direct product, metabelian, soluble, monomial
Aliases: A4×Dic10, C5⋊(Q8×A4), (C5×A4)⋊3Q8, C4.1(D5×A4), C20.1(C2×A4), (C4×A4).3D5, (A4×C20).3C2, (C22×Dic10)⋊C3, C22⋊(C3×Dic10), (C2×A4).13D10, C10.1(C22×A4), (C22×C20).1C6, Dic5.1(C2×A4), (A4×Dic5).2C2, C23.10(C6×D5), (C10×A4).13C22, (C22×Dic5).1C6, (C2×C10)⋊(C3×Q8), C2.3(C2×D5×A4), (C22×C4).(C3×D5), (C22×C10).1(C2×C6), SmallGroup(480,1035)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for A4×Dic10
G = < a,b,c,d,e | a2=b2=c3=d20=1, e2=d10, cac-1=ab=ba, ad=da, ae=ea, cbc-1=a, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >
Subgroups: 480 in 92 conjugacy classes, 27 normal (21 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, C6, C2×C4, Q8, C23, C10, C10, C12, A4, C15, C22×C4, C22×C4, C2×Q8, Dic5, Dic5, C20, C20, C2×C10, C2×C10, C3×Q8, C2×A4, C30, C22×Q8, Dic10, Dic10, C2×Dic5, C2×C20, C22×C10, C4×A4, C4×A4, C3×Dic5, C60, C5×A4, C2×Dic10, C22×Dic5, C22×C20, Q8×A4, C3×Dic10, C10×A4, C22×Dic10, A4×Dic5, A4×C20, A4×Dic10
Quotients: C1, C2, C3, C22, C6, Q8, D5, A4, C2×C6, D10, C3×Q8, C2×A4, C3×D5, Dic10, C22×A4, C6×D5, Q8×A4, C3×Dic10, D5×A4, C2×D5×A4, A4×Dic10
►On 120 points
Generators in S120
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)(101 111)(102 112)(103 113)(104 114)(105 115)(106 116)(107 117)(108 118)(109 119)(110 120)
(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)(81 91)(82 92)(83 93)(84 94)(85 95)(86 96)(87 97)(88 98)(89 99)(90 100)(101 111)(102 112)(103 113)(104 114)(105 115)(106 116)(107 117)(108 118)(109 119)(110 120)
(1 34 64)(2 35 65)(3 36 66)(4 37 67)(5 38 68)(6 39 69)(7 40 70)(8 21 71)(9 22 72)(10 23 73)(11 24 74)(12 25 75)(13 26 76)(14 27 77)(15 28 78)(16 29 79)(17 30 80)(18 31 61)(19 32 62)(20 33 63)(41 115 97)(42 116 98)(43 117 99)(44 118 100)(45 119 81)(46 120 82)(47 101 83)(48 102 84)(49 103 85)(50 104 86)(51 105 87)(52 106 88)(53 107 89)(54 108 90)(55 109 91)(56 110 92)(57 111 93)(58 112 94)(59 113 95)(60 114 96)
(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 43 11 53)(2 42 12 52)(3 41 13 51)(4 60 14 50)(5 59 15 49)(6 58 16 48)(7 57 17 47)(8 56 18 46)(9 55 19 45)(10 54 20 44)(21 110 31 120)(22 109 32 119)(23 108 33 118)(24 107 34 117)(25 106 35 116)(26 105 36 115)(27 104 37 114)(28 103 38 113)(29 102 39 112)(30 101 40 111)(61 82 71 92)(62 81 72 91)(63 100 73 90)(64 99 74 89)(65 98 75 88)(66 97 76 87)(67 96 77 86)(68 95 78 85)(69 94 79 84)(70 93 80 83)
G:=sub<Sym(120)| (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(101,111)(102,112)(103,113)(104,114)(105,115)(106,116)(107,117)(108,118)(109,119)(110,120), (21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80)(81,91)(82,92)(83,93)(84,94)(85,95)(86,96)(87,97)(88,98)(89,99)(90,100)(101,111)(102,112)(103,113)(104,114)(105,115)(106,116)(107,117)(108,118)(109,119)(110,120), (1,34,64)(2,35,65)(3,36,66)(4,37,67)(5,38,68)(6,39,69)(7,40,70)(8,21,71)(9,22,72)(10,23,73)(11,24,74)(12,25,75)(13,26,76)(14,27,77)(15,28,78)(16,29,79)(17,30,80)(18,31,61)(19,32,62)(20,33,63)(41,115,97)(