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 |
Subgroups: 686 in 92 conjugacy classes, 23 normal (17 characteristic)
C1, C2, C2 [×3], C3, C4, C4 [×3], C22 [×5], C5, C6, C2×C4 [×3], D4 [×6], Q8, Q8, C23 [×2], D5 [×2], C10, C10, C12 [×3], C15, C2×D4 [×3], C4○D4, C4○D4 [×3], Dic5 [×2], C20, C20, D10 [×4], C2×C10, SL2(𝔽3), C3×Q8, C30, 2+ (1+4), Dic10, C4×D5 [×2], D20 [×3], C5⋊D4 [×2], C2×C20, C5×D4, C5×Q8, C22×D5 [×2], C4.A4, C4.A4 [×2], C3×Dic5 [×2], C60, C2×D20, C4○D20, D4×D5 [×2], Q8⋊2D5 [×2], C5×C4○D4, Q8.A4, C5×SL2(𝔽3), C3×Dic10, D4⋊8D10, Dic5.A4 [×2], C5×C4.A4, Dic10.A4
Quotients:
C1, C2 [×3], C3, C22, C6 [×3], D5, A4, C2×C6, D10, C2×A4 [×3], C3×D5, C22×A4, C6×D5, Q8.A4, D5×A4, C2×D5×A4, Dic10.A4
Generators and relations
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 >
►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 69 11 79)(2 68 12 78)(3 67 13 77)(4 66 14 76)(5 65 15 75)(6 64 16 74)(7 63 17 73)(8 62 18 72)(9 61 19 71)(10 80 20 70)(21 88 31 98)(22 87 32 97)(23 86 33 96)(24 85 34 95)(25 84 35 94)(26 83 36 93)(27 82 37 92)(28 81 38 91)(29 100 39 90)(30 99 40 89)(41 106 51 116)(42 105 52 115)(43 104 53 114)(44 103 54 113)(45 102 55 112)(46 101 56 111)(47 120 57 110)(48 119 58 109)(49 118 59 108)(50 117 60 107)
(1 79 11 69)(2 80 12 70)(3 61 13 71)(4 62 14 72)(5 63 15 73)(6 64 16 74)(7 65 17 75)(8 66 18 76)(9 67 19 77)(10 68 20 78)(21 36 31 26)(22 37 32 27)(23 38 33 28)(24 39 34 29)(25 40 35 30)(41 117 51 107)(42 118 52 108)(43 119 53 109)(44 120 54 110)(45 101 55 111)(46 102 56 112)(47 103 57 113)(48 104 58 114)(49 105 59 115)(50 106 60 116)(81 86 91 96)(82 87 92 97)(83 88 93 98)(84 89 94 99)(85 90 95 100)
(1 6 11 16)(2 7 12 17)(3 8 13 18)(4 9 14 19)(5 10 15 20)(21 83 31 93)(22 84 32 94)(23 85 33 95)(24 86 34 96)(25 87 35 97)(26 88 36 98)(27 89 37 99)(28 90 38 100)(29 91 39 81)(30 92 40 82)(41 102 51 112)(42 103 52 113)(43 104 53 114)(44 105 54 115)(45 106 55 116)(46 107 56 117)(47 108 57 118)(48 109 58 119)(49 110 59 120)(50 111 60 101)(61 76 71 66)(62 77 72 67)(63 78 73 68)(64 79 74 69)(65 80 75 70)
(1 119 83)(2 120 84)(3 101 85)(4 102 86)(5 103 87)(6 104 88)(7 105 89)(8 106 90)(9 107 91)(10 108 92)(11 109 93)(12 110 94)(13 111 95)(14 112 96)(15 113 97)(16 114 98)(17 115 99)(18 116 100)(19 117 81)(20 118 82)(21 74 43)(22 75 44)(23 76 45)(24 77 46)(25 78 47)(26 79 48)(27 80 49)(28 61 50)(29 62 51)(30 63 52)(31 64 53)(32 65 54)(33 66 55)(34 67 56)(35 68 57)(36 69 58)(37 70 59)(38 71 60)(39 72 41)(40 73 42)
