Aliases: Dic10.A4, SL2(𝔽3)⋊6D10, C5⋊(Q8.A4), D4⋊8D10⋊C3, C4.A4⋊3D5, C4.2(D5×A4), C20.2(C2×A4), Q8.3(C6×D5), Dic5.A4⋊5C2, C10.7(C22×A4), Q8⋊2D5.1C6, Dic5.3(C2×A4), (C5×SL2(𝔽3))⋊7C22, C2.8(C2×D5×A4), C4○D4.(C3×D5), (C5×C4.A4)⋊3C2, (C5×C4○D4).2C6, (C5×Q8).3(C2×C6), SmallGroup(480,1041)
Series: Derived ►Chief ►Lower central ►Upper central
| C5×Q8 — Dic10.A4 |
Generators and relations for Dic10.A4
G = < a,b,c,d,e | a20=e3=1, b2=c2=d2=a10, bab-1=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=a10c, ece-1=a10cd, ede-1=c >
Subgroups: 686 in 92 conjugacy classes, 23 normal (17 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C2×C4, D4, Q8, Q8, C23, D5, C10, C10, C12, C15, C2×D4, C4○D4, C4○D4, Dic5, C20, C20, D10, C2×C10, SL2(𝔽3), C3×Q8, C30, 2+ 1+4, Dic10, C4×D5, D20, C5⋊D4, C2×C20, C5×D4, C5×Q8, C22×D5, C4.A4, C4.A4, C3×Dic5, C60, C2×D20, C4○D20, D4×D5, Q8⋊2D5, C5×C4○D4, Q8.A4, C5×SL2(𝔽3), C3×Dic10, D4⋊8D10, Dic5.A4, C5×C4.A4, Dic10.A4
Quotients: C1, C2, C3, C22, C6, D5, A4, C2×C6, D10, C2×A4, C3×D5, C22×A4, C6×D5, Q8.A4, D5×A4, C2×D5×A4, Dic10.A4
►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 48 11 58)(2 47 12 57)(3 46 13 56)(4 45 14 55)(5 44 15 54)(6 43 16 53)(7 42 17 52)(8 41 18 51)(9 60 19 50)(10 59 20 49)(21 77 31 67)(22 76 32 66)(23 75 33 65)(24 74 34 64)(25 73 35 63)(26 72 36 62)(27 71 37 61)(28 70 38 80)(29 69 39 79)(30 68 40 78)(81 105 91 115)(82 104 92 114)(83 103 93 113)(84 102 94 112)(85 101 95 111)(86 120 96 110)(87 119 97 109)(88 118 98 108)(89 117 99 107)(90 116 100 106)
(1 58 11 48)(2 59 12 49)(3 60 13 50)(4 41 14 51)(5 42 15 52)(6 43 16 53)(7 44 17 54)(8 45 18 55)(9 46 19 56)(10 47 20 57)(21 36 31 26)(22 37 32 27)(23 38 33 28)(24 39 34 29)(25 40 35 30)(61 66 71 76)(62 67 72 77)(63 68 73 78)(64 69 74 79)(65 70 75 80)(81 118 91 108)(82 119 92 109)(83 120 93 110)(84 101 94 111)(85 102 95 112)(86 103 96 113)(87 104 97 114)(88 105 98 115)(89 106 99 116)(90 107 100 117)
(1 6 11 16)(2 7 12 17)(3 8 13 18)(4 9 14 19)(5 10 15 20)(21 70 31 80)(22 71 32 61)(23 72 33 62)(24 73 34 63)(25 74 35 64)(26 75 36 65)(27 76 37 66)(28 77 38 67)(29 78 39 68)(30 79 40 69)(41 56 51 46)(42 57 52 47)(43 58 53 48)(44 59 54 49)(45 60 55 50)(81 103 91 113)(82 104 92 114)(83 105 93 115)(84 106 94 116)(85 107 95 117)(86 108 96 118)(87 109 97 119)(88 110 98 120)(89 111 99 101)(90 112 100 102)
