metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20⋊26D6, D12⋊28D10, Dic10⋊23D6, D60⋊40C22, C30.22C24, D30.8C23, C15⋊22+ (1+4), C60.165C23, Dic30⋊37C22, Dic15.11C23, (C4×D5)⋊1D6, (C2×C20)⋊9D6, C4○D20⋊8S3, (C2×C12)⋊9D10, C5⋊1(D4○D12), C5⋊D4⋊14D6, (C10×D12)⋊3C2, (C2×D12)⋊14D5, (D5×D12)⋊11C2, (C2×C60)⋊3C22, C20⋊D6⋊12C2, C3⋊1(D4⋊6D10), (C22×S3)⋊4D10, C5⋊D12⋊1C22, C15⋊D4⋊2C22, D12⋊5D5⋊12C2, D12⋊D5⋊12C2, D60⋊11C2⋊4C2, (C6×D5).8C23, D6.8(C22×D5), C6.22(C23×D5), (C3×D20)⋊32C22, (C4×D15)⋊13C22, (C5×D12)⋊25C22, (D5×C12)⋊10C22, C15⋊7D4⋊16C22, (S3×C10).8C23, C10.22(S3×C23), (S3×Dic5)⋊1C22, D10.8(C22×S3), C20.131(C22×S3), (C2×C30).241C23, C12.162(C22×D5), (C3×Dic10)⋊29C22, Dic5.10(C22×S3), (C3×Dic5).10C23, (C2×C4)⋊5(S3×D5), C4.88(C2×S3×D5), (S3×C5⋊D4)⋊1C2, (C2×S3×D5)⋊2C22, (C3×C4○D20)⋊3C2, (S3×C2×C10)⋊6C22, C2.25(C22×S3×D5), C22.15(C2×S3×D5), (C3×C5⋊D4)⋊10C22, (C2×C6).12(C22×D5), (C2×C10).248(C22×S3), SmallGroup(480,1094)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1868 in 332 conjugacy classes, 108 normal (36 characteristic)
C1, C2, C2 [×9], C3, C4 [×2], C4 [×4], C22, C22 [×14], C5, S3 [×6], C6, C6 [×3], C2×C4, C2×C4 [×8], D4 [×18], Q8 [×2], C23 [×6], D5 [×4], C10, C10 [×5], Dic3 [×2], C12 [×2], C12 [×2], D6 [×4], D6 [×8], C2×C6, C2×C6 [×2], C15, C2×D4 [×9], C4○D4 [×6], Dic5 [×2], Dic5 [×2], C20 [×2], D10 [×2], D10 [×6], C2×C10, C2×C10 [×6], Dic6, C4×S3 [×6], D12 [×4], D12 [×5], C3⋊D4 [×6], C2×C12, C2×C12 [×2], C3×D4 [×3], C3×Q8, C22×S3 [×2], C22×S3 [×4], C5×S3 [×4], C3×D5 [×2], D15 [×2], C30, C30, 2+ (1+4), Dic10, Dic10, C4×D5 [×2], C4×D5 [×2], D20, D20, C2×Dic5 [×4], C5⋊D4 [×2], C5⋊D4 [×10], C2×C20, C5×D4 [×4], C22×D5 [×4], C22×C10 [×2], C2×D12, C2×D12 [×2], C4○D12 [×3], S3×D4 [×6], Q8⋊3S3 [×2], C3×C4○D4, C3×Dic5 [×2], Dic15 [×2], C60 [×2], S3×D5 [×4], C6×D5 [×2], S3×C10 [×4], S3×C10 [×2], D30 [×2], C2×C30, C4○D20, C4○D20, D4×D5 [×4], D4⋊2D5 [×4], C2×C5⋊D4 [×4], D4×C10, D4○D12, S3×Dic5 [×4], C15⋊D4 [×4], C5⋊D12 [×4], C3×Dic10, D5×C12 [×2], C3×D20, C3×C5⋊D4 [×2], C5×D12 [×4], Dic30, C4×D15 [×2], D60, C15⋊7D4 [×2], C2×C60, C2×S3×D5 [×4], S3×C2×C10 [×2], D4⋊6D10, D12⋊D5 [×2], D12⋊5D5 [×2], D5×D12 [×2], C20⋊D6 [×2], S3×C5⋊D4 [×4], C3×C4○D20, C10×D12, D60⋊11C2, D20⋊26D6
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, D4○D12, C2×S3×D5 [×3], D4⋊6D10, C22×S3×D5, D20⋊26D6
Generators and relations
G = < a,b,c,d | a20=b2=c6=d2=1, bab=a-1, ac=ca, dad=a11, cbc-1=a10b, bd=db, dcd=c-1 >
►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 75)(2 74)(3 73)(4 72)(5 71)(6 70)(7 69)(8 68)(9 67)(10 66)(11 65)(12 64)(13 63)(14 62)(15 61)(16 80)(17 79)(18 78)(19 77)(20 76)(21 55)(22 54)(23 53)(24 52)(25 51)(26 50)(27 49)(28 48)(29 47)(30 46)(31 45)(32 44)(33 43)(34 42)(35 41)(36 60)(37 59)(38 58)(39 57)(40 56)(81 110)(82 109)(83 108)(84 107)(85 106)(86 105)(87 104)(88 103)(89 102)(90 101)(91 120)(92 119)(93 118)(94 117)(95 116)(96 115)(97 114)(98 113)(99 112)(100 111)
(1 26 109)(2 27 110)(3 28 111)(4 29 112)(5 30 113)(6 31 114)(7 32 115)(8 33 116)(9 34 117)(10 35 118)(11 36 119)(12 37 120)(13 38 101)(14 39 102)(15 40 103)(16 21 104)(17 22 105)(18 23 106)(19 24 107)(20 25 108)(41 83 66 51 93 76)(42 84 67 52 94 77)(43 85 68 53 95 78)(44 86 69 54 96 79)(45 87 70 55 97 80)(46 88 71 56 98 61)(47 89 72 57 99 62)(48 90 73 58 100 63)(49 91 74 59 81 64)(50 92 75 60 82 65)
