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G = D65D20order 480 = 25·3·5

2nd semidirect product of D6 and D20 acting via D20/D10=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D65D20, D305D4, D105D12, D6⋊C48D5, (C6×D5)⋊5D4, (C2×C20)⋊2D6, C153C22≀C2, (C2×D60)⋊3C2, (S3×C10)⋊5D4, (C2×C12)⋊2D10, C6.27(D4×D5), C51(D6⋊D4), (C2×C60)⋊1C22, (C2×Dic5)⋊2D6, C30.70(C2×D4), C6.28(C2×D20), C2.29(D5×D12), C10.27(S3×D4), C2.29(S3×D20), D10⋊C48S3, C31(C22⋊D20), (C2×Dic3)⋊2D10, C10.29(C2×D12), D304C424C2, (C6×Dic5)⋊4C22, (C22×D5).60D6, (C2×C30).166C23, (C10×Dic3)⋊4C22, (C22×S3).51D10, (C22×D15)⋊5C22, C2.19(D10⋊D6), (C2×C4)⋊3(S3×D5), (C5×D6⋊C4)⋊8C2, (C22×S3×D5)⋊3C2, (C2×C3⋊D20)⋊10C2, (C2×C5⋊D12)⋊10C2, (C3×D10⋊C4)⋊8C2, (D5×C2×C6).43C22, C22.214(C2×S3×D5), (S3×C2×C10).43C22, (C2×C6).178(C22×D5), (C2×C10).178(C22×S3), SmallGroup(480,552)

Series: Derived Chief Lower central Upper central

C1C2×C30 — D65D20
C1C5C15C30C2×C30D5×C2×C6C22×S3×D5 — D65D20
C15C2×C30 — D65D20
C1C22C2×C4

Generators and relations for D65D20
 G = < a,b,c,d | a6=b2=c20=d2=1, bab=dad=a-1, ac=ca, cbc-1=a3b, dbd=ab, dcd=c-1 >

Subgroups: 1964 in 260 conjugacy classes, 54 normal (44 characteristic)
C1, C2 [×3], C2 [×7], C3, C4 [×3], C22, C22 [×23], C5, S3 [×5], C6 [×3], C6 [×2], C2×C4, C2×C4 [×2], D4 [×6], C23 [×10], D5 [×5], C10 [×3], C10 [×2], Dic3, C12 [×2], D6 [×2], D6 [×17], C2×C6, C2×C6 [×4], C15, C22⋊C4 [×3], C2×D4 [×3], C24, Dic5, C20 [×2], D10 [×2], D10 [×17], C2×C10, C2×C10 [×4], D12 [×4], C2×Dic3, C3⋊D4 [×2], C2×C12, C2×C12, C22×S3, C22×S3 [×8], C22×C6, C5×S3 [×2], C3×D5 [×2], D15 [×3], C30 [×3], C22≀C2, D20 [×4], C2×Dic5, C5⋊D4 [×2], C2×C20, C2×C20, C22×D5, C22×D5 [×8], C22×C10, D6⋊C4, D6⋊C4, C3×C22⋊C4, C2×D12 [×2], C2×C3⋊D4, S3×C23, C5×Dic3, C3×Dic5, C60, S3×D5 [×8], C6×D5 [×2], C6×D5 [×2], S3×C10 [×2], S3×C10 [×2], D30 [×2], D30 [×5], C2×C30, D10⋊C4, D10⋊C4, C5×C22⋊C4, C2×D20 [×2], C2×C5⋊D4, C23×D5, D6⋊D4, C3⋊D20 [×2], C5⋊D12 [×2], C6×Dic5, C10×Dic3, D60 [×2], C2×C60, C2×S3×D5 [×6], D5×C2×C6, S3×C2×C10, C22×D15 [×2], C22⋊D20, D304C4, C3×D10⋊C4, C5×D6⋊C4, C2×C3⋊D20, C2×C5⋊D12, C2×D60, C22×S3×D5, D65D20
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D5, D6 [×3], C2×D4 [×3], D10 [×3], D12 [×2], C22×S3, C22≀C2, D20 [×2], C22×D5, C2×D12, S3×D4 [×2], S3×D5, C2×D20, D4×D5 [×2], D6⋊D4, C2×S3×D5, C22⋊D20, D5×D12, S3×D20, D10⋊D6, D65D20

Smallest permutation representation of D65D20
On 120 points
Generators in S120
(1 78 105 92 30 44)(2 79 106 93 31 45)(3 80 107 94 32 46)(4 61 108 95 33 47)(5 62 109 96 34 48)(6 63 110 97 35 49)(7 64 111 98 36 50)(8 65 112 99 37 51)(9 66 113 100 38 52)(10 67 114 81 39 53)(11 68 115 82 40 54)(12 69 116 83 21 55)(13 70 117 84 22 56)(14 71 118 85 23 57)(15 72 119 86 24 58)(16 73 120 87 25 59)(17 74 101 88 26 60)(18 75 102 89 27 41)(19 76 103 90 28 42)(20 77 104 91 29 43)
(1 115)(2 55)(3 117)(4 57)(5 119)(6 59)(7 101)(8 41)(9 103)(10 43)(11 105)(12 45)(13 107)(14 47)(15 109)(16 49)(17 111)(18 51)(19 113)(20 53)(21 79)(22 32)(23 61)(24 34)(25 63)(26 36)(27 65)(28 38)(29 67)(30 40)(31 69)(33 71)(35 73)(37 75)(39 77)(42 100)(44 82)(46 84)(48 86)(50 88)(52 90)(54 92)(56 94)(58 96)(60 98)(62 72)(64 74)(66 76)(68 78)(70 80)(81 104)(83 106)(85 108)(87 110)(89 112)(91 114)(93 116)(95 118)(97 120)(99 102)
(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 10)(2 9)(3 8)(4 7)(5 6)(11 20)(12 19)(13 18)(14 17)(15 16)(21 103)(22 102)(23 101)(24 120)(25 119)(26 118)(27 117)(28 116)(29 115)(30 114)(31 113)(32 112)(33 111)(34 110)(35 109)(36 108)(37 107)(38 106)(39 105)(40 104)(41 70)(42 69)(43 68)(44 67)(45 66)(46 65)(47 64)(48 63)(49 62)(50 61)(51 80)(52 79)(53 78)(54 77)(55 76)(56 75)(57 74)(58 73)(59 72)(60 71)(81 92)(82 91)(83 90)(84 89)(85 88)(86 87)(93 100)(94 99)(95 98)(96 97)

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

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

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

60 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J 3 4A4B4C5A5B6A6B6C6D6E10A···10F10G10H10I10J12A12B12C12D15A15B20A20B20C20D20E20F20G20H30A···30F60A···60H
order122222222223444556666610···1010101010121212121515202020202020202030···3060···60
size11116610103030602412202222220202···212121212442020444444121212124···44···4

60 irreducible representations

dim1111111122222222222224444444
type++++++++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2S3D4D4D4D5D6D6D6D10D10D10D12D20S3×D4S3×D5D4×D5C2×S3×D5D5×D12S3×D20D10⋊D6
kernelD65D20D304C4C3×D10⋊C4C5×D6⋊C4C2×C3⋊D20C2×C5⋊D12C2×D60C22×S3×D5D10⋊C4C6×D5S3×C10D30D6⋊C4C2×Dic5C2×C20C22×D5C2×Dic3C2×C12C22×S3D10D6C10C2×C4C6C22C2C2C2
# reps1111111112222111222482242444

Matrix representation of D65D20 in GL6(𝔽61)

100000
010000
00606000
001000
0000600
0000060
,
6000000
0600000
00606000
000100
0000600
0000481
,
7320000
2920000
0060000
0006000
00005320
0000128
,
7320000
29540000
0060000
001100
0000841
00005553

G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,1,0,0,0,0,60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,60,1,0,0,0,0,0,0,60,48,0,0,0,0,0,1],[7,29,0,0,0,0,32,2,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,53,12,0,0,0,0,20,8],[7,29,0,0,0,0,32,54,0,0,0,0,0,0,60,1,0,0,0,0,0,1,0,0,0,0,0,0,8,55,0,0,0,0,41,53] >;

D65D20 in GAP, Magma, Sage, TeX

D_6\rtimes_5D_{20}
% in TeX

G:=Group("D6:5D20");
// GroupNames label

G:=SmallGroup(480,552);
// by ID

G=gap.SmallGroup(480,552);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,254,219,142,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^2=c^20=d^2=1,b*a*b=d*a*d=a^-1,a*c=c*a,c*b*c^-1=a^3*b,d*b*d=a*b,d*c*d=c^-1>;
// generators/relations

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