direct product, metabelian, soluble, monomial, A-group
Aliases: A4×C40, C22⋊C120, C23.2C60, (C22×C8)⋊C15, (C22×C40)⋊C3, (C2×C10)⋊5C24, C10.8(C4×A4), C2.1(A4×C20), C4.4(C10×A4), (C4×A4).4C10, (C2×A4).2C20, (C10×A4).6C4, (A4×C20).8C2, C20.10(C2×A4), (C22×C4).2C30, (C22×C20).4C6, (C22×C10).9C12, SmallGroup(480,659)
Series: Derived ►Chief ►Lower central ►Upper central
| C22 — A4×C40 |
Generators and relations for A4×C40
G = < a,b,c,d | a40=b2=c2=d3=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=b >
Smallest permutation representation of A4×C40
►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 21)(2 22)(3 23)(4 24)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)(41 61)(42 62)(43 63)(44 64)(45 65)(46 66)(47 67)(48 68)(49 69)(50 70)(51 71)(52 72)(53 73)(54 74)(55 75)(56 76)(57 77)(58 78)(59 79)(60 80)
(1 21)(2 22)(3 23)(4 24)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)(81 101)(82 102)(83 103)(84 104)(85 105)(86 106)(87 107)(88 108)(89 109)(90 110)(91 111)(92 112)(93 113)(94 114)(95 115)(96 116)(97 117)(98 118)(99 119)(100 120)
(1 119 48)(2 120 49)(3 81 50)(4 82 51)(5 83 52)(6 84 53)(7 85 54)(8 86 55)(9 87 56)(10 88 57)(11 89 58)(12 90 59)(13 91 60)(14 92 61)(15 93 62)(16 94 63)(17 95 64)(18 96 65)(19 97 66)(20 98 67)(21 99 68)(22 100 69)(23 101 70)(24 102 71)(25 103 72)(26 104 73)(27 105 74)(28 106 75)(29 107 76)(30 108 77)(31 109 78)(32 110 79)(33 111 80)(34 112 41)(35 113 42)(36 114 43)(37 115 44)(38 116 45)(39 117 46)(40 118 47)
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), (1,21)(2,22)(3,23)(4,24)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(41,61)(42,62)(43,63)(44,64)(45,65)(46,66)(47,67)(48,68)(49,69)(50,70)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80), (1,21)(2,22)(3,23)(4,24)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(81,101)(82,102)(83,103)(84,104)(85,105)(86,106)(87,107)(88,108)(89,109)(90,110)(91,111)(92,112)(93,113)(94,114)(95,115)(96,116)(97,117)(98,118)(99,119)(100,120), (1,119,48)(2,120,49)(3,81,50)(4,82,51)(5,83,52)(6,84,53)(7,85,54)(8,86,55)(9,87,56)(10,88,57)(11,89,58)(12,90,59)(13,91,60)(14,92,61)(15,93,62)(16,94,63)(17,95,64)(18,96,65)(19,97,66)(20,98,67)(21,99,68)(22,100,69)(23,101,70)(24,102,71)(25,103,72)(26,104,73)(27,105,74)(28,106,75)(29,107,76)(30,108,77)(31,109,78)(32,110,79)(33,111,80)(34,112,41)(35,113,42)(36,114,43)(37,115,44)(38,116,45)(39,117,46)(40,118,47)>;
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), (1,21)(2,22)(3,23)(4,24)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(41,61)(42,62)(43,63)(44,64)(45,65)(46,66)(47,67)(48,68)(49,69)(50,70)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80), (1,21)(2,22)(3,23)(4,24)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(81,101)(82,102)(83,103)(84,104)(85,105)(86,106)(87,107)(88,108)(89,109)(90,110)(91,111)(92,112)(93,113)(94,114)(95,115)(96,116)(97,117)(98,118)(99,119)(100,120), (1,119,48)(2,120,49)(3,81,50)(4,82,51)(5,83,52)(6,84,53)(7,85,54)(8,86,55)(9,87,56)(10,88,57)(11,89,58)(12,90,59)(13,91,60)(14,92,61)(15,93,62)(16,94,63)(17,95,64)(18,96,65)(19,97,66)(20,98,67)(21,99,68)(22,100,69)(23,101,70)(24,102,71)(25,103,72)(26,104,73)(27,105,74)(28,106,75)(29,107,76)(30,108,77)(31,109,78)(32,110,79)(33,111,80)(34,112,41)(35,113,42)(36,114,43)(37,115,44)(38,116,45)(39,117,46)(40,118,47) );