(1 109 71)(2 110 72)(3 111 73)(4 112 74)(5 113 75)(6 114 76)(7 115 77)(8 116 78)(9 117 79)(10 118 80)(11 119 61)(12 120 62)(13 101 63)(14 102 64)(15 103 65)(16 104 66)(17 105 67)(18 106 68)(19 107 69)(20 108 70)(21 52 91)(22 53 92)(23 54 93)(24 55 94)(25 56 95)(26 57 96)(27 58 97)(28 59 98)(29 60 99)(30 41 100)(31 42 81)(32 43 82)(33 44 83)(34 45 84)(35 46 85)(36 47 86)(37 48 87)(38 49 88)(39 50 89)(40 51 90)
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), (1,48,11,58)(2,47,12,57)(3,46,13,56)(4,45,14,55)(5,44,15,54)(6,43,16,53)(7,42,17,52)(8,41,18,51)(9,60,19,50)(10,59,20,49)(21,77,31,67)(22,76,32,66)(23,75,33,65)(24,74,34,64)(25,73,35,63)(26,72,36,62)(27,71,37,61)(28,70,38,80)(29,69,39,79)(30,68,40,78)(81,105,91,115)(82,104,92,114)(83,103,93,113)(84,102,94,112)(85,101,95,111)(86,120,96,110)(87,119,97,109)(88,118,98,108)(89,117,99,107)(90,116,100,106), (1,58,11,48)(2,59,12,49)(3,60,13,50)(4,41,14,51)(5,42,15,52)(6,43,16,53)(7,44,17,54)(8,45,18,55)(9,46,19,56)(10,47,20,57)(21,36,31,26)(22,37,32,27)(23,38,33,28)(24,39,34,29)(25,40,35,30)(61,66,71,76)(62,67,72,77)(63,68,73,78)(64,69,74,79)(65,70,75,80)(81,118,91,108)(82,119,92,109)(83,120,93,110)(84,101,94,111)(85,102,95,112)(86,103,96,113)(87,104,97,114)(88,105,98,115)(89,106,99,116)(90,107,100,117), (1,6,11,16)(2,7,12,17)(3,8,13,18)(4,9,14,19)(5,10,15,20)(21,70,31,80)(22,71,32,61)(23,72,33,62)(24,73,34,63)(25,74,35,64)(26,75,36,65)(27,76,37,66)(28,77,38,67)(29,78,39,68)(30,79,40,69)(41,56,51,46)(42,57,52,47)(43,58,53,48)(44,59,54,49)(45,60,55,50)(81,103,91,113)(82,104,92,114)(83,105,93,115)(84,106,94,116)(85,107,95,117)(86,108,96,118)(87,109,97,119)(88,110,98,120)(89,111,99,101)(90,112,100,102), (1,109,71)(2,110,72)(3,111,73)(4,112,74)(5,113,75)(6,114,76)(7,115,77)(8,116,78)(9,117,79)(10,118,80)(11,119,61)(12,120,62)(13,101,63)(14,102,64)(15,103,65)(16,104,66)(17,105,67)(18,106,68)(19,107,69)(20,108,70)(21,52,91)(22,53,92)(23,54,93)(24,55,94)(25,56,95)(26,57,96)(27,58,97)(28,59,98)(29,60,99)(30,41,100)(31,42,81)(32,43,82)(33,44,83)(34,45,84)(35,46,85)(36,47,86)(37,48,87)(38,49,88)(39,50,89)(40,51,90)>;
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), (1,48,11,58)(2,47,12,57)(3,46,13,56)(4,45,14,55)(5,44,15,54)(6,43,16,53)(7,42,17,52)(8,41,18,51)(9,60,19,50)(10,59,20,49)(21,77,31,67)(22,76,32,66)(23,75,33,65)(24,74,34,64)(25,73,35,63)(26,72,36,62)(27,71,37,61)(28,70,38,80)(29,69,39,79)(30,68,40,78)(81,105,91,115)(82,104,92,114)(83,103,93,113)(84,102,94,112)(85,101,95,111)(86,120,96,110)(87,119,97,109)(88,118,98,108)(89,117,99,107)(90,116,100,106), (1,58,11,48)(2,59,12,49)(3,60,13,50)(4,41,14,51)(5,42,15,52)(6,43,16,53)(7,44,17,54)(8,45,18,55)(9,46,19,56)(10,47,20,57)(21,36,31,26)(22,37