metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D10.4D12, (C2×Dic3)⋊F5, C22⋊F5⋊2S3, C15⋊2(C23⋊C4), (C6×D5).29D4, C22.4(S3×F5), (C10×Dic3)⋊3C4, (C22×D15)⋊3C4, C2.13(D6⋊F5), C10.13(D6⋊C4), D10.4(C3⋊D4), C3⋊1(D10.D4), (C22×D5).35D6, C5⋊1(C23.6D6), C6.13(C22⋊F5), D10.D6⋊2C2, C30.13(C22⋊C4), (C2×C6).2(C2×F5), (C2×C10).9(C4×S3), (C2×C30).7(C2×C4), (C3×C22⋊F5)⋊2C2, (D5×C2×C6).65C22, (C2×C3⋊D20).10C2, SmallGroup(480,249)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D10.4D12
G = < a,b,c,d | a10=b2=c12=1, d2=a4b, bab=a-1, cac-1=dad-1=a3, cbc-1=a7b, dbd-1=a2b, dcd-1=a-1bc-1 >
Subgroups: 836 in 104 conjugacy classes, 26 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C2×C4, D4, C23, D5, C10, C10, Dic3, C12, D6, C2×C6, C2×C6, C15, C22⋊C4, C2×D4, C20, F5, D10, D10, C2×C10, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×C6, C3×D5, D15, C30, C30, C23⋊C4, D20, C2×C20, C2×F5, C22×D5, C22×D5, C6.D4, C3×C22⋊C4, C2×C3⋊D4, C5×Dic3, C3×F5, C3⋊F5, C6×D5, C6×D5, D30, C2×C30, C22⋊F5, C22⋊F5, C2×D20, C23.6D6, C3⋊D20, C10×Dic3, C6×F5, C2×C3⋊F5, D5×C2×C6, C22×D15, D10.D4, C3×C22⋊F5, D10.D6, C2×C3⋊D20, D10.4D12
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, C22⋊C4, F5, C4×S3, D12, C3⋊D4, C23⋊C4, C2×F5, D6⋊C4, C22⋊F5, C23.6D6, S3×F5, D10.D4, D6⋊F5, D10.4D12
►On 120 points
Generators in S120
(1 43 79 46 76 6 82 40 73 37)(2 47 83 38 80 4 74 44 77 41)(3 39 75 42 84 5 78 48 81 45)(7 113 110 54 57 10 51 60 116 119)(8 55 52 120 111 11 117 114 58 49)(9 109 118 50 53 12 59 56 112 115)(13 88 71 27 105 99 33 65 94 19)(14 28 34 20 72 100 95 89 106 66)(15 21 96 67 35 101 107 29 61 90)(16 68 108 91 85 102 62 22 36 30)(17 92 63 31 97 103 25 69 86 23)(18 32 26 24 64 104 87 93 98 70)
(1 49)(2 53)(3 57)(4 115)(5 119)(6 111)(7 84)(8 37)(9 80)(10 45)(11 76)(12 41)(13 22)(14 17)(15 32)(16 65)(18 21)(19 36)(20 69)(23 28)(24 61)(25 72)(26 90)(27 85)(29 64)(30 94)(31 89)(33 68)(34 86)(35 93)(38 109)(39 54)(40 52)(42 113)(43 58)(44 56)(46 117)(47 50)(48 60)(51 81)(55 73)(59 77)(62 88)(63 106)(66 92)(67 98)(70 96)(71 102)(74 112)(75 110)(78 116)(79 114)(82 120)(83 118)(87 101)(91 105)(95 97)(99 108)(100 103)(104 107)
(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 65 11 62)(2 61 9 70)(3 69 7 66)(4 67 12 64)(5 63 10 72)(6 71 8 68)(13 117 91 37)(14 48 25 54)(15 53 93 74)(16 73 27 114)(17 113 95 45)(18 44 29 50)(19 49 85 82)(20 81 31 110)(21 109 87 41)(22 40 33 58)(23 57 89 78)(24 77 35 118)(26 47 101 115)(28 84 103 51)(30 43 105 111)(32 80 107 59)(34 39 97 119)(36 76 99 55)(38 90 56 98)(42 86 60 106)(46 94 52 102)(75 92 116 100)(79 88 120 108)(83 96 112 104)
G:=sub<Sym(120)| (1,43,79,46,76,6,82,40,73,37)(2,47,83,38,80,4,74,44,77,41)(3,39,75,42,84,5,78,48,81,45)(7,113,110,54,57,10,51,60,116,119)(8,55,52,120,111,11,117,114,58,49)(9,109,118,50,53,12,59,56,112,115)(13,88,71,27,105,99,33,65,94,19)(14,28,34,20,72,100,95,89,106,66)(15,21,96,67,35,101,107,29,61,90)(16,68,108,91,85,102,62,22,36,30)(17,92,63,31,97,103,25,69,86,23)(18,32,26,24,64,104,87,93,98,70), (1,49)(2,53)(3,57)(4,115)(5,119)(6,111)(7,84)(8,37)(9,80)(10,