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## G = A4×C40order 480 = 25·3·5

### Direct product of C40 and A4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — A4×C40
 Chief series C1 — C22 — C23 — C22×C4 — C22×C20 — A4×C20 — A4×C40
 Lower central C22 — A4×C40
 Upper central C1 — 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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