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G = D20⋊D6order 480 = 25·3·5

6th semidirect product of D20 and D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D206D6, D12.26D10, C60.26C23, D60.9C22, C3⋊C811D10, Q85(S3×D5), (C5×Q8)⋊5D6, (D5×D12)⋊3C2, C3⋊D405C2, C15⋊D85C2, (C3×Q8)⋊8D10, (C4×D5).9D6, C53(D4⋊D6), C37(D40⋊C2), Q82S31D5, Q82D54S3, (C6×D5).12D4, C6.147(D4×D5), C1522(C8⋊C22), Q82D152C2, C30.188(C2×D4), (C3×D20)⋊6C22, (Q8×C15)⋊2C22, C153C812C22, C20.32D65C2, C20.26(C22×S3), (C3×Dic5).70D4, (C5×D12).9C22, C12.26(C22×D5), D10.19(C3⋊D4), (D5×C12).10C22, Dic5.33(C3⋊D4), C4.26(C2×S3×D5), (C5×C3⋊C8)⋊12C22, C2.29(D5×C3⋊D4), (C5×Q82S3)⋊2C2, (C3×Q82D5)⋊1C2, C10.50(C2×C3⋊D4), SmallGroup(480,578)

Series: Derived Chief Lower central Upper central

C1C60 — D20⋊D6
C1C5C15C30C60D5×C12D5×D12 — D20⋊D6
C15C30C60 — D20⋊D6
C1C2C4Q8

Generators and relations for D20⋊D6
 G = < a,b,c,d | a20=b2=c6=d2=1, bab=dad=a-1, cac-1=a9, cbc-1=a18b, dbd=a3b, dcd=c-1 >

Subgroups: 924 in 136 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2 [×4], C3, C4, C4 [×2], C22 [×6], C5, S3 [×2], C6, C6 [×2], C8 [×2], C2×C4 [×2], D4 [×5], Q8, C23, D5 [×3], C10, C10, C12, C12 [×2], D6 [×4], C2×C6 [×2], C15, M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, Dic5, C20, C20, D10, D10 [×4], C2×C10, C3⋊C8, C3⋊C8, D12, D12 [×2], C2×C12 [×2], C3×D4 [×2], C3×Q8, C22×S3, C5×S3, C3×D5 [×2], D15, C30, C8⋊C22, C52C8, C40, C4×D5, C4×D5, D20, D20 [×2], C5⋊D4, C5×D4, C5×Q8, C22×D5, C4.Dic3, D4⋊S3 [×2], Q82S3, Q82S3, C2×D12, C3×C4○D4, C3×Dic5, C60, C60, S3×D5 [×2], C6×D5, C6×D5, S3×C10, D30, C8⋊D5, D40, D4⋊D5, Q8⋊D5, C5×SD16, D4×D5, Q82D5, D4⋊D6, C5×C3⋊C8, C153C8, C5⋊D12, D5×C12, D5×C12, C3×D20, C3×D20, C5×D12, D60, Q8×C15, C2×S3×D5, D40⋊C2, C20.32D6, C15⋊D8, C3⋊D40, C5×Q82S3, Q82D15, D5×D12, C3×Q82D5, D20⋊D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], C3⋊D4 [×2], C22×S3, C8⋊C22, C22×D5, C2×C3⋊D4, S3×D5, D4×D5, D4⋊D6, C2×S3×D5, D40⋊C2, D5×C3⋊D4, D20⋊D6

Smallest permutation representation of D20⋊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 31)(2 30)(3 29)(4 28)(5 27)(6 26)(7 25)(8 24)(9 23)(10 22)(11 21)(12 40)(13 39)(14 38)(15 37)(16 36)(17 35)(18 34)(19 33)(20 32)(41 84)(42 83)(43 82)(44 81)(45 100)(46 99)(47 98)(48 97)(49 96)(50 95)(51 94)(52 93)(53 92)(54 91)(55 90)(56 89)(57 88)(58 87)(59 86)(60 85)(61 116)(62 115)(63 114)(64 113)(65 112)(66 111)(67 110)(68 109)(69 108)(70 107)(71 106)(72 105)(73 104)(74 103)(75 102)(76 101)(77 120)(78 119)(79 118)(80 117)
(1 81 65)(2 90 66 10 82 74)(3 99 67 19 83 63)(4 88 68 8 84 72)(5 97 69 17 85 61)(6 86 70)(7 95 71 15 87 79)(9 93 73 13 89 77)(11 91 75)(12 100 76 20 92 64)(14 98 78 18 94 62)(16 96 80)(21 56 102 23 54 104)(22 45 103 32 55 113)(24 43 105 30 57 111)(25 52 106 39 58 120)(26 41 107 28 59 109)(27 50 108 37 60 118)(29 48 110 35 42 116)(31 46 112 33 44 114)(34 53 115 40 47 101)(36 51 117 38 49 119)
(1 80)(2 79)(3 78)(4 77)(5 76)(6 75)(7 74)(8 73)(9 72)(10 71)(11 70)(12 69)(13 68)(14 67)(15 66)(16 65)(17 64)(18 63)(19 62)(20 61)(21 104)(22 103)(23 102)(24 101)(25 120)(26 119)(27 118)(28 117)(29 116)(30 115)(31 114)(32 113)(33 112)(34 111)(35 110)(36 109)(37 108)(38 107)(39 106)(40 105)(41 49)(42 48)(43 47)(44 46)(50 60)(51 59)(52 58)(53 57)(54 56)(81 96)(82 95)(83 94)(84 93)(85 92)(86 91)(87 90)(88 89)(97 100)(98 99)

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

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

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

45 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C5A5B6A6B6C6D8A8B10A10B10C10D12A12B12C12D12E15A15B20A20B20C20D30A30B40A40B40C40D60A···60F
order12222234445566668810101010121212121215152020202030304040404060···60
size1110122060224102222020201260222424444101044448844121212128···8

45 irreducible representations

dim1111111122222222222244444448
type+++++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2S3D4D4D5D6D6D6D10D10D10C3⋊D4C3⋊D4C8⋊C22S3×D5D4×D5D4⋊D6C2×S3×D5D40⋊C2D5×C3⋊D4D20⋊D6
kernelD20⋊D6C20.32D6C15⋊D8C3⋊D40C5×Q82S3Q82D15D5×D12C3×Q82D5Q82D5C3×Dic5C6×D5Q82S3C4×D5D20C5×Q8C3⋊C8D12C3×Q8Dic5D10C15Q8C6C5C4C3C2C1
# reps1111111111121112222212222442

Matrix representation of D20⋊D6 in GL8(𝔽241)

2401000000
50190000000
0024000000
0002400000
0000119200
000012324000
000011810240
00000110
,
10000000
191240000000
001711400000
00101700000
000024000192
00001180240240
000012324001
00000001
,
511000000
51190000000
0002400000
0012400000
00001000
00000100
000011802400
000011800240
,
190240000000
19051000000
0024010000
00010000
0000119200
0000024000
00000110
000011810240

G:=sub<GL(8,GF(241))| [240,50,0,0,0,0,0,0,1,190,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,1,123,118,0,0,0,0,0,192,240,1,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0],[1,191,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,171,101,0,0,0,0,0,0,140,70,0,0,0,0,0,0,0,0,240,118,123,0,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,192,240,1,1],[51,51,0,0,0,0,0,0,1,190,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,240,240,0,0,0,0,0,0,0,0,1,0,118,118,0,0,0,0,0,1,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240],[190,190,0,0,0,0,0,0,240,51,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,1,0,0,118,0,0,0,0,192,240,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,240] >;

D20⋊D6 in GAP, Magma, Sage, TeX

D_{20}\rtimes D_6
% in TeX

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

G:=SmallGroup(480,578);
// by ID

G=gap.SmallGroup(480,578);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,422,135,100,346,185,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^20=b^2=c^6=d^2=1,b*a*b=d*a*d=a^-1,c*a*c^-1=a^9,c*b*c^-1=a^18*b,d*b*d=a^3*b,d*c*d=c^-1>;
// generators/relations

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