,32,27)(23,38,33,28)(24,39,34,29)(25,40,35,30)(61,66,71,76)(62,67,72,77)(63,68,73,78)(64,69,74,79)(65,70,75,80)(81,118,91,108)(82,119,92,109)(83,120,93,110)(84,101,94,111)(85,102,95,112)(86,103,96,113)(87,104,97,114)(88,105,98,115)(89,106,99,116)(90,107,100,117), (1,6,11,16)(2,7,12,17)(3,8,13,18)(4,9,14,19)(5,10,15,20)(21,70,31,80)(22,71,32,61)(23,72,33,62)(24,73,34,63)(25,74,35,64)(26,75,36,65)(27,76,37,66)(28,77,38,67)(29,78,39,68)(30,79,40,69)(41,56,51,46)(42,57,52,47)(43,58,53,48)(44,59,54,49)(45,60,55,50)(81,103,91,113)(82,104,92,114)(83,105,93,115)(84,106,94,116)(85,107,95,117)(86,108,96,118)(87,109,97,119)(88,110,98,120)(89,111,99,101)(90,112,100,102), (1,109,71)(2,110,72)(3,111,73)(4,112,74)(5,113,75)(6,114,76)(7,115,77)(8,116,78)(9,117,79)(10,118,80)(11,119,61)(12,120,62)(13,101,63)(14,102,64)(15,103,65)(16,104,66)(17,105,67)(18,106,68)(19,107,69)(20,108,70)(21,52,91)(22,53,92)(23,54,93)(24,55,94)(25,56,95)(26,57,96)(27,58,97)(28,59,98)(29,60,99)(30,41,100)(31,42,81)(32,43,82)(33,44,83)(34,45,84)(35,46,85)(36,47,86)(37,48,87)(38,49,88)(39,50,89)(40,51,90) );
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)], [(1,48,11,58),(2,47,12,57),(3,46,13,56),(4,45,14,55),(5,44,15,54),(6,43,16,53),(7,42,17,52),(8,41,18,51),(9,60,19,50),(10,59,20,49),(21,77,31,67),(22,76,32,66),(23,75,33,65),(24,74,34,64),(25,73,35,63),(26,72,36,62),(27,71,37,61),(28,70,38,80),(29,69,39,79),(30,68,40,78),(81,105,91,115),(82,104,92,114),(83,103,93,113),(84,102,94,112),(85,101,95,111),(86,120,96,110),(87,119,97,109),(88,118,98,108),(89,117,99,107),(90,116,100,106)], [(1,58,11,48),(2,59,12,49),(3,60,13,50),(4,41,14,51),(5,42,15,52),(6,43,16,53),(7,44,17,54),(8,45,18,55),(9,46,19,56),(10,47,20,57),(21,36,31,26),(22,37,32,27),(23,38,33,28),(24,39,34,29),(25,40,35,30),(61,66,71,76),(62,67,72,77),(63,68,73,78),(64,69,74,79),(65,70,75,80),(81,118,91,108),(82,119,92,109),(83,120,93,110),(84,101,94,111),(85,102,95,112),(86,103,96,113),(87,104,97,114),(88,105,98,115),(89,106,99,116),(90,107,100,117)], [(1,6,11,16),(2,7,12,17),(3,8,13,18),(4,9,14,19),(5,10,15,20),(21,70,31,80),(22,71,32,61),(23,72,33,62),(24,73,34,63),(25,74,35,64),(26,75,36,65),(27,76,37,66),(28,77,38,67),(29,78,39,68),(30,79,40,69),(41,56,51,46),(42,57,52,47),(43,58,53,48),(44,59,54,49),(45,60,55,50),(81,103,91,113),(82,104,92,114),(83,105,93,115),(84,106,94,116),(85,107,95,117),(86,108,96,118),(87,109,97,119),(88,110,98,120),(89,111,99,101),(90,112,100,102)], [(1,109,71),(2,110,72),(3,111,73),(4,112,74),(5,113,75),(6,114,