45)(11,76)(12,41)(13,22)(14,17)(15,32)(16,65)(18,21)(19,36)(20,69)(23,28)(24,61)(25,72)(26,90)(27,85)(29,64)(30,94)(31,89)(33,68)(34,86)(35,93)(38,109)(39,54)(40,52)(42,113)(43,58)(44,56)(46,117)(47,50)(48,60)(51,81)(55,73)(59,77)(62,88)(63,106)(66,92)(67,98)(70,96)(71,102)(74,112)(75,110)(78,116)(79,114)(82,120)(83,118)(87,101)(91,105)(95,97)(99,108)(100,103)(104,107), (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), (1,65,11,62)(2,61,9,70)(3,69,7,66)(4,67,12,64)(5,63,10,72)(6,71,8,68)(13,117,91,37)(14,48,25,54)(15,53,93,74)(16,73,27,114)(17,113,95,45)(18,44,29,50)(19,49,85,82)(20,81,31,110)(21,109,87,41)(22,40,33,58)(23,57,89,78)(24,77,35,118)(26,47,101,115)(28,84,103,51)(30,43,105,111)(32,80,107,59)(34,39,97,119)(36,76,99,55)(38,90,56,98)(42,86,60,106)(46,94,52,102)(75,92,116,100)(79,88,120,108)(83,96,112,104)>;
G:=Group( (1,43,79,46,76,6,82,40,73,37)(2,47,83,38,80,4,74,44,77,41)(3,39,75,42,84,5,78,48,81,45)(7,113,110,54,57,10,51,60,116,119)(8,55,52,120,111,11,117,114,58,49)(9,109,118,50,53,12,59,56,112,115)(13,88,71,27,105,99,33,65,94,19)(14,28,34,20,72,100,95,89,106,66)(15,21,96,67,35,101,107,29,61,90)(16,68,108,91,85,102,62,22,36,30)(17,92,63,31,97,103,25,69,86,23)(18,32,26,24,64,104,87,93,98,70), (1,49)(2,53)(3,57)(4,115)(5,119)(6,111)(7,84)(8,37)(9,80)(10,45)(11,76)(12,41)(13,22)(14,17)(15,32)(16,65)(18,21)(19,36)(20,69)(23,28)(24,61)(25,72)(26,90)(27,85)(29,64)(30,94)(31,89)(33,68)(34,86)(35,93)(38,109)(39,54)(40,52)(42,113)(43,58)(44,56)(46,117)(47,50)(48,60)(51,81)(55,73)(59,77)(62,88)(63,106)(66,92)(67,98)(70,96)(71,102)(74,112)(75,110)(78,116)(79,114)(82,120)(83,118)(87,101)(91,105)(95,97)(99,108)(100,103)(104,107), (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), (1,65,11,62)(2,61,9,70)(3,69,7,66)(4,67,12,64)(5,63,10,72)(6,71,8,68)(13,117,91,37)(14,48,25,54)(15,53,93,74)(16,73,27,114)(17,113,95,45)(18,44,29,50)(19,49,85,82)(20,81,31,110)(21,109,87,41)(22,40,33,58)(23,57,89,78)(24,77,35,118)(26,47,101,115)(28,84,103,51)(30,43,105,111)(32,80,107,59)(34,39,97,119)(36,76,99,55)(38,90,56,98)(42,86,60,106)(46,94,52,102)(75,92,116,100)(79,88,120,108)(83,96,112,104) );
G=PermutationGroup([[(1,43,79,46,76,6,82,40,73,37),(2,47,83,38,80,4,74,44,77,41),(3,39,75,42,84,5,78,48,81,45),(7,113,110,54,57,10,51,60,116,119),(8,55,52,120,111,11,117,114,58,49),(9,109,118,50,53,12,59,56,112,115),(13,88,71,27,105,99,33,65,94,19),(14,28,34,20,72,100,95,89,106,66),(15,21,96,67,35,101,107,29,61,90),(16,68,108,91,85,102,62,22,36,30),(17,92,63,31,97,103,25,69,86,23),(18,32,26,24,64,104,87,93,98,70)], [(1,49),(2,53),(3,57),(4,115),(5,119),(6,111),(7,84),(8,37),(9,80),(10,45),(11,76),(12,41),(13,22),(14,17),(15,32),(16,65),(18,21),(19,36),(20,69),(23,28),(24,61),(25,72),(26,90),(27,85),(29,64),(30,94),(31,89),(33,68),(34,86),(35,93),(38,109),(39,54),(40,52),(42,113),(43,58),(44,56),(46,117),(47,50),(48,60),(51,81),(55,73),(59,77),(62,88),(63,106),(66,92),(67,98),(70,96),(71,102),(74,112),(75,110),(78,116),(79,114),(82,120),(83,118),(87,101),(91,105),(95,97),(99,108),(100,103),(104,107)], [(