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)], [(1,21),(2,22),(3,23),(4,24),(5,25),(6,26),(7,27),(8,28),(9,29),(10,30),(11,31),(12,32),(13,33),(14,34),(15,35),(16,36),(17,37),(18,38),(19,39),(20,40),(41,61),(42,62),(43,63),(44,64),(45,65),(46,66),(47,67),(48,68),(49,69),(50,70),(51,71),(52,72),(53,73),(54,74),(55,75),(56,76),(57,77),(58,78),(59,79),(60,80)], [(1,21),(2,22),(3,23),(4,24),(5,25),(6,26),(7,27),(8,28),(9,29),(10,30),(11,31),(12,32),(13,33),(14,34),(15,35),(16,36),(17,37),(18,38),(19,39),(20,40),(81,101),(82,102),(83,103),(84,104),(85,105),(86,106),(87,107),(88,108),(89,109),(90,110),(91,111),(92,112),(93,113),(94,114),(95,115),(96,116),(97,117),(98,118),(99,119),(100,120)], [(1,119,48),(2,120,49),(3,81,50),(4,82,51),(5,83,52),(6,84,53),(7,85,54),(8,86,55),(9,87,56),(10,88,57),(11,89,58),(12,90,59),(13,91,60),(14,92,61),(15,93,62),(16,94,63),(17,95,64),(18,96,65),(19,97,66),(20,98,67),(21,99,68),(22,100,69),(23,101,70),(24,102,71),(25,103,72),(26,104,73),(27,105,74),(28,106,75),(29,107,76),(30,108,77),(31,109,78),(32,110,79),(33,111,80),(34,112,41),(35,113,42),(36,114,43),(37,115,44),(38,116,45),(39,117,46),(40,118,47)]])
160 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 5A | 5B | 5C | 5D | 6A | 6B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | 10B | 10C | 10D | 10E | ··· | 10L | 12A | 12B | 12C | 12D | 15A | ··· | 15H | 20A | ··· | 20H | 20I | ··· | 20P | 24A | ··· | 24H | 30A | ··· | 30H | 40A | ··· | 40P | 40Q | ··· | 40AF | 60A | ··· | 60P | 120A | ··· | 120AF |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 15 | ··· | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 24 | ··· | 24 | 30 | ··· | 30 | 40 | ··· | 40 | 40 | ··· | 40 | 60 | ··· | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 3 | 3 | 4 | 4 | 1 | 1 | 3 | 3 | 1 | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 1 | 1 | 1 | 1 | 3 | ··· | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 1 | ··· | 1 | 3 | ··· | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 1 | ··· | 1 | 3 | ··· | 3 | 4 | ··· | 4 | 4 | ··· | 4 |
160 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |
| type | + | + | + | + | ||||||||||||||||||||
| image | C1 | C2 | C3 | C4 | C5 | C6 | C8 | C10 | C12 | C15 | C20 | C24 | C30 | C40 | C60 | C120 | A4 | C2×A4 | C4×A4 | C5×A4 | C8×A4 | C10×A4 | A4×C20 | A4×C40 |
| kernel | A4×C40 | A4×C20 | C22×C40 | C10×A4 | C8×A4 | C22×C20 | C5×A4 | C4×A4 | C22×C10 | C22×C8 | C2×A4 | C2×C10 | C22×C4 | A4 | C23 | C22 | C40 | C20 | C10 | C8 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 4 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 16 | 16 | 32 | 1 | 1 | 2 | 4 | 4 | 4 | 8 | 16 |
Matrix representation of A4×C40 ►in GL4(𝔽241) generated by
| 8 | 0 | 0 | 0 |
| 0 | 235 | 0 | 0 |
| 0 | 0 | 235 | 0 |
| 0 | 0 | 0 | 235 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 240 | 0 |
| 0 | 16 | 0 | 240 |
| 1 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 15 | 240 |
| 225 | 0 | 0 | 0 |
| 0 | 16 | 15 | 239 |
| 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 225 |
G:=sub<GL(4,GF(241))| [8,0,0,0,0,235,0,0,0,0,235,0,0,0,0,235],[1,0,0,0,0,1,0,16,0,0,240,0,0,0,0,240],[1,0,0,0,0,240,0,0,0,0,1,15,0,0,0,240],[225,0,0,0,0,16,240,0,0,15,0,0,0,239,0,225] >;
A4×C40 in GAP, Magma, Sage, TeX
A_4\times C_{40} % in TeX
G:=Group("A4xC40"); // GroupNames label
G:=SmallGroup(480,659);
// by ID
G=gap.SmallGroup(480,659);
# by ID
G:=PCGroup([7,-2,-3,-5,-2,-2,-2,2,210,80,5052,8833]);
// Polycyclic
G:=Group<a,b,c,d|a^40=b^2=c^2=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,d*c*d^-1=b>;
// generators/relations
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