(1 109)(2 120)(3 111)(4 102)(5 113)(6 104)(7 115)(8 106)(9 117)(10 108)(11 119)(12 110)(13 101)(14 112)(15 103)(16 114)(17 105)(18 116)(19 107)(20 118)(21 31)(23 33)(25 35)(27 37)(29 39)(41 51)(43 53)(45 55)(47 57)(49 59)(61 88)(62 99)(63 90)(64 81)(65 92)(66 83)(67 94)(68 85)(69 96)(70 87)(71 98)(72 89)(73 100)(74 91)(75 82)(76 93)(77 84)(78 95)(79 86)(80 97)
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), (1,75)(2,74)(3,73)(4,72)(5,71)(6,70)(7,69)(8,68)(9,67)(10,66)(11,65)(12,64)(13,63)(14,62)(15,61)(16,80)(17,79)(18,78)(19,77)(20,76)(21,55)(22,54)(23,53)(24,52)(25,51)(26,50)(27,49)(28,48)(29,47)(30,46)(31,45)(32,44)(33,43)(34,42)(35,41)(36,60)(37,59)(38,58)(39,57)(40,56)(81,110)(82,109)(83,108)(84,107)(85,106)(86,105)(87,104)(88,103)(89,102)(90,101)(91,120)(92,119)(93,118)(94,117)(95,116)(96,115)(97,114)(98,113)(99,112)(100,111), (1,26,109)(2,27,110)(3,28,111)(4,29,112)(5,30,113)(6,31,114)(7,32,115)(8,33,116)(9,34,117)(10,35,118)(11,36,119)(12,37,120)(13,38,101)(14,39,102)(15,40,103)(16,21,104)(17,22,105)(18,23,106)(19,24,107)(20,25,108)(41,83,66,51,93,76)(42,84,67,52,94,77)(43,85,68,53,95,78)(44,86,69,54,96,79)(45,87,70,55,97,80)(46,88,71,56,98,61)(47,89,72,57,99,62)(48,90,73,58,100,63)(49,91,74,59,81,64)(50,92,75,60,82,65), (1,109)(2,120)(3,111)(4,102)(5,113)(6,104)(7,115)(8,106)(9,117)(10,108)(11,119)(12,110)(13,101)(14,112)(15,103)(16,114)(17,105)(18,116)(19,107)(20,118)(21,31)(23,33)(25,35)(27,37)(29,39)(41,51)(43,53)(45,55)(47,57)(49,59)(61,88)(62,99)(63,90)(64,81)(65,92)(66,83)(67,94)(68,85)(69,96)(70,87)(71,98)(72,89)(73,100)(74,91)(75,82)(76,93)(77,84)(78,95)(79,86)(80,97)>;
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), (1,75)(2,74)(3,73)(4,72)(5,71)(6,70)(7,69)(8,68)(9,67)(10,66)(11,65)(12,64)(13,63)(14,62)(15,61)(16,80)(17,79)(18,78)(19,77)(20,76)(21,55)(22,54)(23,53)(24,52)(25,51)(26,50)(27,49)(28,48)(29,47)(30,46)(31,45)(32,44)(33,43)(34,42)(35,41)(36,60)(37,59)(38,58)(39,57)(40,56)(81,110)(82,109)(83,108)(84,107)(85,106)(86,105)(87,104)(88,103)(89,102)(90,101)(91,120)(92,119)(93,118)(94,117)(95,116)(96,115)(97,114)(98,113)(99,112)(100,111), (1,26,109)(2,27,110)(3,28,111)(4,29,112)(5,30,113)(6,31,114)(7,32,115)(8,33,116)(9,34,117)(10,35,118)(11,36,119)(12,37,120)(13,38,101)(14,39,102)(15,40,103)(16,21,104)(17,22,105)(18,23,106)(19,24,107)(20,25,108)(41,83,66,51,93,76)(42,84,67,52,94,77)(43,85,68,53,95,78)(44,86,69,54,96,79)(45,87,70,55,97,80)(46,88,71,56,98,61)(47,89,72,57,99,62)(48,90,73,58,100,63)(49,91,74,59,81,64)(50,92,75,60,82,65), (1,109)(2,120)(3,111)(4,102)(5,113)(6,104)(7,115)(8,106)(9,117)(10,108)(11,119)(12,110)(13,101)(14,112)(15,103)(16,114)(17,105)(18,116)(19,107)(20,118)(21,31)(23,33)(25,35)(27,37)(29,39)(41,51)(43,53)(45,55)(47,57)(49,59)(61,88)(62,99)(63,90)(64,81)(65,92)(66,83)(67,94)(68,85)(69,96)(70,87)(71,98)(72,89)(73,100)(74,91)(75,82)(76,93)(77,84)(78,95)(79,86)(80,97) );
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)], [(1,75),(2,74),(3,73),(4,72),(5,71),(6,70),(7,69),(8,68),(9,67),(10,66),(11,65),(12,64),(13,63),(14,62),(15,61),(16,80),(17,79),(18,78),(19,77),(20,76),(21,55),(22,54),(23,53),(24,52),(25,51),(26,50),(27,49),(28,48),(29,47),(30,46),(31,45),(32,44),(33,43),(34,42),(35,41),(36,60),(37,59),(38,58),(39,57),(40,56),(81,110),(82,109),(83,108),(84,107),(85,106),(86,105),(87,104),(88,103),(89,102),(90,101),(91,120),(92,119),(93,118),(94,117),(95,116),(96,115),(97,114),(98,113),(99,112),(100,111)], [(1,26,109),(2,27,110),(3,28,111),(4,29,112),(5,30,113),(6,31,114),(7,32,115),(8,33,116),(9,34,117),(10,35,118),(11,36,119),(12,37,120),(13,38,101),(14,39,102),(15,40,103),(16,21,104),(17,22,105),(18,23,106),(19,24,107),(20,25,108),(41,83,66,51,93,76),(42,84,67,52,94,77),(43,85,68,53,95,78),(44,86,69,54,96,79),(45,87,70,55,97,80),(46,88,71,56,98,61),(47,89,72,57,99,62),(48,90,73,58,100,63),(49,91,74,59,81,64),(50,92,75,60,82,65)], [(1,109),(2,120),(3,111),(4,102),(5,113),(6,104),(7,115),(8,106),(9,117),(10,108),(11,119),(12,110),(13,101),(14,112),(15,103),(16,114),(17,105),(18,116),(19,107),(20,118),(21,31),(23,33),(25,35),(27,37),(29,39),(41,51),(43,53),(45,55),(47,57),(49,59),(61,88),(62,99),(63,90),(64,81),(65,92),(66,83),(67,94),(68,85),(69,96),(70,87),(71,98),(72,89),(73,100),(74,91),(75,82),(76,93),(77,84),(78,95),(79,86),(80,97)])
Matrix representation ►G ⊆ GL6(𝔽61)
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 58 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 41 |
| 0 | 0 | 0 | 0 | 20 | 0 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 41 |
| 0 | 0 | 0 | 0 | 20 | 0 |
| 0 | 0 | 0 | 58 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 1 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 1 | 60 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,3,0,0,0,0,58,0,0,0,0,0,0,0,0,20,0,0,0,0,41,0],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,3,0,0,0,0,58,0,0,0,0,20,0,0,0,0,41,0,0,0],[0,1,0,0,0,0,60,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[1,0,0,0,0,0,60,60,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,1] >;
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 | 10A | ··· | 10F | 10G | ··· | 10N | 12A | 12B | 12C | 12D | 12E | 15A | 15B | 20A | 20B | 20C | 20D | 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 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 6 | 6 | 6 | 6 | 10 | 10 | 30 | 30 | 2 | 2 | 2 | 10 | 10 | 30 | 30 | 2 | 2 | 2 | 4 | 20 | 20 | 2 | ··· | 2 | 12 | ··· | 12 | 2 | 2 | 4 | 20 | 20 | 4 | 4 | 4 | 4 | 4 | 4 | 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 | D6 | D6 | D10 | D10 | D10 | 2+ (1+4) | S3×D5 | D4○D12 | C2×S3×D5 | C2×S3×D5 | D4⋊6D10 | D20⋊26D6 |
| kernel | D20⋊26D6 | D12⋊D5 | D12⋊5D5 | D5×D12 | C20⋊D6 | S3×C5⋊D4 | C3×C4○D20 | C10×D12 | D60⋊11C2 | C4○D20 | C2×D12 | Dic10 | C4×D5 | D20 | C5⋊D4 | C2×C20 | D12 | C2×C12 | C22×S3 | C15 | C2×C4 | C5 | C4 | C22 | C3 | C1 |
| # reps | 1 | 2 | 2 | 2 | 2 | 4 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 8 | 2 | 4 | 1 | 2 | 2 | 4 | 2 | 4 | 8 |
In GAP, Magma, Sage, TeX
D_{20}\rtimes_{26}D_6 % in TeX
G:=Group("D20:26D6"); // GroupNames label
G:=SmallGroup(480,1094);
// by ID
G=gap.SmallGroup(480,1094);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,219,100,675,1356,18822]);
// 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^11,c*b*c^-1=a^10*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations