metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D6⋊5D20, D30⋊5D4, D10⋊5D12, D6⋊C4⋊8D5, (C6×D5)⋊5D4, (C2×C20)⋊2D6, C15⋊3C22≀C2, (C2×D60)⋊3C2, (S3×C10)⋊5D4, (C2×C12)⋊2D10, C6.27(D4×D5), C5⋊1(D6⋊D4), (C2×C60)⋊1C22, (C2×Dic5)⋊2D6, C30.70(C2×D4), C6.28(C2×D20), C2.29(D5×D12), C10.27(S3×D4), C2.29(S3×D20), D10⋊C4⋊8S3, C3⋊1(C22⋊D20), (C2×Dic3)⋊2D10, C10.29(C2×D12), D30⋊4C4⋊24C2, (C6×Dic5)⋊4C22, (C22×D5).60D6, (C2×C30).166C23, (C10×Dic3)⋊4C22, (C22×S3).51D10, (C22×D15)⋊5C22, C2.19(D10⋊D6), (C2×C4)⋊3(S3×D5), (C5×D6⋊C4)⋊8C2, (C22×S3×D5)⋊3C2, (C2×C3⋊D20)⋊10C2, (C2×C5⋊D12)⋊10C2, (C3×D10⋊C4)⋊8C2, (D5×C2×C6).43C22, C22.214(C2×S3×D5), (S3×C2×C10).43C22, (C2×C6).178(C22×D5), (C2×C10).178(C22×S3), SmallGroup(480,552)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D6⋊5D20
G = < a,b,c,d | a6=b2=c20=d2=1, bab=dad=a-1, ac=ca, cbc-1=a3b, dbd=ab, dcd=c-1 >
Subgroups: 1964 in 260 conjugacy classes, 54 normal (44 characteristic)
C1, C2 [×3], C2 [×7], C3, C4 [×3], C22, C22 [×23], C5, S3 [×5], C6 [×3], C6 [×2], C2×C4, C2×C4 [×2], D4 [×6], C23 [×10], D5 [×5], C10 [×3], C10 [×2], Dic3, C12 [×2], D6 [×2], D6 [×17], C2×C6, C2×C6 [×4], C15, C22⋊C4 [×3], C2×D4 [×3], C24, Dic5, C20 [×2], D10 [×2], D10 [×17], C2×C10, C2×C10 [×4], D12 [×4], C2×Dic3, C3⋊D4 [×2], C2×C12, C2×C12, C22×S3, C22×S3 [×8], C22×C6, C5×S3 [×2], C3×D5 [×2], D15 [×3], C30 [×3], C22≀C2, D20 [×4], C2×Dic5, C5⋊D4 [×2], C2×C20, C2×C20, C22×D5, C22×D5 [×8], C22×C10, D6⋊C4, D6⋊C4, C3×C22⋊C4, C2×D12 [×2], C2×C3⋊D4, S3×C23, C5×Dic3, C3×Dic5, C60, S3×D5 [×8], C6×D5 [×2], C6×D5 [×2], S3×C10 [×2], S3×C10 [×2], D30 [×2], D30 [×5], C2×C30, D10⋊C4, D10⋊C4, C5×C22⋊C4, C2×D20 [×2], C2×C5⋊D4, C23×D5, D6⋊D4, C3⋊D20 [×2], C5⋊D12 [×2], C6×Dic5, C10×Dic3, D60 [×2], C2×C60, C2×S3×D5 [×6], D5×C2×C6, S3×C2×C10, C22×D15 [×2], C22⋊D20, D30⋊4C4, C3×D10⋊C4, C5×D6⋊C4, C2×C3⋊D20, C2×C5⋊D12, C2×D60, C22×S3×D5, D6⋊5D20
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D5, D6 [×3], C2×D4 [×3], D10 [×3], D12 [×2], C22×S3, C22≀C2, D20 [×2], C22×D5, C2×D12, S3×D4 [×2], S3×D5, C2×D20, D4×D5 [×2], D6⋊D4, C2×S3×D5, C22⋊D20, D5×D12, S3×D20, D10⋊D6, D6⋊5D20
(1 78 105 92 30 44)(2 79 106 93 31 45)(3 80 107 94 32 46)(4 61 108 95 33 47)(5 62 109 96 34 48)(6 63 110 97 35 49)(7 64 111 98 36 50)(8 65 112 99 37 51)(9 66 113 100 38 52)(10 67 114 81 39 53)(11 68 115 82 40 54)(12 69 116 83 21 55)(13 70 117 84 22 56)(14 71 118 85 23 57)(15 72 119 86 24 58)(16 73 120 87 25 59)(17 74 101 88 26 60)(18 75 102 89 27 41)(19 76 103 90 28 42)(20 77 104 91 29 43)
