metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20⋊25D6, D12⋊25D10, Dic6⋊23D10, D60⋊34C22, C30.21C24, D30.7C23, C15⋊12+ 1+4, C60.164C23, Dic30⋊31C22, Dic15.10C23, (C2×C20)⋊8D6, C4○D12⋊8D5, (C6×D20)⋊3C2, (C4×S3)⋊1D10, (C2×C12)⋊8D10, (C2×D20)⋊14S3, (S3×D20)⋊11C2, C3⋊D4⋊13D10, C5⋊1(D4⋊6D6), (C2×C60)⋊4C22, (C22×D5)⋊5D6, C20⋊D6⋊11C2, C3⋊1(D4⋊8D10), C3⋊D20⋊1C22, C15⋊D4⋊1C22, D20⋊5S3⋊12C2, D20⋊S3⋊12C2, D60⋊11C2⋊7C2, D6.7(C22×D5), (C6×D5).7C23, C6.21(C23×D5), (S3×C20)⋊10C22, (C3×D20)⋊25C22, (C5×D12)⋊32C22, (C4×D15)⋊12C22, C15⋊7D4⋊15C22, (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
Generators and relations for D20⋊25D6
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, C3, C4, C4, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, Dic3, Dic3, C12, D6, D6, C2×C6, C2×C6, C15, C2×D4, C4○D4, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, Dic6, Dic6, C4×S3, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, C5×S3, C3×D5, D15, C30, C30, 2+ 1+4, Dic10, C4×D5, D20, D20, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×Q8, C22×D5, C22×D5, C4○D12, C4○D12, S3×D4, D4⋊2S3, C2×C3⋊D4, C6×D4, C5×Dic3, Dic15, C60, S3×D5, C6×D5, C6×D5, S3×C10, D30, C2×C30, C2×D20, C2×D20, C4○D20, D4×D5, Q8⋊2D5, C5×C4○D4, D4⋊6D6, D5×Dic3, C15⋊D4, C3⋊D20, C3×D20, C5×Dic6, S3×C20, C5×D12, C5×C3⋊D4, Dic30, C4×D15, D60, C15⋊7D4, C2×C60, C2×S3×D5, D5×C2×C6, D4⋊8D10, D20⋊5S3, D20⋊S3, S3×D20, C20⋊D6, D5×C3⋊D4, C6×D20, C5×C4○D12, D60⋊11C2, D20⋊25D6
Quotients: C1, C2, C22, S3, C23, D5, D6, C24, D10, C22×S3, 2+ 1+4, C22×D5, S3×C23, S3×D5, C23×D5, D4⋊6D6, C2×S3×D5, D4⋊8D10, C22×S3×D5, D20⋊25D6
►On 120 points
(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 32)(22 31)(23 30)(24 29)(25 28)(26 27)(33 40)(34 39)(35 38)(36 37)(41 48)(42 47)(43 46)(44 45)(49 60)(50 59)(51 58)(52 57)(53 56)(54 55)(61 76)(62 75)(63 74)(64 73)(65 72)(66 71)(67 70)(68 69)(77 80)(78 79)(81 84)(82 83)(85 100)(86 99)(87 98)(88 97)(89 96)(90 95)(91 94)(92 93)(101 110)(102 109)(103 108)(104 107)(105 106)(111 120)(112 119)(113 118)(114 117)(115 116)
(1 74 55 6 69 60)(2 73 56 5 70 59)(3 72 57 4 71 58)(7 68 41 20 75 54)(8 67 42 19 76 53)(9 66 43 18 77 52)(10 65 44 17 78 51)(11 64 45 16 79 50)(12 63 46 15 80 49)(13 62 47 14 61 48)(21 84 110 28 97 117)(22 83 111 27 98 116)(23 82 112 26 99 115)(24 81 113 25 100 114)(29 96 118 40 85 109)(30 95 119 39 86 108)(31 94 120 38 87 107)(32 93 101 37 88 106)(33 92 102 36 89 105)(34 91 103 35 90 104)
(1 88)(2 97)(3 86)(4 95)(5 84)(6 93)(7 82)(8 91)(9 100)(10 89)(11 98)(12 87)(13 96)(14 85)(15 94)(16 83)(17 92)(18 81)(19 90)(20 99)(21 70)(22 79)(23 68)(24 77)(25 66)(26 75)(27 64)(28 73)(29 62)(30 71)(31 80)(32 69)(33 78)(34 67)(35 76)(36 65)(37 74)(38 63)(39 72)(40 61)(41 115)(42 104)(43 113)(44 102)(45 111)(46 120)(47 109)(48 118)(49 107)(50 116)(51 105)(52 114)(53 103)(54 112)(55 101)(56 110)(57 119)(58 108)(59 117)(60 106)
