metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20.8D6, D12⋊10D10, C60.13C23, Dic30⋊4C22, C3⋊C8⋊7D10, D4⋊S3⋊4D5, (D4×D5)⋊2S3, C15⋊D8⋊3C2, (C5×D4).5D6, (C4×D5).7D6, D4.9(S3×D5), C3⋊6(D8⋊D5), D4.D15⋊5C2, (C6×D5).62D4, C6.142(D4×D5), D12⋊5D5⋊1C2, C15⋊16(C8⋊C22), C15⋊3C8⋊7C22, (C3×D4).20D10, C30.175(C2×D4), C6.D20⋊3C2, C5⋊2(D12⋊6C22), (C5×D12)⋊4C22, C20.32D6⋊3C2, C20.13(C22×S3), (C3×Dic5).13D4, (D5×C12).5C22, (C3×D20).4C22, (D4×C15).7C22, C12.13(C22×D5), D10.28(C3⋊D4), Dic5.22(C3⋊D4), (C3×D4×D5)⋊2C2, C4.13(C2×S3×D5), (C5×D4⋊S3)⋊5C2, (C5×C3⋊C8)⋊7C22, C2.24(D5×C3⋊D4), C10.45(C2×C3⋊D4), SmallGroup(480,565)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12⋊10D10
G = < a,b,c,d | a12=b2=c10=d2=1, bab=a-1, cac-1=dad=a7, cbc-1=a3b, dbd=a9b, dcd=c-1 >
Subgroups: 748 in 136 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2 [×4], C3, C4, C4 [×2], C22 [×6], C5, S3, C6, C6 [×3], C8 [×2], C2×C4 [×2], D4, D4 [×4], Q8, C23, D5 [×2], C10, C10 [×2], Dic3, C12, C12, D6, C2×C6 [×5], C15, M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, Dic5, Dic5, C20, D10, D10 [×3], C2×C10 [×2], C3⋊C8, C3⋊C8, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C3×D4, C3×D4 [×2], C22×C6, C5×S3, C3×D5 [×2], C30, C30, C8⋊C22, C5⋊2C8, C40, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4 [×2], C5×D4, C5×D4, C22×D5, C4.Dic3, D4⋊S3, D4⋊S3, D4.S3 [×2], C4○D12, C6×D4, C3×Dic5, Dic15, C60, C6×D5, C6×D5 [×3], S3×C10, C2×C30, C8⋊D5, C40⋊C2, D4⋊D5, D4.D5, C5×D8, D4×D5, D4⋊2D5, D12⋊6C22, C5×C3⋊C8, C15⋊3C8, S3×Dic5, C15⋊D4, D5×C12, C3×D20, C3×C5⋊D4, C5×D12, Dic30, D4×C15, D5×C2×C6, D8⋊D5, C20.32D6, C15⋊D8, C6.D20, C5×D4⋊S3, D4.D15, D12⋊5D5, C3×D4×D5, D12⋊10D10
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], C3⋊D4 [×2], C22×S3, C8⋊C22, C22×D5, C2×C3⋊D4, S3×D5, D4×D5, D12⋊6C22, C2×S3×D5, D8⋊D5, D5×C3⋊D4, D12⋊10D10
►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 91)(2 90)(3 89)(4 88)(5 87)(6 86)(7 85)(8 96)(9 95)(10 94)(11 93)(12 92)(13 115)(14 114)(15 113)(16 112)(17 111)(18 110)(19 109)(20 120)(21 119)(22 118)(23 117)(24 116)(25 37)(26 48)(27 47)(28 46)(29 45)(30 44)(31 43)(32 42)(33 41)(34 40)(35 39)(36 38)(49 108)(50 107)(51 106)(52 105)(53 104)(54 103)(55 102)(56 101)(57 100)(58 99)(59 98)(60 97)(61 83)(62 82)(63 81)(64 80)(65 79)(66 78)(67 77)(68 76)(69 75)(70 74)(71 73)(72 84)
(1 98 23 78 27 4 107 14 75 30)(2 105 24 73 28 11 108 21 76 25)(3 100 13 80 29 6 97 16 77 32)(5 102 15 82 31 8 99 18 79 34)(7 104 17 84 33 10 101 20 81 36)(9 106 19 74 35 12 103 22 83 26)(37 87 52 113 71 43 93 58 119 65)(38 94 53 120 72)(39 89 54 115 61 45 95 60 109 67)(40 96 55 110 62)(41 91 56 117 63 47 85 50 111 69)(42 86 57 112 64)(44 88 59 114 66)(46 90 49 116 68)(48 92 51 118 70)
(1 30)(2 25)(3 32)(4 27)(5 34)(6 29)(7 36)(8 31)(9 26)(10 33)(11 28)(12 35)(13 16)(14 23)(15 18)(17 20)(19 22)(21 24)(37 93)(38 88)(39 95)(40 90)(41 85)(42 92)(43 87)(44 94)(45 89)(46 96)(47 91)(48 86)(49 62)(50 69)(51 64)(52 71)(53 66)(54 61)(55 68)(56 63)(57 70)(58 65)(59 72)(60 67)(73 108)(74 103)(75 98)(76 105)(77 100)(78 107)(79 102)(80 97)(81 104)(82 99)(83 106)(84 101)(110 116)(112 118)(114 120)