42,116,98)(43,117,99)(44,118,100)(45,119,81)(46,120,82)(47,101,83)(48,102,84)(49,103,85)(50,104,86)(51,105,87)(52,106,88)(53,107,89)(54,108,90)(55,109,91)(56,110,92)(57,111,93)(58,112,94)(59,113,95)(60,114,96), (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), (1,43,11,53)(2,42,12,52)(3,41,13,51)(4,60,14,50)(5,59,15,49)(6,58,16,48)(7,57,17,47)(8,56,18,46)(9,55,19,45)(10,54,20,44)(21,110,31,120)(22,109,32,119)(23,108,33,118)(24,107,34,117)(25,106,35,116)(26,105,36,115)(27,104,37,114)(28,103,38,113)(29,102,39,112)(30,101,40,111)(61,82,71,92)(62,81,72,91)(63,100,73,90)(64,99,74,89)(65,98,75,88)(66,97,76,87)(67,96,77,86)(68,95,78,85)(69,94,79,84)(70,93,80,83)>;
G:=Group( (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(101,111)(102,112)(103,113)(104,114)(105,115)(106,116)(107,117)(108,118)(109,119)(110,120), (21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80)(81,91)(82,92)(83,93)(84,94)(85,95)(86,96)(87,97)(88,98)(89,99)(90,100)(101,111)(102,112)(103,113)(104,114)(105,115)(106,116)(107,117)(108,118)(109,119)(110,120), (1,34,64)(2,35,65)(3,36,66)(4,37,67)(5,38,68)(6,39,69)(7,40,70)(8,21,71)(9,22,72)(10,23,73)(11,24,74)(12,25,75)(13,26,76)(14,27,77)(15,28,78)(16,29,79)(17,30,80)(18,31,61)(19,32,62)(20,33,63)(41,115,97)(42,116,98)(43,117,99)(44,118,100)(45,119,81)(46,120,82)(47,101,83)(48,102,84)(49,103,85)(50,104,86)(51,105,87)(52,106,88)(53,107,89)(54,108,90)(55,109,91)(56,110,92)(57,111,93)(58,112,94)(59,113,95)(60,114,96), (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), (1,43,11,53)(2,42,12,52)(3,41,13,51)(4,60,14,50)(5,59,15,49)(6,58,16,48)(7,57,17,47)(8,56,18,46)(9,55,19,45)(10,54,20,44)(21,110,31,120)(22,109,32,119)(23,108,33,118)(24,107,34,117)(25,106,35,116)(26,105,36,115)(27,104,37,114)(28,103,38,113)(29,102,39,112)(30,101,40,111)(61,82,71,92)(62,81,72,91)(63,100,73,90)(64,99,74,89)(65,98,75,88)(66,97,76,87)(67,96,77,86)(68,95,78,85)(69,94,79,84)(70,93,80,83) );
G=PermutationGroup([[(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,19),(10,20),(21,31),(22,32),(23,33),(24,34),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40),(41,51),(42,52),(43,53),(44,54),(45,55),(46,56),(47,57),(48,58),(49,59),(50,60),(101,111),(102,112),(103,113),(104,114),(105,115),(106,116),(107,117),(108,118),(109,119),(110,120)], [(21,31),(22,32),(23,33),(24,34),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40),(61,71),(62,72),(63,73),(64,74),(65,75),(66,76),(67,77),(68,78),(69,79),(70,80),(81,91),(82,92),(83,93),(84,94),(85,95),(86,96),(87,97),(88,98),(89,99),(90,100),(101,111),(102,112),(103,113),(104,114),(105,115),(106,116),(107,117),(108,118),(109,119),(110,120)], [(1,34,64),(2,35,65),(3,36,66),(4,37,67),(5,38,68),(6,39,69),(7,40,70),(8,21,71),(9,22,72),(10,23,73),(11,24,74),(12,25,75),(13,26,76),(14,27,77),(15,28,78),(16,29,79),(17,30,80),(18,31,61),(19,32,62),(20,33,63),(41,115,97),(42,116,98),(43,117,99),(44,118,100),(45,119,81),(46,120,82),(47,101,83),(48,102,84),(49,103,85),(50,104,86),(51,105,87),(52,