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)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)(33,40)(34,39)(35,38)(36,37)(41,48)(42,47)(43,46)(44,45)(49,60)(50,59)(51,58)(52,57)(53,56)(54,55)(61,76)(62,75)(63,74)(64,73)(65,72)(66,71)(67,70)(68,69)(77,80)(78,79)(81,84)(82,83)(85,100)(86,99)(87,98)(88,97)(89,96)(90,95)(91,94)(92,93)(101,110)(102,109)(103,108)(104,107)(105,106)(111,120)(112,119)(113,118)(114,117)(115,116), (1,74,55,6,69,60)(2,73,56,5,70,59)(3,72,57,4,71,58)(7,68,41,20,75,54)(8,67,42,19,76,53)(9,66,43,18,77,52)(10,65,44,17,78,51)(11,64,45,16,79,50)(12,63,46,15,80,49)(13,62,47,14,61,48)(21,84,110,28,97,117)(22,83,111,27,98,116)(23,82,112,26,99,115)(24,81,113,25,100,114)(29,96,118,40,85,109)(30,95,119,39,86,108)(31,94,120,38,87,107)(32,93,101,37,88,106)(33,92,102,36,89,105)(34,91,103,35,90,104), (1,88)(2,97)(3,86)(4,95)(5,84)(6,93)(7,82)(8,91)(9,100)(10,89)(11,98)(12,87)(13,96)(14,85)(15,94)(16,83)(17,92)(18,81)(19,90)(20,99)(21,70)(22,79)(23,68)(24,77)(25,66)(26,75)(27,64)(28,73)(29,62)(30,71)(31,80)(32,69)(33,78)(34,67)(35,76)(36,65)(37,74)(38,63)(39,72)(40,61)(41,115)(42,104)(43,113)(44,102)(45,111)(46,120)(47,109)(48,118)(49,107)(50,116)(51,105)(52,114)(53,103)(54,112)(55,101)(56,110)(57,119)(58,108)(59,117)(60,106)>;
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)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)(33,40)(34,39)(35,38)(36,37)(41,48)(42,47)(43,46)(44,45)(49,60)(50,59)(51,58)(52,57)(53,56)(54,55)(61,76)(62,75)(63,74)(64,73)(65,72)(66,71)(67,70)(68,69)(77,80)(78,79)(81,84)(82,83)(85,100)(86,99)(87,98)(88,97)(89,96)(90,95)(91,94)(92,93)(101,110)(102,109)(103,108)(104,107)(105,106)(111,120)(112,119)(113,118)(114,117)(115,116), (1,74,55,6,69,60)(2,73,56,5,70,59)(3,72,57,4,71,58)(7,68,41,20,75,54)(8,67,42,19,76,53)(9,66,43,18,77,52)(10,65,44,17,78,51)(11,64,45,16,79,50)(12,63,46,15,80,49)(13,62,47,14,61,48)(21,84,110,28,97,117)(22,83,111,27,98,116)(23,82,112,26,99,115)(24,81,113,25,100,114)(29,96,118,40,85,109)(30,95,119,39,86,108)(31,94,120,38,87,107)(32,93,101,37,88,106)(33,92,102,36,89,105)(34,91,103,35,90,104), (1,88)(2,97)(3,86)(4,95)(5,84)(6,93)(7,82)(8,91)(9,100)(10,89)(11,98)(12,87)(13,96)(14,85)(15,94)(16,83)(17,92)(18,81)(19,90)(20,99)(21,70)(22,79)(23,68)(24,77)(25,66)(26,75)(27,64)(28,73)(29,62)(30,71)(31,80)(32,69)(33,78)(34,67)(35,76)(36,65)(37,74)(38,63)(39,72)(40,61)(41,115)(42,104)(43,113)(44,102)(45,111)(46,120)(47,109)(48,118)(49,107)(50,116)(51,105)(52,114)(53,103)(54,112)(55,101)(56,110)(57,119)(58,108)(59,117)(60,106) );