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), (1,91)(2,90)(3,89)(4,88)(5,87)(6,86)(7,85)(8,96)(9,95)(10,94)(11,93)(12,92)(13,115)(14,114)(15,113)(16,112)(17,111)(18,110)(19,109)(20,120)(21,119)(22,118)(23,117)(24,116)(25,37)(26,48)(27,47)(28,46)(29,45)(30,44)(31,43)(32,42)(33,41)(34,40)(35,39)(36,38)(49,108)(50,107)(51,106)(52,105)(53,104)(54,103)(55,102)(56,101)(57,100)(58,99)(59,98)(60,97)(61,83)(62,82)(63,81)(64,80)(65,79)(66,78)(67,77)(68,76)(69,75)(70,74)(71,73)(72,84), (1,98,23,78,27,4,107,14,75,30)(2,105,24,73,28,11,108,21,76,25)(3,100,13,80,29,6,97,16,77,32)(5,102,15,82,31,8,99,18,79,34)(7,104,17,84,33,10,101,20,81,36)(9,106,19,74,35,12,103,22,83,26)(37,87,52,113,71,43,93,58,119,65)(38,94,53,120,72)(39,89,54,115,61,45,95,60,109,67)(40,96,55,110,62)(41,91,56,117,63,47,85,50,111,69)(42,86,57,112,64)(44,88,59,114,66)(46,90,49,116,68)(48,92,51,118,70), (1,30)(2,25)(3,32)(4,27)(5,34)(6,29)(7,36)(8,31)(9,26)(10,33)(11,28)(12,35)(13,16)(14,23)(15,18)(17,20)(19,22)(21,24)(37,93)(38,88)(39,95)(40,90)(41,85)(42,92)(43,87)(44,94)(45,89)(46,96)(47,91)(48,86)(49,62)(50,69)(51,64)(52,71)(53,66)(54,61)(55,68)(56,63)(57,70)(58,65)(59,72)(60,67)(73,108)(74,103)(75,98)(76,105)(77,100)(78,107)(79,102)(80,97)(81,104)(82,99)(83,106)(84,101)(110,116)(112,118)(114,120)>;
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), (1,91)(2,90)(3,89)(4,88)(5,87)(6,86)(7,85)(8,96)(9,95)(10,94)(11,93)(12,92)(13,115)(14,114)(15,113)(16,112)(17,111)(18,110)(19,109)(20,120)(21,119)(22,118)(23,117)(24,116)(25,37)(26,48)(27,47)(28,46)(29,45)(30,44)(31,43)(32,42)(33,41)(34,40)(35,39)(36,38)(49,108)(50,107)(51,106)(52,105)(53,104)(54,103)(55,102)(56,101)(57,100)(58,99)(59,98)(60,97)(61,83)(62,82)(63,81)(64,80)(65,79)(66,78)(67,77)(68,76)(69,75)(70,74)(71,73)(72,84), (1,98,23,78,27,4,107,14,75,30)(2,105,24,73,28,11,108,21,76,25)(3,100,13,80,29,6,97,16,77,32)(5,102,15,82,31,8,99,18,79,34)(7,104,17,84,33,10,101,20,81,36)(9,106,19,74,35,12,103,22,83,26)(37,87,52,113,71,43,93,58,119,65)(38,94,53,120,72)(39,89,54,115,61,45,95,60,109,67)(40,96,55,110,62)(41,91,56,117,63,47,85,50,111,69)(42,86,57,112,64)(44,88,59,114,66)(46,90,49,116,68)(48,92,51,118,70), (1,30)(2,25)(3,32)(4,27)(5,34)(6,29)(7,36)(8,31)(9,26)(10,33)(11,28)(12,35)(13,16)(14,23)(15,18)(17,20)(19,22)(21,24)(37,93)(38,88)(39,95)(40,90)(41,85)(42,92)(43,87)(44,94)(45,89)(46,96)(47,91)(48,86)(49,62)(50,69)(51,64)(52,71)(53,66)(54,61)(55,68)(56,63)(57,70)(58,65)(59,72)(60,67)(73,108)(74,103)(75,98)(76,105)(77,100)(78,107)(79,102)(80,97)(81,104)(82,99)(83,106)(84,101)(110,116)(112,118)(114,120) );
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)], [(1,91),(2,90),(3,89),(4,88),(5,87),(6,86),(7,85),(8,96),(9,95),(10,94),(11,93),(12,92),(13,115),(14,114),(15,113),(16,112),(17,111),(18,110),(19,109),(20,120),(21,119),(22,118),(23,117),(24,116),(25,37),(26,48),(27,47),(28,46),(29,45),(30,44),(31,43),(32,42),(33,41),(34,40),(35,39),(36,38),(49,108),(50,107),(51,106),(52,105),(53,104),(54,103),(55,102),(56,101),(57,100),(58,99),(59,98),(60,97),(61,83),(62,82),(63,81),(64,80),(65,79),(66,78),(67,77),(68,76),(69,75),(70,74),(71,73),(72,84)], [(1,98,23,78,27,4,107,14,75,30),(2,105,24,73,28,11,108,21,76,25),(3,100,13,80,29,6,97,16,77,32),(5,102,15,82,31,8,99,18