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), (1,69,11,79)(2,68,12,78)(3,67,13,77)(4,66,14,76)(5,65,15,75)(6,64,16,74)(7,63,17,73)(8,62,18,72)(9,61,19,71)(10,80,20,70)(21,88,31,98)(22,87,32,97)(23,86,33,96)(24,85,34,95)(25,84,35,94)(26,83,36,93)(27,82,37,92)(28,81,38,91)(29,100,39,90)(30,99,40,89)(41,106,51,116)(42,105,52,115)(43,104,53,114)(44,103,54,113)(45,102,55,112)(46,101,56,111)(47,120,57,110)(48,119,58,109)(49,118,59,108)(50,117,60,107), (1,79,11,69)(2,80,12,70)(3,61,13,71)(4,62,14,72)(5,63,15,73)(6,64,16,74)(7,65,17,75)(8,66,18,76)(9,67,19,77)(10,68,20,78)(21,36,31,26)(22,37,32,27)(23,38,33,28)(24,39,34,29)(25,40,35,30)(41,117,51,107)(42,118,52,108)(43,119,53,109)(44,120,54,110)(45,101,55,111)(46,102,56,112)(47,103,57,113)(48,104,58,114)(49,105,59,115)(50,106,60,116)(81,86,91,96)(82,87,92,97)(83,88,93,98)(84,89,94,99)(85,90,95,100), (1,6,11,16)(2,7,12,17)(3,8,13,18)(4,9,14,19)(5,10,15,20)(21,83,31,93)(22,84,32,94)(23,85,33,95)(24,86,34,96)(25,87,35,97)(26,88,36,98)(27,89,37,99)(28,90,38,100)(29,91,39,81)(30,92,40,82)(41,102,51,112)(42,103,52,113)(43,104,53,114)(44,105,54,115)(45,106,55,116)(46,107,56,117)(47,108,57,118)(48,109,58,119)(49,110,59,120)(50,111,60,101)(61,76,71,66)(62,77,72,67)(63,78,73,68)(64,79,74,69)(65,80,75,70), (1,119,83)(2,120,84)(3,101,85)(4,102,86)(5,103,87)(6,104,88)(7,105,89)(8,106,90)(9,107,91)(10,108,92)(11,109,93)(12,110,94)(13,111,95)(14,112,96)(15,113,97)(16,114,98)(17,115,99)(18,116,100)(19,117,81)(20,118,82)(21,74,43)(22,75,44)(23,76,45)(24,77,46)(25,78,47)(26,79,48)(27,80,49)(28,61,50)(29,62,51)(30,63,52)(31,64,53)(32,65,54)(33,66,55)(34,67,56)(35,68,57)(36,69,58)(37,70,59)(38,71,60)(39,72,41)(40,73,42)>;
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), (1,69,11,79)(2,68,12,78)(3,67,13,77)(4,66,14,76)(5,65,15,75)(6,64,16,74)(7,63,17,73)(8,62,18,72)(9,61,19,71)(10,80,20,70)(21,88,31,98)(22,87,32,97)(23,86,33,96)(24,85,34,95)(25,84,35,94)(26,83,36,93)(27,82,37,92)(28,81,38,91)(29,100,39,90)(30,99,40,89)(41,106,51,116)(42,105,52,115)(43,104,53,114)(44,103,54,113)(45,102,55,112)(46,101,56,111)(47,120,57,110)(48,119,58,109)(49,118,59,108)(50,117,60,107), (1,79,11,69)(2,80,12,70)(3,61,13,71)(4,62,14,72)(5,63,15,73)(6,64,16,74)(7,65,17,75)(8,66,18,76)(9,67,19,77)(10,68,20,78)(21,36,31,26)(22,37,32,27)(23,38,33,28)(24,39,34,29)(25,40,35,30)(41,117,51,107)(42,118,52,108)(43,119,53,109)(44,120,54,110)(45,101,55,111)(46,102,56,112)(47,103,57,113)(48,104,58,114)(49,105,59,115)(50,106,60,116)(81,86,91,96)(82,87,92,97)(83,88,93,98)(84,89,94,99)(85,90,95,100), (1,6,11,16)(2,7,12,17)(3,8,13,18)(4,9,14,19)(5,10,15,20)(21,83,31,93)(22,84,32,94)(23,85,33,95)(24,86,34,96)(25,87,35,97)(26,88,36,98)(27,89,37,99)(28,90,38,100)(29,91,39,81)(30,92,40,82)(41,102,51,112)(42,103,52,113)(43,104,53,114)(44,105