(1 115)(2 55)(3 117)(4 57)(5 119)(6 59)(7 101)(8 41)(9 103)(10 43)(11 105)(12 45)(13 107)(14 47)(15 109)(16 49)(17 111)(18 51)(19 113)(20 53)(21 79)(22 32)(23 61)(24 34)(25 63)(26 36)(27 65)(28 38)(29 67)(30 40)(31 69)(33 71)(35 73)(37 75)(39 77)(42 100)(44 82)(46 84)(48 86)(50 88)(52 90)(54 92)(56 94)(58 96)(60 98)(62 72)(64 74)(66 76)(68 78)(70 80)(81 104)(83 106)(85 108)(87 110)(89 112)(91 114)(93 116)(95 118)(97 120)(99 102)
(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 10)(2 9)(3 8)(4 7)(5 6)(11 20)(12 19)(13 18)(14 17)(15 16)(21 103)(22 102)(23 101)(24 120)(25 119)(26 118)(27 117)(28 116)(29 115)(30 114)(31 113)(32 112)(33 111)(34 110)(35 109)(36 108)(37 107)(38 106)(39 105)(40 104)(41 70)(42 69)(43 68)(44 67)(45 66)(46 65)(47 64)(48 63)(49 62)(50 61)(51 80)(52 79)(53 78)(54 77)(55 76)(56 75)(57 74)(58 73)(59 72)(60 71)(81 92)(82 91)(83 90)(84 89)(85 88)(86 87)(93 100)(94 99)(95 98)(96 97)
G:=sub<Sym(120)| (1,78,105,92,30,44)(2,79,106,93,31,45)(3,80,107,94,32,46)(4,61,108,95,33,47)(5,62,109,96,34,48)(6,63,110,97,35,49)(7,64,111,98,36,50)(8,65,112,99,37,51)(9,66,113,100,38,52)(10,67,114,81,39,53)(11,68,115,82,40,54)(12,69,116,83,21,55)(13,70,117,84,22,56)(14,71,118,85,23,57)(15,72,119,86,24,58)(16,73,120,87,25,59)(17,74,101,88,26,60)(18,75,102,89,27,41)(19,76,103,90,28,42)(20,77,104,91,29,43), (1,115)(2,55)(3,117)(4,57)(5,119)(6,59)(7,101)(8,41)(9,103)(10,43)(11,105)(12,45)(13,107)(14,47)(15,109)(16,49)(17,111)(18,51)(19,113)(20,53)(21,79)(22,32)(23,61)(24,34)(25,63)(26,36)(27,65)(28,38)(29,67)(30,40)(31,69)(33,71)(35,73)(37,75)(39,77)(42,100)(44,82)(46,84)(48,86)(50,88)(52,90)(54,92)(56,94)(58,96)(60,98)(62,72)(64,74)(66,76)(68,78)(70,80)(81,104)(83,106)(85,108)(87,110)(89,112)(91,114)(93,116)(95,118)(97,120)(99,102), (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), (1,10)(2,9)(3,8)(4,7)(5,6)(11,20)(12,19)(13,18)(14,17)(15,16)(21,103)(22,102)(23,101)(24,120)(25,119)(26,118)(27,117)(28,116)(29,115)(30,114)(31,113)(32,112)(33,111)(34,110)(35,109)(36,108)(37,107)(38,106)(39,105)(40,104)(41,70)(42,69)(43,68)(44,67)(45,66)(46,65)(47,64)(48,63)(49,62)(50,61)(51,80)(52,79)(53,78)(54,77)(55,76)(56,75)(57,74)(58,73)(59,72)(60,71)(81,92)(82,91)(83,90)(84,89)(85,88)(86,87)(93,100)(94,99)(95,98)(96,97)>;
