non-abelian, soluble, monomial
Aliases: A4⋊1Dic10, Dic5.3S4, (C5×A4)⋊Q8, A4⋊C4.D5, C5⋊1(A4⋊Q8), (C2×C10)⋊Dic6, C22⋊(C15⋊Q8), C10.9(C2×S4), C2.12(D5×S4), (C2×A4).1D10, C23.1(S3×D5), A4⋊Dic5.2C2, (A4×Dic5).1C2, (C22×C10).1D6, (C10×A4).1C22, (C22×Dic5).1S3, (C5×A4⋊C4).1C2, SmallGroup(480,975)
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=dad-1=ab=ba, ae=ea, cbc-1=a, bd=db, be=eb, dcd-1=c-1, ce=ec, ede-1=d-1 >
Subgroups: 548 in 84 conjugacy classes, 19 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, C6, C2×C4, Q8, C23, C10, C10, Dic3, C12, A4, C15, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, Dic5, Dic5, C20, C2×C10, C2×C10, Dic6, C2×A4, C30, C22⋊Q8, Dic10, C2×Dic5, C2×C20, C22×C10, A4⋊C4, A4⋊C4, C4×A4, C5×Dic3, C3×Dic5, Dic15, C5×A4, C10.D4, C4⋊Dic5, C23.D5, C5×C22⋊C4, C2×Dic10, C22×Dic5, A4⋊Q8, C15⋊Q8, C10×A4, Dic5.14D4, C5×A4⋊C4, A4⋊Dic5, A4×Dic5, A4⋊Dic10
Quotients: C1, C2, C22, S3, Q8, D5, D6, D10, Dic6, S4, Dic10, C2×S4, S3×D5, A4⋊Q8, C15⋊Q8, D5×S4, A4⋊Dic10
►On 120 points
Generators in S120
(1 11)(3 13)(5 15)(7 17)(9 19)(22 32)(24 34)(26 36)(28 38)(30 40)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)(62 72)(64 74)(66 76)(68 78)(70 80)(81 91)(82 92)(83 93)(84 94)(85 95)(86 96)(87 97)(88 98)(89 99)(90 100)(102 112)(104 114)(106 116)(108 118)(110 120)
(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)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)(101 111)(102 112)(103 113)(104 114)(105 115)(106 116)(107 117)(108 118)(109 119)(110 120)
(1 103 93)(2 94 104)(3 105 95)(4 96 106)(5 107 97)(6 98 108)(7 109 99)(8 100 110)(9 111 81)(10 82 112)(11 113 83)(12 84 114)(13 115 85)(14 86 116)(15 117 87)(16 88 118)(17 119 89)(18 90 120)(19 101 91)(20 92 102)(21 43 62)(22 63 44)(23 45 64)(24 65 46)(25 47 66)(26 67 48)(27 49 68)(28 69 50)(29 51 70)(30 71 52)(31 53 72)(32 73 54)(33 55 74)(34 75 56)(35 57 76)(36 77 58)(37 59 78)(38 79 60)(39 41 80)(40 61 42)
(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 80 11 70)(2 79 12 69)(3 78 13 68)(4 77 14 67)(5 76 15 66)(6 75 16 65)(7 74 17 64)(8 73 18 63)(9 72 19 62)(10 71 20 61)(21 111 31 101)(22 110 32 120)(23 109 33 119)(24 108 34 118)(25 107 35 117)(26 106 36 116)(27 105 37 115)(28 104 38 114)(29 103 39 113)(30 102 40 112)(41 83 51 93)(42 82 52 92)(43 81 53 91)(44 100 54 90)(45 99 55 89)(46 98 56 88)(47 97 57 87)(48 96 58 86)(49 95 59 85)(50 94 60 84)
G:=sub<Sym(120)| (1,11)(3,13)(5,15)(7,17)(9,19)(22,32)(24,34)(26,36)(28,38)(30,40)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(62,72)(64,74)(66,76)(68,78)(70,80)(81,91)(82,92)(83,93)(84,94)(85,95)(86,96)(87,97)(88,98)(89,99)(90,100)(102,112)(104,114)(106,116)(108,118)(110,120), (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)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80)(101,111)(102,112)(103,113)(104,114)(105,115)(106,116)(107,117)(108,118)(109,119)(110,120), (1,103,93)(2,94,104