,54,115)(45,106,55,116)(46,107,56,117)(47,108,57,118)(48,109,58,119)(49,110,59,120)(50,111,60,101)(61,76,71,66)(62,77,72,67)(63,78,73,68)(64,79,74,69)(65,80,75,70), (1,119,83)(2,120,84)(3,101,85)(4,102,86)(5,103,87)(6,104,88)(7,105,89)(8,106,90)(9,107,91)(10,108,92)(11,109,93)(12,110,94)(13,111,95)(14,112,96)(15,113,97)(16,114,98)(17,115,99)(18,116,100)(19,117,81)(20,118,82)(21,74,43)(22,75,44)(23,76,45)(24,77,46)(25,78,47)(26,79,48)(27,80,49)(28,61,50)(29,62,51)(30,63,52)(31,64,53)(32,65,54)(33,66,55)(34,67,56)(35,68,57)(36,69,58)(37,70,59)(38,71,60)(39,72,41)(40,73,42) );
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)], [(1,69,11,79),(2,68,12,78),(3,67,13,77),(4,66,14,76),(5,65,15,75),(6,64,16,74),(7,63,17,73),(8,62,18,72),(9,61,19,71),(10,80,20,70),(21,88,31,98),(22,87,32,97),(23,86,33,96),(24,85,34,95),(25,84,35,94),(26,83,36,93),(27,82,37,92),(28,81,38,91),(29,100,39,90),(30,99,40,89),(41,106,51,116),(42,105,52,115),(43,104,53,114),(44,103,54,113),(45,102,55,112),(46,101,56,111),(47,120,57,110),(48,119,58,109),(49,118,59,108),(50,117,60,107)], [(1,79,11,69),(2,80,12,70),(3,61,13,71),(4,62,14,72),(5,63,15,73),(6,64,16,74),(7,65,17,75),(8,66,18,76),(9,67,19,77),(10,68,20,78),(21,36,31,26),(22,37,32,27),(23,38,33,28),(24,39,34,29),(25,40,35,30),(41,117,51,107),(42,118,52,108),(43,119,53,109),(44,120,54,110),(45,101,55,111),(46,102,56,112),(47,103,57,113),(48,104,58,114),(49,105,59,115),(50,106,60,116),(81,86,91,96),(82,87,92,97),(83,88,93,98),(84,89,94,99),(85,90,95,100)], [(1,6,11,16),(2,7,12,17),(3,8,13,18),(4,9,14,19),(5,10,15,20),(21,83,31,93),(22,84,32,94),(23,85,33,95),(24,86,34,96),(25,87,35,97),(26,88,36,98),(27,89,37,99),(28,90,38,100),(29,91,39,81),(30,92,40,82),(41,102,51,112),(42,103,52,113),(43,104,53,114),(44,105,54,115),(45,106,55,116),(46,107,56,117),(47,108,57,118),(48,109,58,119),(49,110,59,120),(50,111,60,101),(61,76,71,66),(62,77,72,67),(63,78,73,68),(64,79,74,69),(65,80,75,70)], [(1,119,83),(2,120,84),(3,101,85),(4,102,86),(5,103,87),(6,104,88),(7,105,89),(8,106,90),(9,107,91),(10,108,92),(11,109,93),(12,110,94),(13,111,95),(14,112,96),(15,113,97),(16,114,98),(17,115,99),(18,116,100),(19,117,81),(20,118,82),(21,74,43),(22,75,44),(23,76,45),(24,77,46),(25,78,47),(26,79,48),(27,80,49),(28,61,50),(29,62,51),(30,63,52),(31,64,53),(32,65,54),(33,66,55),(34,67,56),(35,68,57),(36,69,58),(37,70,59),(38,71,60),(39,72,41),(40,73,42)])
Matrix representation ►G ⊆ 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] >;
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 |
In GAP, Magma, Sage, TeX
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
×
⋊
ℤ
𝔽
○
≀