,0,0,60,0,0,0,0,0,32,0,0,21,0],[43,18,0,0,44,0,0,0,0,0,11,0,0,0,0,11],[43,18,0,0,60,18,0,0,0,0,53,3,0,0,19,8] >;

D6.D10 in GAP, Magma, Sage, TeX

D_6.D_{10}
% in TeX

G:=Group("D6.D10");
// GroupNames label

G:=SmallGroup(240,132);
// by ID

G=gap.SmallGroup(240,132);
# by ID

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

G:=Group<a,b,c,d|a^6=b^2=1,c^10=d^2=a^3,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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