106,88),(53,107,89),(54,108,90),(55,109,91),(56,110,92),(57,111,93),(58,112,94),(59,113,95),(60,114,96)], [(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)], [(1,43,11,53),(2,42,12,52),(3,41,13,51),(4,60,14,50),(5,59,15,49),(6,58,16,48),(7,57,17,47),(8,56,18,46),(9,55,19,45),(10,54,20,44),(21,110,31,120),(22,109,32,119),(23,108,33,118),(24,107,34,117),(25,106,35,116),(26,105,36,115),(27,104,37,114),(28,103,38,113),(29,102,39,112),(30,101,40,111),(61,82,71,92),(62,81,72,91),(63,100,73,90),(64,99,74,89),(65,98,75,88),(66,97,76,87),(67,96,77,86),(68,95,78,85),(69,94,79,84),(70,93,80,83)]])
52 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 6A | 6B | 10A | 10B | 10C | 10D | 10E | 10F | 12A | 12B | 12C | 12D | 12E | 12F | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 30A | 30B | 30C | 30D | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 30 | 30 | 30 | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 3 | 3 | 4 | 4 | 2 | 6 | 10 | 10 | 30 | 30 | 2 | 2 | 4 | 4 | 2 | 2 | 6 | 6 | 6 | 6 | 8 | 8 | 40 | 40 | 40 | 40 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | ··· | 8 |
52 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 6 | 6 | 6 | 6 |
| type | + | + | + | - | + | + | - | + | + | + | - | + | + | - | |||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | Q8 | D5 | D10 | C3×Q8 | C3×D5 | Dic10 | C6×D5 | C3×Dic10 | A4 | C2×A4 | C2×A4 | Q8×A4 | D5×A4 | C2×D5×A4 | A4×Dic10 |
| kernel | A4×Dic10 | A4×Dic5 | A4×C20 | C22×Dic10 | C22×Dic5 | C22×C20 | C5×A4 | C4×A4 | C2×A4 | C2×C10 | C22×C4 | A4 | C23 | C22 | Dic10 | Dic5 | C20 | C5 | C4 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 4 | 2 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 1 | 2 | 1 | 1 | 2 | 2 | 4 |
Matrix representation of A4×Dic10 ►in GL5(𝔽61)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 60 | 60 | 60 |
| 0 | 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 60 | 60 | 60 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 47 | 0 | 0 |
| 0 | 0 | 0 | 0 | 47 |
| 0 | 0 | 14 | 14 | 14 |
| 27 | 31 | 0 | 0 | 0 |
| 31 | 4 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 60 |
| 56 | 14 | 0 | 0 | 0 |
| 33 | 5 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(5,GF(61))| [1,0,0,0,0,0,1,0,0,0,0,0,0,60,1,0,0,0,60,0,0,0,1,60,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,60,0,0,1,0,60,0,0,0,0,60],[1,0,0,0,0,0,1,0,0,0,0,0,47,0,14,0,0,0,0,14,0,0,0,47,14],[27,31,0,0,0,31,4,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,60],[56,33,0,0,0,14,5,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1] >;
A4×Dic10 in GAP, Magma, Sage, TeX
A_4\times {\rm Dic}_{10} % in TeX
G:=Group("A4xDic10"); // GroupNames label
G:=SmallGroup(480,1035);
// by ID
G=gap.SmallGroup(480,1035);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,2,-5,84,197,92,648,271,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^3=d^20=1,e^2=d^10,c*a*c^-1=a*b=b*a,a*d=d*a,a*e=e*a,c*b*c^-1=a,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations