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G = D20:24D6order 480 = 25·3·5

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D20:24D6, D12:24D10, Dic6:22D10, Dic10:22D6, C30.20C24, C60.163C23, D30.38C23, Dic15.38C23, (C2xC20):7D6, C5:D4:8D6, C4oD20:7S3, C4oD12:7D5, (C4xD5):13D6, (C2xC12):7D10, C3:D4:8D10, (C4xS3):13D10, D15:Q8:13C2, C15:Q8:11C22, D15:1(C4oD4), D10:D6:7C2, C20:D6:13C2, (C2xC60):17C22, D20:S3:13C2, D12:D5:13C2, D6.6(C22xD5), (C6xD5).6C23, C6.20(C23xD5), (S3xC20):13C22, C30.C23:7C2, (C3xD20):31C22, (C5xD12):31C22, (D5xC12):13C22, (C4xD15):24C22, C3:D20:13C22, C15:D4:13C22, C5:D12:13C22, (S3xC10).6C23, C10.20(S3xC23), (D5xDic3):8C22, (S3xDic5):8C22, D10.6(C22xS3), D6.D10:11C2, (C2xC30).239C23, C20.189(C22xS3), (C5xDic6):28C22, C12.189(C22xD5), (C3xDic5).9C23, Dic3.9(C22xD5), (C5xDic3).9C23, Dic5.9(C22xS3), (C3xDic10):28C22, (C2xDic15):34C22, D30.C2.10C22, (C22xD15).122C22, C3:2(D5xC4oD4), C5:2(S3xC4oD4), (C4xS3xD5):10C2, (C2xC4xD15):27C2, (C2xC4):15(S3xD5), C15:10(C2xC4oD4), C4.162(C2xS3xD5), (C3xC4oD20):10C2, (C5xC4oD12):10C2, (C2xS3xD5).7C22, C2.23(C22xS3xD5), C22.18(C2xS3xD5), (C5xC3:D4):9C22, (C3xC5:D4):9C22, (C2xC6).11(C22xD5), (C2xC10).11(C22xS3), SmallGroup(480,1092)

Series: Derived Chief Lower central Upper central

C1C30 — D20:24D6
C1C5C15C30C6xD5C2xS3xD5C4xS3xD5 — D20:24D6
C15C30 — D20:24D6
C1C4C2xC4

Generators and relations for D20:24D6
 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, C3, C4, C4, C22, C22, C5, S3, C6, C6, C2xC4, C2xC4, D4, Q8, C23, D5, C10, C10, Dic3, Dic3, C12, C12, D6, D6, C2xC6, C2xC6, C15, C22xC4, C2xD4, C2xQ8, C4oD4, Dic5, Dic5, C20, C20, D10, D10, C2xC10, C2xC10, Dic6, Dic6, C4xS3, C4xS3, D12, D12, C2xDic3, C3:D4, C3:D4, C2xC12, C2xC12, C3xD4, C3xQ8, C22xS3, C5xS3, C3xD5, D15, D15, C30, C30, C2xC4oD4, Dic10, Dic10, C4xD5, C4xD5, D20, D20, C2xDic5, C5:D4, C5:D4, C2xC20, C2xC20, C5xD4, C5xQ8, C22xD5, S3xC2xC4, C4oD12, C4oD12, S3xD4, D4:2S3, S3xQ8, Q8:3S3, C3xC4oD4, C5xDic3, C3xDic5, Dic15, C60, S3xD5, C6xD5, S3xC10, D30, D30, C2xC30, C2xC4xD5, C4oD20, C4oD20, D4xD5, D4:2D5, Q8xD5, Q8:2D5, C5xC4oD4, S3xC4oD4, D5xDic3, S3xDic5, D30.C2, C15:D4, C3:D20, C5:D12, C15:Q8, C3xDic10, D5xC12, C3xD20, C3xC5:D4, C5xDic6, S3xC20, C5xD12, C5xC3:D4, C4xD15, C2xDic15, C2xC60, C2xS3xD5, C22xD15, D5xC4oD4, D20:S3, D12:D5, D15:Q8, D6.D10, C4xS3xD5, C20:D6, C30.C23, D10:D6, C3xC4oD20, C5xC4oD12, C2xC4xD15, D20:24D6
Quotients: C1, C2, C22, S3, C23, D5, D6, C4oD4, C24, D10, C22xS3, C2xC4oD4, C22xD5, S3xC23, S3xD5, C23xD5, S3xC4oD4, C2xS3xD5, D5xC4oD4, C22xS3xD5, D20:24D6

Smallest permutation representation of D20:24D6
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 28)(22 27)(23 26)(24 25)(29 40)(30 39)(31 38)(32 37)(33 36)(34 35)(41 58)(42 57)(43 56)(44 55)(45 54)(46 53)(47 52)(48 51)(49 50)(59 60)(61 66)(62 65)(63 64)(67 80)(68 79)(69 78)(70 77)(71 76)(72 75)(73 74)(81 88)(82 87)(83 86)(84 85)(89 100)(90 99)(91 98)(92 97)(93 96)(94 95)(101 108)(102 107)(103 106)(104 105)(109 120)(110 119)(111 118)(112 117)(113 116)(114 115)
(1 120 85 40 64 55)(2 101 86 21 65 56)(3 102 87 22 66 57)(4 103 88 23 67 58)(5 104 89 24 68 59)(6 105 90 25 69 60)(7 106 91 26 70 41)(8 107 92 27 71 42)(9 108 93 28 72 43)(10 109 94 29 73 44)(11 110 95 30 74 45)(12 111 96 31 75 46)(13 112 97 32 76 47)(14 113 98 33 77 48)(15 114 99 34 78 49)(16 115 100 35 79 50)(17 116 81 36 80 51)(18 117 82 37 61 52)(19 118 83 38 62 53)(20 119 84 39 63 54)
(1 45)(2 54)(3 43)(4 52)(5 41)(6 50)(7 59)(8 48)(9 57)(10 46)(11 55)(12 44)(13 53)(14 42)(15 51)(16 60)(17 49)(18 58)(19 47)(20 56)(21 84)(22 93)(23 82)(24 91)(25 100)(26 89)(27 98)(28 87)(29 96)(30 85)(31 94)(32 83)(33 92)(34 81)(35 90)(36 99)(37 88)(38 97)(39 86)(40 95)(61 103)(62 112)(63 101)(64 110)(65 119)(66 108)(67 117)(68 106)(69 115)(70 104)(71 113)(72 102)(73 111)(74 120)(75 109)(76 118)(77 107)(78 116)(79 105)(80 114)

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

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

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

66 conjugacy classes

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

66 irreducible representations

dim1111111111112222222222222444444
type+++++++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2S3D5D6D6D6D6D6C4oD4D10D10D10D10D10S3xD5S3xC4oD4C2xS3xD5C2xS3xD5D5xC4oD4D20:24D6
kernelD20:24D6D20:S3D12:D5D15:Q8D6.D10C4xS3xD5C20:D6C30.C23D10:D6C3xC4oD20C5xC4oD12C2xC4xD15C4oD20C4oD12Dic10C4xD5D20C5:D4C2xC20D15Dic6C4xS3D12C3:D4C2xC12C2xC4C5C4C22C3C1
# reps1111221221111212121424242224248

Matrix representation of D20:24D6 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] >;

D20:24D6 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

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