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G = A4×C40order 480 = 25·3·5

Direct product of C40 and A4

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

C1C22 — A4×C40
C1C22C23C22×C4C22×C20A4×C20 — A4×C40
C22 — A4×C40
C1C40

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 >

3C2
3C2
4C3
3C4
3C22
3C22
4C6
3C10
3C10
4C15
3C2×C4
3C8
3C2×C4
4C12
3C2×C10
3C2×C10
3C20
4C30
3C2×C8
3C2×C8
4C24
3C2×C20
3C2×C20
3C40
4C60
3C2×C40
3C2×C40
4C120

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 2A2B2C3A3B4A4B4C4D5A5B5C5D6A6B8A8B8C8D8E8F8G8H10A10B10C10D10E···10L12A12B12C12D15A···15H20A···20H20I···20P24A···24H30A···30H40A···40P40Q···40AF60A···60P120A···120AF
order1222334444555566888888881010101010···101212121215···1520···2020···2024···2430···3040···4040···4060···60120···120
size11334411331111441111333311113···344444···41···13···34···44···41···13···34···44···4

160 irreducible representations

dim111111111111111133333333
type++++
imageC1C2C3C4C5C6C8C10C12C15C20C24C30C40C60C120A4C2×A4C4×A4C5×A4C8×A4C10×A4A4×C20A4×C40
kernelA4×C40A4×C20C22×C40C10×A4C8×A4C22×C20C5×A4C4×A4C22×C10C22×C8C2×A4C2×C10C22×C4A4C23C22C40C20C10C8C5C4C2C1
# reps1122424448888161632112444816

Matrix representation of A4×C40 in GL4(𝔽241) generated by

8000
023500
002350
000235
,
1000
0100
002400
0160240
,
1000
024000
0010
0015240
,
225000
01615239
024000
000225
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

Export

Subgroup lattice of A4×C40 in TeX

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