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
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
(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 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 6A | 6B | 6C | 6D | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 12A | 12B | 12C | 12D | 12E | 15A | 15B | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 20I | 20J | 30A | ··· | 30F | 60A | ··· | 60H |
order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
size | 1 | 1 | 2 | 6 | 6 | 10 | 10 | 15 | 15 | 30 | 2 | 1 | 1 | 2 | 6 | 6 | 10 | 10 | 15 | 15 | 30 | 2 | 2 | 2 | 4 | 20 | 20 | 2 | 2 | 4 | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 4 | 20 | 20 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 12 | 12 | 12 | 12 | 4 | ··· | 4 | 4 | ··· | 4 |
66 irreducible representations
dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D5 | D6 | D6 | D6 | D6 | D6 | C4oD4 | D10 | D10 | D10 | D10 | D10 | S3xD5 | S3xC4oD4 | C2xS3xD5 | C2xS3xD5 | D5xC4oD4 | D20:24D6 |
kernel | D20:24D6 | D20:S3 | D12:D5 | D15:Q8 | D6.D10 | C4xS3xD5 | C20:D6 | C30.C23 | D10:D6 | C3xC4oD20 | C5xC4oD12 | C2xC4xD15 | C4oD20 | C4oD12 | Dic10 | C4xD5 | D20 | C5:D4 | C2xC20 | D15 | Dic6 | C4xS3 | D12 | C3:D4 | C2xC12 | C2xC4 | C5 | C4 | C22 | C3 | C1 |
# reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 4 | 2 | 4 | 2 | 4 | 2 | 2 | 2 | 4 | 2 | 4 | 8 |
Matrix representation of D20:24D6 ►in GL6(F61)
44 | 60 | 0 | 0 | 0 | 0 |
1 | 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 | 44 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 50 | 0 | 0 |
0 | 0 | 11 | 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 | 41 |
0 | 0 | 0 | 0 | 52 | 59 |
44 | 17 | 0 | 0 | 0 | 0 |
1 | 17 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 60 | 0 | 0 |
0 | 0 | 60 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 2 | 41 |
0 | 0 | 0 | 0 | 52 | 59 |
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