)(3,105,95)(4,96,106)(5,107,97)(6,98,108)(7,109,99)(8,100,110)(9,111,81)(10,82,112)(11,113,83)(12,84,114)(13,115,85)(14,86,116)(15,117,87)(16,88,118)(17,119,89)(18,90,120)(19,101,91)(20,92,102)(21,43,62)(22,63,44)(23,45,64)(24,65,46)(25,47,66)(26,67,48)(27,49,68)(28,69,50)(29,51,70)(30,71,52)(31,53,72)(32,73,54)(33,55,74)(34,75,56)(35,57,76)(36,77,58)(37,59,78)(38,79,60)(39,41,80)(40,61,42), (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), (1,80,11,70)(2,79,12,69)(3,78,13,68)(4,77,14,67)(5,76,15,66)(6,75,16,65)(7,74,17,64)(8,73,18,63)(9,72,19,62)(10,71,20,61)(21,111,31,101)(22,110,32,120)(23,109,33,119)(24,108,34,118)(25,107,35,117)(26,106,36,116)(27,105,37,115)(28,104,38,114)(29,103,39,113)(30,102,40,112)(41,83,51,93)(42,82,52,92)(43,81,53,91)(44,100,54,90)(45,99,55,89)(46,98,56,88)(47,97,57,87)(48,96,58,86)(49,95,59,85)(50,94,60,84)>;
G:=Group( (1,11)(3,13)(5,15)(7,17)(9,19)(22,32)(24,34)(26,36)(28,38)(30,40)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(62,72)(64,74)(66,76)(68,78)(70,80)(81,91)(82,92)(83,93)(84,94)(85,95)(86,96)(87,97)(88,98)(89,99)(90,100)(102,112)(104,114)(106,116)(108,118)(110,120), (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)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80)(101,111)(102,112)(103,113)(104,114)(105,115)(106,116)(107,117)(108,118)(109,119)(110,120), (1,103,93)(2,94,104)(3,105,95)(4,96,106)(5,107,97)(6,98,108)(7,109,99)(8,100,110)(9,111,81)(10,82,112)(11,113,83)(12,84,114)(13,115,85)(14,86,116)(15,117,87)(16,88,118)(17,119,89)(18,90,120)(19,101,91)(20,92,102)(21,43,62)(22,63,44)(23,45,64)(24,65,46)(25,47,66)(26,67,48)(27,49,68)(28,69,50)(29,51,70)(30,71,52)(31,53,72)(32,73,54)(33,55,74)(34,75,56)(35,57,76)(36,77,58)(37,59,78)(38,79,60)(39,41,80)(40,61,42), (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), (1,80,11,70)(2,79,12,69)(3,78,13,68)(4,77,14,67)(5,76,15,66)(6,75,16,65)(7,74,17,64)(8,73,18,63)(9,72,19,62)(10,71,20,61)(21,111,31,101)(22,110,32,120)(23,109,33,119)(24,108,34,118)(25,107,35,117)(26,106,36,116)(27,105,37,115)(28,104,38,114)(29,103,39,113)(30,102,40,112)(41,83,51,93)(42,82,52,92)(43,81,53,91)(44,100,54,90)(45,99,55,89)(46,98,56,88)(47,97,57,87)(48,96,58,86)(49,95,59,85)(50,94,60,84) );
G=PermutationGroup([[(1,11),(3,13),(5,15),(7,17),(9,19),(22,32),(24,34),(26,36),(28,38),(30,40),(41,51),(42,52),(43,53),(44,54),(45,55),(46,56),(47,57),(48,58),(49,59),(50,60),(62,72),(64,74),(66,76),(68,78),(70,80),(81,91),(82,92),(83,93),(84,94),(85,95),(86,96),(87,97),(88,98),(89,99),(90,100),(102,112),(104,114),(106,116),(108,118),(110,120)], [(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),(61,71),(62,72),(63,73),(64,74),(65,75),(66,76),(67,77),(68,78),(69,79),(70,80),(101,111),(102,112),(103,113),(104,114),(105,115),(106,116),(107,117),(108,118),(109,119),(110,120)], [(1,103,93),(2,94,104),(3,105,95),(4,96,106),(5,107,97),(6,98,108),(7,109