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G = D2024D6order 480 = 25·3·5

7th semidirect product of D20 and D6 acting via D6/C6=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D2024D6, D1224D10, Dic622D10, Dic1022D6, C30.20C24, C60.163C23, D30.38C23, Dic15.38C23, (C2×C20)⋊7D6, C5⋊D48D6, C4○D207S3, C4○D127D5, (C4×D5)⋊13D6, (C2×C12)⋊7D10, C3⋊D48D10, (C4×S3)⋊13D10, D15⋊Q813C2, C15⋊Q811C22, D151(C4○D4), D10⋊D67C2, C20⋊D613C2, (C2×C60)⋊17C22, D20⋊S313C2, D12⋊D513C2, D6.6(C22×D5), (C6×D5).6C23, C6.20(C23×D5), (S3×C20)⋊13C22, C30.C237C2, (C3×D20)⋊31C22, (C5×D12)⋊31C22, (D5×C12)⋊13C22, (C4×D15)⋊24C22, C3⋊D2013C22, C15⋊D413C22, C5⋊D1213C22, (S3×C10).6C23, C10.20(S3×C23), (D5×Dic3)⋊8C22, (S3×Dic5)⋊8C22, D10.6(C22×S3), D6.D1011C2, (C2×C30).239C23, C20.189(C22×S3), (C5×Dic6)⋊28C22, C12.189(C22×D5), (C3×Dic5).9C23, Dic3.9(C22×D5), (C5×Dic3).9C23, Dic5.9(C22×S3), (C3×Dic10)⋊28C22, (C2×Dic15)⋊34C22, D30.C2.10C22, (C22×D15).122C22, C32(D5×C4○D4), C52(S3×C4○D4), (C4×S3×D5)⋊10C2, (C2×C4×D15)⋊27C2, (C2×C4)⋊15(S3×D5), C1510(C2×C4○D4), C4.162(C2×S3×D5), (C3×C4○D20)⋊10C2, (C5×C4○D12)⋊10C2, (C2×S3×D5).7C22, C2.23(C22×S3×D5), C22.18(C2×S3×D5), (C5×C3⋊D4)⋊9C22, (C3×C5⋊D4)⋊9C22, (C2×C6).11(C22×D5), (C2×C10).11(C22×S3), SmallGroup(480,1092)

Series: Derived Chief Lower central Upper central

C1C30 — D2024D6
C1C5C15C30C6×D5C2×S3×D5C4×S3×D5 — D2024D6
C15C30 — D2024D6
C1C4C2×C4

Generators and relations for D2024D6
 G = < a,b,c,d | a20=b2=c6=d2=1, bab=a-1, ac=ca, dad=a9, cbc-1=a10b, dbd=a18b, dcd=c-1 >

Subgroups: 1676 in 328 conjugacy classes, 110 normal (52 characteristic)
C1, C2, C2 [×8], C3, C4 [×2], C4 [×6], C22, C22 [×12], C5, S3 [×5], C6, C6 [×3], C2×C4, C2×C4 [×15], D4 [×12], Q8 [×4], C23 [×3], D5 [×5], C10, C10 [×3], Dic3 [×2], Dic3 [×2], C12 [×2], C12 [×2], D6 [×2], D6 [×8], C2×C6, C2×C6 [×2], C15, C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], Dic5 [×2], Dic5 [×2], C20 [×2], C20 [×2], D10 [×2], D10 [×8], C2×C10, C2×C10 [×2], Dic6, Dic6 [×2], C4×S3 [×2], C4×S3 [×8], D12, D12 [×2], C2×Dic3 [×3], C3⋊D4 [×2], C3⋊D4 [×4], C2×C12, C2×C12 [×2], C3×D4 [×3], C3×Q8, C22×S3 [×3], C5×S3 [×2], C3×D5 [×2], D15 [×2], D15, C30, C30, C2×C4○D4, Dic10, Dic10 [×2], C4×D5 [×2], C4×D5 [×8], D20, D20 [×2], C2×Dic5 [×3], C5⋊D4 [×2], C5⋊D4 [×4], C2×C20, C2×C20 [×2], C5×D4 [×3], C5×Q8, C22×D5 [×3], S3×C2×C4 [×3], C4○D12, C4○D12 [×2], S3×D4 [×3], D42S3 [×3], S3×Q8, Q83S3, C3×C4○D4, C5×Dic3 [×2], C3×Dic5 [×2], Dic15 [×2], C60 [×2], S3×D5 [×4], C6×D5 [×2], S3×C10 [×2], D30 [×2], D30 [×2], C2×C30, C2×C4×D5 [×3], C4○D20, C4○D20 [×2], D4×D5 [×3], D42D5 [×3], Q8×D5, Q82D5, C5×C4○D4, S3×C4○D4, D5×Dic3 [×2], S3×Dic5 [×2], D30.C2 [×2], C15⋊D4 [×2], C3⋊D20 [×2], C5⋊D12 [×2], C15⋊Q8 [×2], C3×Dic10, D5×C12 [×2], C3×D20, C3×C5⋊D4 [×2], C5×Dic6, S3×C20 [×2], C5×D12, C5×C3⋊D4 [×2], C4×D15 [×4], C2×Dic15, C2×C60, C2×S3×D5 [×2], C22×D15, D5×C4○D4, D20⋊S3, D12⋊D5, D15⋊Q8, D6.D10 [×2], C4×S3×D5 [×2], C20⋊D6, C30.C23 [×2], D10⋊D6 [×2], C3×C4○D20, C5×C4○D12, C2×C4×D15, D2024D6
Quotients: C1, C2 [×15], C22 [×35], S3, C23 [×15], D5, D6 [×7], C4○D4 [×2], C24, D10 [×7], C22×S3 [×7], C2×C4○D4, C22×D5 [×7], S3×C23, S3×D5, C23×D5, S3×C4○D4, C2×S3×D5 [×3], D5×C4○D4, C22×S3×D5, D2024D6

Smallest permutation representation of D2024D6
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 26)(22 25)(23 24)(27 40)(28 39)(29 38)(30 37)(31 36)(32 35)(33 34)(41 52)(42 51)(43 50)(44 49)(45 48)(46 47)(53 60)(54 59)(55 58)(56 57)(61 72)(62 71)(63 70)(64 69)(65 68)(66 67)(73 80)(74 79)(75 78)(76 77)(81 82)(83 100)(84 99)(85 98)(86 97)(87 96)(88 95)(89 94)(90 93)(91 92)(101 110)(102 109)(103 108)(104 107)(105 106)(111 120)(112 119)(113 118)(114 117)(115 116)
(1 29 67 97 116 52)(2 30 68 98 117 53)(3 31 69 99 118 54)(4 32 70 100 119 55)(5 33 71 81 120 56)(6 34 72 82 101 57)(7 35 73 83 102 58)(8 36 74 84 103 59)(9 37 75 85 104 60)(10 38 76 86 105 41)(11 39 77 87 106 42)(12 40 78 88 107 43)(13 21 79 89 108 44)(14 22 80 90 109 45)(15 23 61 91 110 46)(16 24 62 92 111 47)(17 25 63 93 112 48)(18 26 64 94 113 49)(19 27 65 95 114 50)(20 28 66 96 115 51)
(1 42)(2 51)(3 60)(4 49)(5 58)(6 47)(7 56)(8 45)(9 54)(10 43)(11 52)(12 41)(13 50)(14 59)(15 48)(16 57)(17 46)(18 55)(19 44)(20 53)(21 114)(22 103)(23 112)(24 101)(25 110)(26 119)(27 108)(28 117)(29 106)(30 115)(31 104)(32 113)(33 102)(34 111)(35 120)(36 109)(37 118)(38 107)(39 116)(40 105)(61 93)(62 82)(63 91)(64 100)(65 89)(66 98)(67 87)(68 96)(69 85)(70 94)(71 83)(72 92)(73 81)(74 90)(75 99)(76 88)(77 97)(78 86)(79 95)(80 84)

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

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

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

66 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I 3 4A4B4C4D4E4F4G4H4I4J5A5B6A6B6C6D10A10B10C10D10E10F10G10H12A12B12C12D12E15A15B20A20B20C20D20E20F20G20H20I20J30A···30F60A···60H
order1222222222344444444445566661010101010101010121212121215152020202020202020202030···3060···60
size112661010151530211266101015153022242020224412121212224202044222244121212124···44···4

66 irreducible representations

dim1111111111112222222222222444444
type+++++++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2S3D5D6D6D6D6D6C4○D4D10D10D10D10D10S3×D5S3×C4○D4C2×S3×D5C2×S3×D5D5×C4○D4D2024D6
kernelD2024D6D20⋊S3D12⋊D5D15⋊Q8D6.D10C4×S3×D5C20⋊D6C30.C23D10⋊D6C3×C4○D20C5×C4○D12C2×C4×D15C4○D20C4○D12Dic10C4×D5D20C5⋊D4C2×C20D15Dic6C4×S3D12C3⋊D4C2×C12C2×C4C5C4C22C3C1
# reps1111221221111212121424242224248

Matrix representation of D2024D6 in GL6(𝔽61)

44600000
100000
0005000
0050000
000010
000001
,
60440000
010000
0005000
0011000
000010
000001
,
6000000
0600000
000100
001000
0000141
00005259
,
44170000
1170000
0006000
0060000
0000241
00005259

G:=sub<GL(6,GF(61))| [44,1,0,0,0,0,60,0,0,0,0,0,0,0,0,50,0,0,0,0,50,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,0,0,0,0,0,44,1,0,0,0,0,0,0,0,11,0,0,0,0,50,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,52,0,0,0,0,41,59],[44,1,0,0,0,0,17,17,0,0,0,0,0,0,0,60,0,0,0,0,60,0,0,0,0,0,0,0,2,52,0,0,0,0,41,59] >;

D2024D6 in GAP, Magma, Sage, TeX

D_{20}\rtimes_{24}D_6
% in TeX

G:=Group("D20:24D6");
// GroupNames label

G:=SmallGroup(480,1092);
// by ID

G=gap.SmallGroup(480,1092);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,100,675,346,1356,18822]);
// 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^9,c*b*c^-1=a^10*b,d*b*d=a^18*b,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