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)], [(1,65,11,62),(2,61,9,70),(3,69,7,66),(4,67,12,64),(5,63,10,72),(6,71,8,68),(13,117,91,37),(14,48,25,54),(15,53,93,74),(16,73,27,114),(17,113,95,45),(18,44,29,50),(19,49,85,82),(20,81,31,110),(21,109,87,41),(22,40,33,58),(23,57,89,78),(24,77,35,118),(26,47,101,115),(28,84,103,51),(30,43,105,111),(32,80,107,59),(34,39,97,119),(36,76,99,55),(38,90,56,98),(42,86,60,106),(46,94,52,102),(75,92,116,100),(79,88,120,108),(83,96,112,104)]])
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 5 | 6A | 6B | 6C | 6D | 6E | 10A | 10B | 10C | 12A | 12B | 12C | 12D | 15 | 20A | 20B | 20C | 20D | 30A | 30B | 30C |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 6 | 6 | 6 | 6 | 6 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 15 | 20 | 20 | 20 | 20 | 30 | 30 | 30 |
| size | 1 | 1 | 2 | 10 | 10 | 60 | 2 | 12 | 20 | 20 | 60 | 60 | 4 | 2 | 4 | 10 | 10 | 20 | 4 | 4 | 4 | 20 | 20 | 20 | 20 | 8 | 12 | 12 | 12 | 12 | 8 | 8 | 8 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | S3 | D4 | D6 | D12 | C3⋊D4 | C4×S3 | F5 | C23⋊C4 | C2×F5 | C22⋊F5 | C23.6D6 | D10.D4 | S3×F5 | D6⋊F5 | D10.4D12 |
| kernel | D10.4D12 | C3×C22⋊F5 | D10.D6 | C2×C3⋊D20 | C10×Dic3 | C22×D15 | C22⋊F5 | C6×D5 | C22×D5 | D10 | D10 | C2×C10 | C2×Dic3 | C15 | C2×C6 | C6 | C5 | C3 | C22 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 4 | 1 | 1 | 2 |
Matrix representation of D10.4D12 ►in GL8(𝔽61)
| 60 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 60 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 60 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 | 0 | 0 |
| 60 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 | 1 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 60 |
| 13 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 48 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 14 | 0 | 0 | 0 | 0 |
| 0 | 0 | 47 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 14 | 0 | 0 | 0 | 0 |
| 0 | 0 | 47 | 0 | 0 | 0 | 0 | 0 |
| 48 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 47 | 54 | 0 |
| 0 | 0 | 0 | 0 | 7 | 47 | 0 | 14 |
| 0 | 0 | 0 | 0 | 14 | 0 | 47 | 7 |
| 0 | 0 | 0 | 0 | 0 | 54 | 47 | 14 |
G:=sub<GL(8,GF(61))| [60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,60,60,60,60,0,0,0,0,1,0,0,0],[0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,1,1,1,1,0,0,0,0,0,0,0,60],[13,0,0,0,0,0,0,0,0,48,0,0,0,0,0,0,0,0,0,47,0,0,0,0,0,0,14,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0],[0,0,48,0,0,0,0,0,0,0,0,13,0,0,0,0,0,47,0,0,0,0,0,0,14,0,0,0,0,0,0,0,0,0,0,0,14,7,14,0,0,0,0,0,47,47,0,54,0,0,0,0,54,0,47,47,0,0,0,0,0,14,7,14] >;
D10.4D12 in GAP, Magma, Sage, TeX
D_{10}._4D_{12} % in TeX
G:=Group("D10.4D12"); // GroupNames label
G:=SmallGroup(480,249);
// by ID
G=gap.SmallGroup(480,249);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,120,219,675,1356,9414,4724]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^2=c^12=1,d^2=a^4*b,b*a*b=a^-1,c*a*c^-1=d*a*d^-1=a^3,c*b*c^-1=a^7*b,d*b*d^-1=a^2*b,d*c*d^-1=a^-1*b*c^-1>;
// generators/relations