,79,34),(7,104,17,84,33,10,101,20,81,36),(9,106,19,74,35,12,103,22,83,26),(37,87,52,113,71,43,93,58,119,65),(38,94,53,120,72),(39,89,54,115,61,45,95,60,109,67),(40,96,55,110,62),(41,91,56,117,63,47,85,50,111,69),(42,86,57,112,64),(44,88,59,114,66),(46,90,49,116,68),(48,92,51,118,70)], [(1,30),(2,25),(3,32),(4,27),(5,34),(6,29),(7,36),(8,31),(9,26),(10,33),(11,28),(12,35),(13,16),(14,23),(15,18),(17,20),(19,22),(21,24),(37,93),(38,88),(39,95),(40,90),(41,85),(42,92),(43,87),(44,94),(45,89),(46,96),(47,91),(48,86),(49,62),(50,69),(51,64),(52,71),(53,66),(54,61),(55,68),(56,63),(57,70),(58,65),(59,72),(60,67),(73,108),(74,103),(75,98),(76,105),(77,100),(78,107),(79,102),(80,97),(81,104),(82,99),(83,106),(84,101),(110,116),(112,118),(114,120)])
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 10A | 10B | 10C | 10D | 10E | 10F | 12A | 12B | 15A | 15B | 20A | 20B | 30A | 30B | 30C | 30D | 30E | 30F | 40A | 40B | 40C | 40D | 60A | 60B |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 15 | 15 | 20 | 20 | 30 | 30 | 30 | 30 | 30 | 30 | 40 | 40 | 40 | 40 | 60 | 60 |
| size | 1 | 1 | 4 | 10 | 12 | 20 | 2 | 2 | 10 | 60 | 2 | 2 | 2 | 4 | 4 | 10 | 10 | 20 | 20 | 12 | 60 | 2 | 2 | 8 | 8 | 24 | 24 | 4 | 20 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 8 | 8 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D6 | D10 | D10 | D10 | C3⋊D4 | C3⋊D4 | C8⋊C22 | S3×D5 | D4×D5 | D12⋊6C22 | C2×S3×D5 | D8⋊D5 | D5×C3⋊D4 | D12⋊10D10 |
| kernel | D12⋊10D10 | C20.32D6 | C15⋊D8 | C6.D20 | C5×D4⋊S3 | D4.D15 | D12⋊5D5 | C3×D4×D5 | D4×D5 | C3×Dic5 | C6×D5 | D4⋊S3 | C4×D5 | D20 | C5×D4 | C3⋊C8 | D12 | C3×D4 | Dic5 | D10 | C15 | D4 | C6 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 2 |
Matrix representation of D12⋊10D10 ►in GL6(𝔽241)
| 15 | 0 | 0 | 0 | 0 | 0 |
| 239 | 225 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 |
| 131 | 223 | 0 | 0 | 0 | 0 |
| 150 | 110 | 0 | 0 | 0 | 0 |
| 0 | 0 | 113 | 57 | 128 | 184 |
| 0 | 0 | 184 | 128 | 57 | 113 |
| 0 | 0 | 128 | 184 | 128 | 184 |
| 0 | 0 | 57 | 113 | 57 | 113 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 190 | 190 |
| 0 | 0 | 0 | 0 | 51 | 240 |
| 0 | 0 | 190 | 190 | 0 | 0 |
| 0 | 0 | 51 | 240 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 190 | 190 |
| 0 | 0 | 0 | 0 | 240 | 51 |
| 0 | 0 | 190 | 190 | 0 | 0 |
| 0 | 0 | 240 | 51 | 0 | 0 |
G:=sub<GL(6,GF(241))| [15,239,0,0,0,0,0,225,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,1,0,0,0,0,0,0,1,0,0],[131,150,0,0,0,0,223,110,0,0,0,0,0,0,113,184,128,57,0,0,57,128,184,113,0,0,128,57,128,57,0,0,184,113,184,113],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,190,51,0,0,0,0,190,240,0,0,190,51,0,0,0,0,190,240,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,190,240,0,0,0,0,190,51,0,0,190,240,0,0,0,0,190,51,0,0] >;
D12⋊10D10 in GAP, Magma, Sage, TeX
D_{12}\rtimes_{10}D_{10} % in TeX
G:=Group("D12:10D10"); // GroupNames label
G:=SmallGroup(480,565);
// by ID
G=gap.SmallGroup(480,565);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,135,346,185,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=c^10=d^2=1,b*a*b=a^-1,c*a*c^-1=d*a*d=a^7,c*b*c^-1=a^3*b,d*b*d=a^9*b,d*c*d=c^-1>;
// generators/relations