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G = D2026D6order 480 = 25·3·5

9th semidirect product of D20 and D6 acting via D6/C6=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D2026D6, D1228D10, Dic1023D6, D6040C22, C30.22C24, D30.8C23, C1522+ 1+4, C60.165C23, Dic3037C22, Dic15.11C23, (C4×D5)⋊1D6, (C2×C20)⋊9D6, C4○D208S3, (C2×C12)⋊9D10, C51(D4○D12), C5⋊D414D6, (C10×D12)⋊3C2, (D5×D12)⋊11C2, (C2×D12)⋊14D5, (C2×C60)⋊3C22, C20⋊D612C2, C31(D46D10), (C22×S3)⋊4D10, C5⋊D121C22, C15⋊D42C22, D12⋊D512C2, D125D512C2, D6011C24C2, D6.8(C22×D5), (C6×D5).8C23, C6.22(C23×D5), (C3×D20)⋊32C22, (C4×D15)⋊13C22, (C5×D12)⋊25C22, (D5×C12)⋊10C22, C157D416C22, (S3×C10).8C23, C10.22(S3×C23), (S3×Dic5)⋊1C22, D10.8(C22×S3), (C2×C30).241C23, C20.131(C22×S3), C12.162(C22×D5), (C3×Dic10)⋊29C22, Dic5.10(C22×S3), (C3×Dic5).10C23, (C2×C4)⋊5(S3×D5), C4.88(C2×S3×D5), (S3×C5⋊D4)⋊1C2, (C2×S3×D5)⋊2C22, (C3×C4○D20)⋊3C2, (S3×C2×C10)⋊6C22, C2.25(C22×S3×D5), C22.15(C2×S3×D5), (C3×C5⋊D4)⋊10C22, (C2×C6).12(C22×D5), (C2×C10).248(C22×S3), SmallGroup(480,1094)

Series: Derived Chief Lower central Upper central

C1C30 — D2026D6
C1C5C15C30C6×D5C2×S3×D5D5×D12 — D2026D6
C15C30 — D2026D6
C1C2C2×C4

Generators and relations for D2026D6
 G = < a,b,c,d | a20=b2=c6=d2=1, bab=a-1, ac=ca, dad=a11, cbc-1=a10b, bd=db, dcd=c-1 >

Subgroups: 1868 in 332 conjugacy classes, 108 normal (36 characteristic)
C1, C2, C2 [×9], C3, C4 [×2], C4 [×4], C22, C22 [×14], C5, S3 [×6], C6, C6 [×3], C2×C4, C2×C4 [×8], D4 [×18], Q8 [×2], C23 [×6], D5 [×4], C10, C10 [×5], Dic3 [×2], C12 [×2], C12 [×2], D6 [×4], D6 [×8], C2×C6, C2×C6 [×2], C15, C2×D4 [×9], C4○D4 [×6], Dic5 [×2], Dic5 [×2], C20 [×2], D10 [×2], D10 [×6], C2×C10, C2×C10 [×6], Dic6, C4×S3 [×6], D12 [×4], D12 [×5], C3⋊D4 [×6], C2×C12, C2×C12 [×2], C3×D4 [×3], C3×Q8, C22×S3 [×2], C22×S3 [×4], C5×S3 [×4], C3×D5 [×2], D15 [×2], C30, C30, 2+ 1+4, Dic10, Dic10, C4×D5 [×2], C4×D5 [×2], D20, D20, C2×Dic5 [×4], C5⋊D4 [×2], C5⋊D4 [×10], C2×C20, C5×D4 [×4], C22×D5 [×4], C22×C10 [×2], C2×D12, C2×D12 [×2], C4○D12 [×3], S3×D4 [×6], Q83S3 [×2], C3×C4○D4, C3×Dic5 [×2], Dic15 [×2], C60 [×2], S3×D5 [×4], C6×D5 [×2], S3×C10 [×4], S3×C10 [×2], D30 [×2], C2×C30, C4○D20, C4○D20, D4×D5 [×4], D42D5 [×4], C2×C5⋊D4 [×4], D4×C10, D4○D12, S3×Dic5 [×4], C15⋊D4 [×4], C5⋊D12 [×4], C3×Dic10, D5×C12 [×2], C3×D20, C3×C5⋊D4 [×2], C5×D12 [×4], Dic30, C4×D15 [×2], D60, C157D4 [×2], C2×C60, C2×S3×D5 [×4], S3×C2×C10 [×2], D46D10, D12⋊D5 [×2], D125D5 [×2], D5×D12 [×2], C20⋊D6 [×2], S3×C5⋊D4 [×4], C3×C4○D20, C10×D12, D6011C2, D2026D6
Quotients: C1, C2 [×15], C22 [×35], S3, C23 [×15], D5, D6 [×7], C24, D10 [×7], C22×S3 [×7], 2+ 1+4, C22×D5 [×7], S3×C23, S3×D5, C23×D5, D4○D12, C2×S3×D5 [×3], D46D10, C22×S3×D5, D2026D6