,99),(8,100,110),(9,111,81),(10,82,112),(11,113,83),(12,84,114),(13,115,85),(14,86,116),(15,117,87),(16,88,118),(17,119,89),(18,90,120),(19,101,91),(20,92,102),(21,43,62),(22,63,44),(23,45,64),(24,65,46),(25,47,66),(26,67,48),(27,49,68),(28,69,50),(29,51,70),(30,71,52),(31,53,72),(32,73,54),(33,55,74),(34,75,56),(35,57,76),(36,77,58),(37,59,78),(38,79,60),(39,41,80),(40,61,42)], [(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)], [(1,80,11,70),(2,79,12,69),(3,78,13,68),(4,77,14,67),(5,76,15,66),(6,75,16,65),(7,74,17,64),(8,73,18,63),(9,72,19,62),(10,71,20,61),(21,111,31,101),(22,110,32,120),(23,109,33,119),(24,108,34,118),(25,107,35,117),(26,106,36,116),(27,105,37,115),(28,104,38,114),(29,103,39,113),(30,102,40,112),(41,83,51,93),(42,82,52,92),(43,81,53,91),(44,100,54,90),(45,99,55,89),(46,98,56,88),(47,97,57,87),(48,96,58,86),(49,95,59,85),(50,94,60,84)]])
34 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 6 | 10A | 10B | 10C | 10D | 10E | 10F | 12A | 12B | 15A | 15B | 20A | ··· | 20H | 30A | 30B |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 15 | 15 | 20 | ··· | 20 | 30 | 30 |
| size | 1 | 1 | 3 | 3 | 8 | 10 | 12 | 12 | 30 | 60 | 60 | 2 | 2 | 8 | 2 | 2 | 6 | 6 | 6 | 6 | 40 | 40 | 16 | 16 | 12 | ··· | 12 | 16 | 16 |
34 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 6 | 6 | 6 |
| type | + | + | + | + | + | - | + | + | + | - | - | + | + | + | - | - | + | - |
| image | C1 | C2 | C2 | C2 | S3 | Q8 | D5 | D6 | D10 | Dic6 | Dic10 | S4 | C2×S4 | S3×D5 | C15⋊Q8 | A4⋊Q8 | D5×S4 | A4⋊Dic10 |
| kernel | A4⋊Dic10 | C5×A4⋊C4 | A4⋊Dic5 | A4×Dic5 | C22×Dic5 | C5×A4 | A4⋊C4 | C22×C10 | C2×A4 | C2×C10 | A4 | Dic5 | C10 | C23 | C22 | C5 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 1 | 4 | 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 | 60 | 60 | 60 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 60 | 60 | 60 |
| 0 | 0 | 0 | 1 | 0 |
| 12 | 34 | 0 | 0 | 0 |
| 33 | 58 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 |
| 0 | 0 | 0 | 60 | 0 |
| 55 | 30 | 0 | 0 | 0 |
| 13 | 6 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 60 |
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],[1,0,0,0,0,0,1,0,0,0,0,0,60,0,0,0,0,60,0,1,0,0,60,1,0],[1,0,0,0,0,0,1,0,0,0,0,0,1,60,0,0,0,0,60,1,0,0,0,60,0],[12,33,0,0,0,34,58,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,60,0],[55,13,0,0,0,30,6,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,60] >;
A4⋊Dic10 in GAP, Magma, Sage, TeX
A_4\rtimes {\rm Dic}_{10} % in TeX
G:=Group("A4:Dic10"); // GroupNames label
G:=SmallGroup(480,975);
// by ID
G=gap.SmallGroup(480,975);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-5,-2,2,28,85,36,234,3364,5052,1286,2953,2232]);
// 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=d*a*d^-1=a*b=b*a,a*e=e*a,c*b*c^-1=a,b*d=d*b,b*e=e*b,d*c*d^-1=c^-1,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations