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⋊(C3×Dic10), (C22×Dic10)⋊C3, (C2×A4).13D10, (C22×C20).1C6, C10.1(C22×A4), (A4×Dic5).2C2, Dic5.1(C2×A4), 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
Subgroups: 480 in 92 conjugacy classes, 27 normal (21 characteristic)
C1, C2, C2 [×2], C3, C4, C4 [×5], C22, C22 [×2], C5, C6, C2×C4 [×6], Q8 [×6], C23, C10, C10 [×2], C12 [×3], A4, C15, C22×C4, C22×C4 [×2], C2×Q8 [×4], Dic5 [×2], Dic5 [×2], C20, C20, C2×C10, C2×C10 [×2], C3×Q8, C2×A4, C30, C22×Q8, Dic10, Dic10 [×5], C2×Dic5 [×4], C2×C20 [×2], C22×C10, C4×A4, C4×A4 [×2], C3×Dic5 [×2], C60, C5×A4, C2×Dic10 [×4], C22×Dic5 [×2], C22×C20, Q8×A4, C3×Dic10, C10×A4, C22×Dic10, A4×Dic5 [×2], A4×C20, A4×Dic10
Quotients:
C1, C2 [×3], C3, C22, C6 [×3], Q8, D5, A4, C2×C6, D10, C3×Q8, C2×A4 [×3], C3×D5, Dic10, C22×A4, C6×D5, Q8×A4, C3×Dic10, D5×A4, C2×D5×A4, A4×Dic10
Generators and relations
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 >
►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)(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)
(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)(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)
(1 73 47)(2 74 48)(3 75 49)(4 76 50)(5 77 51)(6 78 52)(7 79 53)(8 80 54)(9 61 55)(10 62 56)(11 63 57)(12 64 58)(13 65 59)(14 66 60)(15 67 41)(16 68 42)(17 69 43)(18 70 44)(19 71 45)(20 72 46)(21 96 102)(22 97 103)(23 98 104)(24 99 105)(25 100 106)(26 81 107)(27 82 108)(28 83 109)(29 84 110)(30 85 111)(31 86 112)(32 87 113)(33 88 114)(34 89 115)(35 90 116)(36 91 117)(37 92 118)(38 93 119)(39 94 120)(40 95 101)
(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)
(1 107 11 117)(2 106 12 116)(3 105 13 115)(4 104 14 114)(5 103 15 113)(6 102 16 112)(7 101 17 111)(8 120 18 110)(9 119 19 109)(10 118 20 108)(21 68 31 78)(22 67 32 77)(23 66 33 76)(24 65 34 75)(25 64 35 74)(26 63 36 73)(27 62 37 72)(28 61 38 71)(29 80 39 70)(30 79 40 69)(41 87 51 97)(42 86 52 96)(43 85 53 95)(44 84 54 94)(45 83 55 93)(46 82 56 92)(47 81 57 91)(48 100 58 90)(49 99 59 89)(50 98 60 88)
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)(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), (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)(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), (1,73,47)(2,74,48)(3,75,49)(4,76,50)(5,77,51)(6,78,52)(7,79,53)(8,80,54)(9,61,55)(10,62,56)(11,63,57)(12,64,58)(13,65,59)(14,66,60)(15,67,41)(16,68,42)(17,69,43)(18,70,44)(19,71,45)(20,72,46)(21,96,102)(22,97,103)(23,98,104)(24,99,105)(25,100,106)(26,81,107)(27,82,108)(28,83,109)(29,84,110)(30,85,111)(31,86,112)(32,87,113)(33,88,114)(34,89,115)(35,90,116)(36,91,117)(37,92,118)(38,93,119)(39,94,120)(40,95,101), (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), (1,107,11,117)(2,106,12,116)(3,105,13,115)(4,104,14,114)(5,103,15,113)(6,102,16,112)(7,101,17,111)(8,120,18,110)(9,119,19,109)(10,118,20,108)(21,68,31,78)(22,67,32,77)(23,66,33,76)(24,65,34,75)(25,64,35,74)(26,63,36,73)(27,62,37,72)(28,61,38,71)(29,80,39,70)(30,79,40,69)(41,87,51,97)(42,86,52,96)(43,85,53,95)(44,84,54,94)(45,83,55,93)(46,82,56,92)(47,81,57,91)(48,100,58,90)(49,99,59,89)(50,98,60,88)>;
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)(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), (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)(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), (1,73,47)(2,74,48)(3,75,49)(4,76,50)(5,77,51)(6,78,52)(7,79,53)(8,80,54)(9,61,55)(10,62,56)(11,63,57)(12,64,58)(13,65,59)(14,66,60)(15,67,41)(16,68,42)(17,69,43)(18,70,44)(19,71,45)(20,72,46)(21,96,102)(22,97,103)(23,98,104)(24,99,105)(25,100,106)(26,81,107)(27,82,108)(28,83,109)(29,84,110)(30,85,111)(31,86,112)(32,87,113)(33,88,114)(34,89,115)(35,90,116)(36,91,117)(37,92,118)(38,93,119)(39,94,120)(40,95,101), (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), (1,107,11,117)(2,106,12,116)(3,105,13,115)(4,104,14,114)(5,103,15,113)(6,102,16,112)(7,101,17,111)(8,120,18,110)(9,119,19,109)(10,118,20,108)(21,68,31,78)(22,67,32,77)(23,66,33,76)(24,65,34,75)(25,64,35,74)(26,63,36,73)(27,62,37,72)(28,61,38,71)(29,80,39,70)(30,79,40,69)(41,87,51,97)(42,86,52,96)(43,85,53,95)(44,84,54,94)(45,83,55,93)(46,82,56,92)(47,81,57,91)(48,100,58,90)(49,99,59,89)(50,98,60,88) );
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),(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)], [(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),(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)], [(1,73,47),(2,74,48),(3,75,49),(4,76,50),(5,77,51),(6,78,52),(7,79,53),(8,80,54),(9,61,55),(10,62,56),(11,63,57),(12,64,58),(13,65,59),(14,66,60),(15,67,41),(16,68,42),(17,69,43),(18,70,44),(19,71,45),(20,72,46),(21,96,102),(22,97,103),(23,98,104),(24,99,105),(25,100,106),(26,81,107),(27,82,108),(28,83,109),(29,84,110),(30,85,111),(31,86,112),(32,87,113),(33,88,114),(34,89,115),(35,90,116),(36,91,117),(37,92,118),(38,93,119),(39,94,120),(40,95,101)], [(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)], [(1,107,11,117),(2,106,12,116),(3,105,13,115),(4,104,14,114),(5,103,15,113),(6,102,16,112),(7,101,17,111),(8,120,18,110),(9,119,19,109),(10,118,20,108),(21,68,31,78),(22,67,32,77),(23,66,33,76),(24,65,34,75),(25,64,35,74),(26,63,36,73),(27,62,37,72),(28,61,38,71),(29,80,39,70),(30,79,40,69),(41,87,51,97),(42,86,52,96),(43,85,53,95),(44,84,54,94),(45,83,55,93),(46,82,56,92),(47,81,57,91),(48,100,58,90),(49,99,59,89),(50,98,60,88)])
Matrix representation ►G ⊆ 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] >;
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 |
In GAP, Magma, Sage, TeX
A_4\times 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
×
⋊
ℤ
𝔽
○
≀