Smallest permutation representation of D2026D6
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 81)(2 100)(3 99)(4 98)(5 97)(6 96)(7 95)(8 94)(9 93)(10 92)(11 91)(12 90)(13 89)(14 88)(15 87)(16 86)(17 85)(18 84)(19 83)(20 82)(21 49)(22 48)(23 47)(24 46)(25 45)(26 44)(27 43)(28 42)(29 41)(30 60)(31 59)(32 58)(33 57)(34 56)(35 55)(36 54)(37 53)(38 52)(39 51)(40 50)(61 115)(62 114)(63 113)(64 112)(65 111)(66 110)(67 109)(68 108)(69 107)(70 106)(71 105)(72 104)(73 103)(74 102)(75 101)(76 120)(77 119)(78 118)(79 117)(80 116)
(1 68 28)(2 69 29)(3 70 30)(4 71 31)(5 72 32)(6 73 33)(7 74 34)(8 75 35)(9 76 36)(10 77 37)(11 78 38)(12 79 39)(13 80 40)(14 61 21)(15 62 22)(16 63 23)(17 64 24)(18 65 25)(19 66 26)(20 67 27)(41 90 107 51 100 117)(42 91 108 52 81 118)(43 92 109 53 82 119)(44 93 110 54 83 120)(45 94 111 55 84 101)(46 95 112 56 85 102)(47 96 113 57 86 103)(48 97 114 58 87 104)(49 98 115 59 88 105)(50 99 116 60 89 106)
(1 28)(2 39)(3 30)(4 21)(5 32)(6 23)(7 34)(8 25)(9 36)(10 27)(11 38)(12 29)(13 40)(14 31)(15 22)(16 33)(17 24)(18 35)(19 26)(20 37)(41 90)(42 81)(43 92)(44 83)(45 94)(46 85)(47 96)(48 87)(49 98)(50 89)(51 100)(52 91)(53 82)(54 93)(55 84)(56 95)(57 86)(58 97)(59 88)(60 99)(61 71)(63 73)(65 75)(67 77)(69 79)(101 111)(103 113)(105 115)(107 117)(109 119)

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

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

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

63 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J 3 4A4B4C4D4E4F5A5B6A6B6C6D10A···10F10G···10N12A12B12C12D12E15A15B20A20B20C20D30A···30F60A···60H
order12222222222344444455666610···1010···10121212121215152020202030···3060···60
size11266661010303022210103030222420202···212···1222420204444444···44···4

63 irreducible representations

dim11111111122222222224444444
type++++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2S3D5D6D6D6D6D6D10D10D102+ 1+4S3×D5D4○D12C2×S3×D5C2×S3×D5D46D10D2026D6
kernelD2026D6D12⋊D5D125D5D5×D12C20⋊D6S3×C5⋊D4C3×C4○D20C10×D12D6011C2C4○D20C2×D12Dic10C4×D5D20C5⋊D4C2×C20D12C2×C12C22×S3C15C2×C4C5C4C22C3C1
# reps12222411112121218241224248

Matrix representation of D2026D6 in GL6(𝔽61)

6000000
0600000
0005800
003000
0000041
0000200
,
6000000
0600000
0000041
0000200
0005800
003000
,
0600000
1600000
001000
000100
0000600
0000060
,
1600000
0600000
001000
0006000
0000600
000001

G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,3,0,0,0,0,58,0,0,0,0,0,0,0,0,20,0,0,0,0,41,0],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,3,0,0,0,0,58,0,0,0,0,20,0,0,0,0,41,0,0,0],[0,1,0,0,0,0,60,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[1,0,0,0,0,0,60,60,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,1] >;

D2026D6 in GAP, Magma, Sage, TeX

D_{20}\rtimes_{26}D_6
% in TeX

G:=Group("D20:26D6");
// GroupNames label

G:=SmallGroup(480,1094);
// by ID

G=gap.SmallGroup(480,1094);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,219,100,675,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^20=b^2=c^6=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^11,c*b*c^-1=a^10*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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