G:=Group( (1,78,105,92,30,44)(2,79,106,93,31,45)(3,80,107,94,32,46)(4,61,108,95,33,47)(5,62,109,96,34,48)(6,63,110,97,35,49)(7,64,111,98,36,50)(8,65,112,99,37,51)(9,66,113,100,38,52)(10,67,114,81,39,53)(11,68,115,82,40,54)(12,69,116,83,21,55)(13,70,117,84,22,56)(14,71,118,85,23,57)(15,72,119,86,24,58)(16,73,120,87,25,59)(17,74,101,88,26,60)(18,75,102,89,27,41)(19,76,103,90,28,42)(20,77,104,91,29,43), (1,115)(2,55)(3,117)(4,57)(5,119)(6,59)(7,101)(8,41)(9,103)(10,43)(11,105)(12,45)(13,107)(14,47)(15,109)(16,49)(17,111)(18,51)(19,113)(20,53)(21,79)(22,32)(23,61)(24,34)(25,63)(26,36)(27,65)(28,38)(29,67)(30,40)(31,69)(33,71)(35,73)(37,75)(39,77)(42,100)(44,82)(46,84)(48,86)(50,88)(52,90)(54,92)(56,94)(58,96)(60,98)(62,72)(64,74)(66,76)(68,78)(70,80)(81,104)(83,106)(85,108)(87,110)(89,112)(91,114)(93,116)(95,118)(97,120)(99,102), (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), (1,10)(2,9)(3,8)(4,7)(5,6)(11,20)(12,19)(13,18)(14,17)(15,16)(21,103)(22,102)(23,101)(24,120)(25,119)(26,118)(27,117)(28,116)(29,115)(30,114)(31,113)(32,112)(33,111)(34,110)(35,109)(36,108)(37,107)(38,106)(39,105)(40,104)(41,70)(42,69)(43,68)(44,67)(45,66)(46,65)(47,64)(48,63)(49,62)(50,61)(51,80)(52,79)(53,78)(54,77)(55,76)(56,75)(57,74)(58,73)(59,72)(60,71)(81,92)(82,91)(83,90)(84,89)(85,88)(86,87)(93,100)(94,99)(95,98)(96,97) );
G=PermutationGroup([(1,78,105,92,30,44),(2,79,106,93,31,45),(3,80,107,94,32,46),(4,61,108,95,33,47),(5,62,109,96,34,48),(6,63,110,97,35,49),(7,64,111,98,36,50),(8,65,112,99,37,51),(9,66,113,100,38,52),(10,67,114,81,39,53),(11,68,115,82,40,54),(12,69,116,83,21,55),(13,70,117,84,22,56),(14,71,118,85,23,57),(15,72,119,86,24,58),(16,73,120,87,25,59),(17,74,101,88,26,60),(18,75,102,89,27,41),(19,76,103,90,28,42),(20,77,104,91,29,43)], [(1,115),(2,55),(3,117),(4,57),(5,119),(6,59),(7,101),(8,41),(9,103),(10,43),(11,105),(12,45),(13,107),(14,47),(15,109),(16,49),(17,111),(18,51),(19,113),(20,53),(21,79),(22,32),(23,61),(24,34),(25,63),(26,36),(27,65),(28,38),(29,67),(30,40),(31,69),(33,71),(35,73),(37,75),(39,77),(42,100),(44,82),(46,84),(48,86),(50,88),(52,90),(54,92),(56,94),(58,96),(60,98),(62,72),(64,74),(66,76),(68,78),(70,80),(81,104),(83,106),(85,108),(87,110),(89,112),(91,114),(93,116),(95,118),(97,120),(99,102)], [(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)], [(1,10),(2,9),(3,8),(4,7),(5,6),(11,20),(12,19),(13,18),(14,17),(15,16),(21,103),(22,102),(23,101),(24,120),(25,119),(26,118),(27,117),(28,116),(29,115),(30,114),(31,113),(32,112),(33,111),(34,110),(35,109),(36,108),(37,107),(38,106),(39,105),(40,104),(41,70),(42,69),(43,68),(44,67),(45,66),(46,65),(47,64),(48,63),(49,62),(50,61),(51,80),(52,79),(53,78),(54,77),(55,76),(56,75),(57,74),(58,73),(59,72),(60,71),(81,92),(82,91),(83,90),(84,89),(85,88),(86,87),(93,100),(94,99),(95,98),(96,97)])
60 conjugacy classes
