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G = D2025D6order 480 = 25·3·5

8th semidirect product of D20 and D6 acting via D6/C6=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D2025D6, D1225D10, Dic623D10, D6034C22, C30.21C24, D30.7C23, C1512+ 1+4, C60.164C23, Dic3031C22, Dic15.10C23, (C2×C20)⋊8D6, C4○D128D5, (C6×D20)⋊3C2, (C4×S3)⋊1D10, (C2×C12)⋊8D10, (C2×D20)⋊14S3, (S3×D20)⋊11C2, C3⋊D413D10, C51(D46D6), (C2×C60)⋊4C22, (C22×D5)⋊5D6, C20⋊D611C2, C31(D48D10), C3⋊D201C22, C15⋊D41C22, D205S312C2, D20⋊S312C2, D6011C27C2, D6.7(C22×D5), (C6×D5).7C23, C6.21(C23×D5), (S3×C20)⋊10C22, (C3×D20)⋊25C22, (C5×D12)⋊32C22, (C4×D15)⋊12C22, C157D415C22, (S3×C10).7C23, C10.21(S3×C23), (D5×Dic3)⋊1C22, D10.7(C22×S3), C20.130(C22×S3), (C2×C30).240C23, (C5×Dic6)⋊29C22, C12.129(C22×D5), (C5×Dic3).10C23, Dic3.10(C22×D5), (C2×C4)⋊4(S3×D5), (D5×C3⋊D4)⋊1C2, C4.135(C2×S3×D5), (C2×S3×D5)⋊1C22, (D5×C2×C6)⋊6C22, (C5×C4○D12)⋊3C2, C22.10(C2×S3×D5), C2.24(C22×S3×D5), (C5×C3⋊D4)⋊10C22, (C2×C10).12(C22×S3), (C2×C6).248(C22×D5), SmallGroup(480,1093)

Series: Derived Chief Lower central Upper central

C1C30 — D2025D6
C1C5C15C30C6×D5C2×S3×D5D5×C3⋊D4 — D2025D6
C15C30 — D2025D6
C1C2C2×C4

Generators and relations for D2025D6
 G = < a,b,c,d | a20=b2=c6=d2=1, bab=cac-1=a-1, dad=a9, cbc-1=a8b, dbd=a18b, dcd=c-1 >

Subgroups: 1916 in 332 conjugacy classes, 108 normal (36 characteristic)
C1, C2, C2 [×9], C3, C4 [×2], C4 [×4], C22, C22 [×14], C5, S3 [×4], C6, C6 [×5], C2×C4, C2×C4 [×8], D4 [×18], Q8 [×2], C23 [×6], D5 [×6], C10, C10 [×3], Dic3 [×2], Dic3 [×2], C12 [×2], D6 [×2], D6 [×6], C2×C6, C2×C6 [×6], C15, C2×D4 [×9], C4○D4 [×6], Dic5 [×2], C20 [×2], C20 [×2], D10 [×4], D10 [×8], C2×C10, C2×C10 [×2], Dic6, Dic6, C4×S3 [×2], C4×S3 [×2], D12, D12, C2×Dic3 [×4], C3⋊D4 [×2], C3⋊D4 [×10], C2×C12, C3×D4 [×4], C22×S3 [×4], C22×C6 [×2], C5×S3 [×2], C3×D5 [×4], D15 [×2], C30, C30, 2+ 1+4, Dic10, C4×D5 [×6], D20 [×4], D20 [×5], C5⋊D4 [×6], C2×C20, C2×C20 [×2], C5×D4 [×3], C5×Q8, C22×D5 [×2], C22×D5 [×4], C4○D12, C4○D12, S3×D4 [×4], D42S3 [×4], C2×C3⋊D4 [×4], C6×D4, C5×Dic3 [×2], Dic15 [×2], C60 [×2], S3×D5 [×4], C6×D5 [×4], C6×D5 [×2], S3×C10 [×2], D30 [×2], C2×C30, C2×D20, C2×D20 [×2], C4○D20 [×3], D4×D5 [×6], Q82D5 [×2], C5×C4○D4, D46D6, D5×Dic3 [×4], C15⋊D4 [×4], C3⋊D20 [×4], C3×D20 [×4], C5×Dic6, S3×C20 [×2], C5×D12, C5×C3⋊D4 [×2], Dic30, C4×D15 [×2], D60, C157D4 [×2], C2×C60, C2×S3×D5 [×4], D5×C2×C6 [×2], D48D10, D205S3 [×2], D20⋊S3 [×2], S3×D20 [×2], C20⋊D6 [×2], D5×C3⋊D4 [×4], C6×D20, C5×C4○D12, D6011C2, D2025D6
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, D46D6, C2×S3×D5 [×3], D48D10, C22×S3×D5, D2025D6

Smallest permutation representation of D2025D6
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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 34)(22 33)(23 32)(24 31)(25 30)(26 29)(27 28)(35 40)(36 39)(37 38)(41 46)(42 45)(43 44)(47 60)(48 59)(49 58)(50 57)(51 56)(52 55)(53 54)(61 68)(62 67)(63 66)(64 65)(69 80)(70 79)(71 78)(72 77)(73 76)(74 75)(81 92)(82 91)(83 90)(84 89)(85 88)(86 87)(93 100)(94 99)(95 98)(96 97)(101 114)(102 113)(103 112)(104 111)(105 110)(106 109)(107 108)(115 120)(116 119)(117 118)
(1 82 65 6 97 70)(2 81 66 5 98 69)(3 100 67 4 99 68)(7 96 71 20 83 64)(8 95 72 19 84 63)(9 94 73 18 85 62)(10 93 74 17 86 61)(11 92 75 16 87 80)(12 91 76 15 88 79)(13 90 77 14 89 78)(21 110 57 30 101 46)(22 109 58 29 102 45)(23 108 59 28 103 44)(24 107 60 27 104 43)(25 106 41 26 105 42)(31 120 47 40 111 56)(32 119 48 39 112 55)(33 118 49 38 113 54)(34 117 50 37 114 53)(35 116 51 36 115 52)
(1 33)(2 22)(3 31)(4 40)(5 29)(6 38)(7 27)(8 36)(9 25)(10 34)(11 23)(12 32)(13 21)(14 30)(15 39)(16 28)(17 37)(18 26)(19 35)(20 24)(41 85)(42 94)(43 83)(44 92)(45 81)(46 90)(47 99)(48 88)(49 97)(50 86)(51 95)(52 84)(53 93)(54 82)(55 91)(56 100)(57 89)(58 98)(59 87)(60 96)(61 117)(62 106)(63 115)(64 104)(65 113)(66 102)(67 111)(68 120)(69 109)(70 118)(71 107)(72 116)(73 105)(74 114)(75 103)(76 112)(77 101)(78 110)(79 119)(80 108)

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

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

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

63 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J 3 4A4B4C4D4E4F5A5B6A6B6C6D6E6F6G10A10B10C10D10E10F10G10H12A12B15A15B20A20B20C20D20E20F20G20H20I20J30A···30F60A···60H
order1222222222234444445566666661010101010101010121215152020202020202020202030···3060···60
size1126610101010303022266303022222202020202244121212124444222244121212124···44···4

63 irreducible representations

dim11111111122222222224444444
type++++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2S3D5D6D6D6D10D10D10D10D102+ 1+4S3×D5D46D6C2×S3×D5C2×S3×D5D48D10D2025D6
kernelD2025D6D205S3D20⋊S3S3×D20C20⋊D6D5×C3⋊D4C6×D20C5×C4○D12D6011C2C2×D20C4○D12D20C2×C20C22×D5Dic6C4×S3D12C3⋊D4C2×C12C15C2×C4C5C4C22C3C1
# reps12222411112412242421224248

Matrix representation of D2025D6 in GL6(𝔽61)

0600000
1180000
000100
0060000
0000060
000010
,
0600000
6000000
000100
001000
0000060
0000600
,
100000
43600000
0004800
0048000
0000014
0000140
,
6000000
1810000
0000014
0000140
0004800
0048000

G:=sub<GL(6,GF(61))| [0,1,0,0,0,0,60,18,0,0,0,0,0,0,0,60,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,60,0],[0,60,0,0,0,0,60,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,60,0,0,0,0,60,0],[1,43,0,0,0,0,0,60,0,0,0,0,0,0,0,48,0,0,0,0,48,0,0,0,0,0,0,0,0,14,0,0,0,0,14,0],[60,18,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,48,0,0,0,0,48,0,0,0,0,14,0,0,0,0,14,0,0,0] >;

D2025D6 in GAP, Magma, Sage, TeX

D_{20}\rtimes_{25}D_6
% in TeX

G:=Group("D20:25D6");
// GroupNames label

G:=SmallGroup(480,1093);
// by ID

G=gap.SmallGroup(480,1093);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,219,675,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^20=b^2=c^6=d^2=1,b*a*b=c*a*c^-1=a^-1,d*a*d=a^9,c*b*c^-1=a^8*b,d*b*d=a^18*b,d*c*d=c^-1>;
// generators/relations

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