76),(7,115,77),(8,116,78),(9,117,79),(10,118,80),(11,119,61),(12,120,62),(13,101,63),(14,102,64),(15,103,65),(16,104,66),(17,105,67),(18,106,68),(19,107,69),(20,108,70),(21,52,91),(22,53,92),(23,54,93),(24,55,94),(25,56,95),(26,57,96),(27,58,97),(28,59,98),(29,60,99),(30,41,100),(31,42,81),(32,43,82),(33,44,83),(34,45,84),(35,46,85),(36,47,86),(37,48,87),(38,49,88),(39,50,89),(40,51,90)]])
47 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 4A | 4B | 4C | 4D | 5A | 5B | 6A | 6B | 10A | 10B | 10C | 10D | 12A | 12B | 12C | 12D | 12E | 12F | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 20E | 20F | 30A | 30B | 30C | 30D | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 30 | 30 | 30 | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 6 | 30 | 30 | 4 | 4 | 2 | 6 | 10 | 10 | 2 | 2 | 4 | 4 | 2 | 2 | 12 | 12 | 8 | 8 | 40 | 40 | 40 | 40 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 12 | 12 | 8 | 8 | 8 | 8 | 8 | ··· | 8 |
47 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | D5 | D10 | C3×D5 | C6×D5 | A4 | C2×A4 | C2×A4 | Q8.A4 | Q8.A4 | Dic10.A4 | Dic10.A4 | D5×A4 | C2×D5×A4 |
| kernel | Dic10.A4 | Dic5.A4 | C5×C4.A4 | D4⋊8D10 | Q8⋊2D5 | C5×C4○D4 | C4.A4 | SL2(𝔽3) | C4○D4 | Q8 | Dic10 | Dic5 | C20 | C5 | C5 | C1 | C1 | C4 | C2 |
| # reps | 1 | 2 | 1 | 2 | 4 | 2 | 2 | 2 | 4 | 4 | 1 | 2 | 1 | 1 | 2 | 4 | 8 | 2 | 2 |
Matrix representation of Dic10.A4 ►in GL4(𝔽61) generated by
| 0 | 50 | 0 | 0 |
| 50 | 31 | 0 | 0 |
| 0 | 25 | 2 | 36 |
| 50 | 2 | 27 | 29 |
| 45 | 0 | 34 | 36 |
| 59 | 0 | 32 | 16 |
| 45 | 18 | 17 | 18 |
| 30 | 44 | 16 | 60 |
| 1 | 0 | 59 | 0 |
| 0 | 0 | 17 | 1 |
| 1 | 0 | 60 | 0 |
| 44 | 60 | 17 | 0 |
| 39 | 8 | 0 | 0 |
| 8 | 22 | 0 | 0 |
| 39 | 4 | 29 | 4 |
| 0 | 54 | 3 | 32 |
| 60 | 57 | 30 | 4 |
| 0 | 19 | 18 | 41 |
| 41 | 57 | 0 | 0 |
| 43 | 49 | 18 | 41 |
G:=sub<GL(4,GF(61))| [0,50,0,50,50,31,25,2,0,0,2,27,0,0,36,29],[45,59,45,30,0,0,18,44,34,32,17,16,36,16,18,60],[1,0,1,44,0,0,0,60,59,17,60,17,0,1,0,0],[39,8,39,0,8,22,4,54,0,0,29,3,0,0,4,32],[60,0,41,43,57,19,57,49,30,18,0,18,4,41,0,41] >;
Dic10.A4 in GAP, Magma, Sage, TeX
{\rm Dic}_{10}.A_4 % in TeX
G:=Group("Dic10.A4"); // GroupNames label
G:=SmallGroup(480,1041);
// by ID
G=gap.SmallGroup(480,1041);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,2,-5,-2,1680,3389,1688,269,584,123,795,382,8069]);
// Polycyclic
G:=Group<a,b,c,d,e|a^20=e^3=1,b^2=c^2=d^2=a^10,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=a^10*c,e*c*e^-1=a^10*c*d,e*d*e^-1=c>;
// generators/relations