class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 3 | 4A | 4B | 4C | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 12A | 12B | 12C | 12D | 15A | 15B | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 30A | ··· | 30F | 60A | ··· | 60H |
order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
size | 1 | 1 | 1 | 1 | 6 | 6 | 10 | 10 | 30 | 30 | 60 | 2 | 4 | 12 | 20 | 2 | 2 | 2 | 2 | 2 | 20 | 20 | 2 | ··· | 2 | 12 | 12 | 12 | 12 | 4 | 4 | 20 | 20 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 4 | ··· | 4 | 4 | ··· | 4 |
60 irreducible representations
dim | 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 | 4 |
type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D4 | D5 | D6 | D6 | D6 | D10 | D10 | D10 | D12 | D20 | S3×D4 | S3×D5 | D4×D5 | C2×S3×D5 | D5×D12 | S3×D20 | D10⋊D6 |
kernel | D6⋊5D20 | D30⋊4C4 | C3×D10⋊C4 | C5×D6⋊C4 | C2×C3⋊D20 | C2×C5⋊D12 | C2×D60 | C22×S3×D5 | D10⋊C4 | C6×D5 | S3×C10 | D30 | D6⋊C4 | C2×Dic5 | C2×C20 | C22×D5 | C2×Dic3 | C2×C12 | C22×S3 | D10 | D6 | C10 | C2×C4 | C6 | C22 | C2 | C2 | C2 |
# reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 8 | 2 | 2 | 4 | 2 | 4 | 4 | 4 |
Matrix representation of D6⋊5D20 ►in GL6(𝔽61)
1 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 60 | 60 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 60 | 0 |
0 | 0 | 0 | 0 | 0 | 60 |
60 | 0 | 0 | 0 | 0 | 0 |
0 | 60 | 0 | 0 | 0 | 0 |
0 | 0 | 60 | 60 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 0 | 60 | 0 |
0 | 0 | 0 | 0 | 48 | 1 |
7 | 32 | 0 | 0 | 0 | 0 |
29 | 2 | 0 | 0 | 0 | 0 |
0 | 0 | 60 | 0 | 0 | 0 |
0 | 0 | 0 | 60 | 0 | 0 |
0 | 0 | 0 | 0 | 53 | 20 |
0 | 0 | 0 | 0 | 12 | 8 |
7 | 32 | 0 | 0 | 0 | 0 |
29 | 54 | 0 | 0 | 0 | 0 |
0 | 0 | 60 | 0 | 0 | 0 |
0 | 0 | 1 | 1 | 0 | 0 |
0 | 0 | 0 | 0 | 8 | 41 |
0 | 0 | 0 | 0 | 55 | 53 |
G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,1,0,0,0,0,60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,60,1,0,0,0,0,0,0,60,48,0,0,0,0,0,1],[7,29,0,0,0,0,32,2,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,53,12,0,0,0,0,20,8],[7,29,0,0,0,0,32,54,0,0,0,0,0,0,60,1,0,0,0,0,0,1,0,0,0,0,0,0,8,55,0,0,0,0,41,53] >;
D6⋊5D20 in GAP, Magma, Sage, TeX
D_6\rtimes_5D_{20}
% in TeX
G:=Group("D6:5D20");
// GroupNames label
G:=SmallGroup(480,552);
// by ID
G=gap.SmallGroup(480,552);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,254,219,142,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^2=c^20=d^2=1,b*a*b=d*a*d=a^-1,a*c=c*a,c*b*c^-1=a^3*b,d*b*d=a*b,d*c*d=c^-1>;
// generators/relations