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),(21,32),(22,31),(23,30),(24,29),(25,28),(26,27),(33,40),(34,39),(35,38),(36,37),(41,48),(42,47),(43,46),(44,45),(49,60),(50,59),(51,58),(52,57),(53,56),(54,55),(61,76),(62,75),(63,74),(64,73),(65,72),(66,71),(67,70),(68,69),(77,80),(78,79),(81,84),(82,83),(85,100),(86,99),(87,98),(88,97),(89,96),(90,95),(91,94),(92,93),(101,110),(102,109),(103,108),(104,107),(105,106),(111,120),(112,119),(113,118),(114,117),(115,116)], [(1,74,55,6,69,60),(2,73,56,5,70,59),(3,72,57,4,71,58),(7,68,41,20,75,54),(8,67,42,19,76,53),(9,66,43,18,77,52),(10,65,44,17,78,51),(11,64,45,16,79,50),(12,63,46,15,80,49),(13,62,47,14,61,48),(21,84,110,28,97,117),(22,83,111,27,98,116),(23,82,112,26,99,115),(24,81,113,25,100,114),(29,96,118,40,85,109),(30,95,119,39,86,108),(31,94,120,38,87,107),(32,93,101,37,88,106),(33,92,102,36,89,105),(34,91,103,35,90,104)], [(1,88),(2,97),(3,86),(4,95),(5,84),(6,93),(7,82),(8,91),(9,100),(10,89),(11,98),(12,87),(13,96),(14,85),(15,94),(16,83),(17,92),(18,81),(19,90),(20,99),(21,70),(22,79),(23,68),(24,77),(25,66),(26,75),(27,64),(28,73),(29,62),(30,71),(31,80),(32,69),(33,78),(34,67),(35,76),(36,65),(37,74),(38,63),(39,72),(40,61),(41,115),(42,104),(43,113),(44,102),(45,111),(46,120),(47,109),(48,118),(49,107),(50,116),(51,105),(52,114),(53,103),(54,112),(55,101),(56,110),(57,119),(58,108),(59,117),(60,106)]])
63 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 12A | 12B | 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 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 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 | 10 | 10 | 30 | 30 | 2 | 2 | 2 | 6 | 6 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | 2 | 4 | 4 | 12 | 12 | 12 | 12 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 12 | 12 | 12 | 12 | 4 | ··· | 4 | 4 | ··· | 4 |
63 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D5 | D6 | D6 | D6 | D10 | D10 | D10 | D10 | D10 | 2+ 1+4 | S3×D5 | D4⋊6D6 | C2×S3×D5 | C2×S3×D5 | D4⋊8D10 | D20⋊25D6 |
| kernel | D20⋊25D6 | D20⋊5S3 | D20⋊S3 | S3×D20 | C20⋊D6 | D5×C3⋊D4 | C6×D20 | C5×C4○D12 | D60⋊11C2 | C2×D20 | C4○D12 | D20 | C2×C20 | C22×D5 | Dic6 | C4×S3 | D12 | C3⋊D4 | C2×C12 | C15 | C2×C4 | C5 | C4 | C22 | C3 | C1 |
| # reps | 1 | 2 | 2 | 2 | 2 | 4 | 1 | 1 | 1 | 1 | 2 | 4 | 1 | 2 | 2 | 4 | 2 | 4 | 2 | 1 | 2 | 2 | 4 | 2 | 4 | 8 |
Matrix representation of D20⋊25D6 ►in GL6(𝔽61)
| 0 | 60 | 0 | 0 | 0 | 0 |
| 1 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 0 | 0 | 0 | 0 | 1 | 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 | 0 | 0 | 0 | 0 | 0 |
| 43 | 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 | 0 | 0 | 0 | 0 | 0 |
| 18 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 14 |
| 0 | 0 | 0 | 0 | 14 | 0 |
| 0 | 0 | 0 | 48 | 0 | 0 |
| 0 | 0 | 48 | 0 | 0 | 0 |
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] >;
D20⋊25D6 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