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## G = A4×C5⋊C8order 480 = 25·3·5

### Direct product of A4 and C5⋊C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — A4×C5⋊C8
 Chief series C1 — C5 — C2×C10 — C22×C10 — C22×Dic5 — A4×Dic5 — A4×C5⋊C8
 Lower central C2×C10 — A4×C5⋊C8
 Upper central C1 — C2

Generators and relations for A4×C5⋊C8
G = < a,b,c,d,e | a2=b2=c3=d5=e8=1, cac-1=ab=ba, ad=da, ae=ea, cbc-1=a, bd=db, be=eb, cd=dc, ce=ec, ede-1=d3 >

Smallest permutation representation of A4×C5⋊C8
On 120 points
Generators in S120
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)(65 69)(66 70)(67 71)(68 72)(73 77)(74 78)(75 79)(76 80)(89 93)(90 94)(91 95)(92 96)(97 101)(98 102)(99 103)(100 104)(105 109)(106 110)(107 111)(108 112)
(9 13)(10 14)(11 15)(12 16)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(57 61)(58 62)(59 63)(60 64)(65 69)(66 70)(67 71)(68 72)(73 77)(74 78)(75 79)(76 80)(81 85)(82 86)(83 87)(84 88)(89 93)(90 94)(91 95)(92 96)(97 101)(98 102)(99 103)(100 104)(113 117)(114 118)(115 119)(116 120)
(1 100 60)(2 101 61)(3 102 62)(4 103 63)(5 104 64)(6 97 57)(7 98 58)(8 99 59)(9 86 51)(10 87 52)(11 88 53)(12 81 54)(13 82 55)(14 83 56)(15 84 49)(16 85 50)(17 94 119)(18 95 120)(19 96 113)(20 89 114)(21 90 115)(22 91 116)(23 92 117)(24 93 118)(25 45 66)(26 46 67)(27 47 68)(28 48 69)(29 41 70)(30 42 71)(31 43 72)(32 44 65)(33 108 74)(34 109 75)(35 110 76)(36 111 77)(37 112 78)(38 105 79)(39 106 80)(40 107 73)
(1 53 23 46 109)(2 47 54 110 24)(3 111 48 17 55)(4 18 112 56 41)(5 49 19 42 105)(6 43 50 106 20)(7 107 44 21 51)(8 22 108 52 45)(9 98 73 65 90)(10 66 99 91 74)(11 92 67 75 100)(12 76 93 101 68)(13 102 77 69 94)(14 70 103 95 78)(15 96 71 79 104)(16 80 89 97 72)(25 59 116 33 87)(26 34 60 88 117)(27 81 35 118 61)(28 119 82 62 36)(29 63 120 37 83)(30 38 64 84 113)(31 85 39 114 57)(32 115 86 58 40)
(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)

G:=sub<Sym(120)| (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(65,69)(66,70)(67,71)(68,72)(73,77)(74,78)(75,79)(76,80)(89,93)(90,94)(91,95)(92,96)(97,101)(98,102)(99,103)(100,104)(105,109)(106,110)(107,111)(108,112), (9,13)(10,14)(11,15)(12,16)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(57,61)(58,62)(59,63)(60,64)(65,69)(66,70)(67,71)(68,72)(73,77)(74,78)(75,79)(76,80)(81,85)(82,86)(83,87)(84,88)(89,93)(90,94)(91,95)(92,96)(97,101)(98,102)(99,103)(100,104)(113,117)(114,118)(115,119)(116,120), (1,100,60)(2,101,61)(3,102,62)(4,103,63)(5,104,64)(6,97,57)(7,98,58)(8,99,59)(9,86,51)(10,87,52)(11,88,53)(12,81,54)(13,82,55)(14,83,56)(15,84,49)(16,85,50)(17,94,119)(18,95,120)(19,96,113)(20,89,114)(21,90,115)(22,91,116)(23,92,117)(24,93,118)(25,45,66)(26,46,67)(27,47,68)(28,48,69)(29,41,70)(30,42,71)(31,43,72)(32,44,65)(33,108,74)(34,109,75)(35,110,76)(36,111,77)(37,112,78)(38,105,79)(39,106,80)(40,107,73), (1,53,23,46,109)(2,47,54,110,24)(3,111,48,17,55)(4,18,112,56,41)(5,49,19,42,105)(6,43,50,106,20)(7,107,44,21,51)(8,22,108,52,45)(9,98,73,65,90)(10,66,99,91,74)(11,92,67,75,100)(12,76,93,101,68)(13,102,77,69,94)(14,70,103,95,78)(15,96,71,79,104)(16,80,89,97,72)(25,59,116,33,87)(26,34,60,88,117)(27,81,35,118,61)(28,119,82,62,36)(29,63,120,37,83)(30,38,64,84,113)(31,85,39,114,57)(32,115,86,58,40), (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)>;

G:=Group( (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(65,69)(66,70)(67,71)(68,72)(73,77)(74,78)(75,79)(76,80)(89,93)(90,94)(91,95)(92,96)(97,101)(98,102)(99,103)(100,104)(105,109)(106,110)(107,111)(108,112), (9,13)(10,14)(11,15)(12,16)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(57,61)(58,62)(59,63)(60,64)(65,69)(66,70)(67,71)(68,72)(73,77)(74,78)(75,79)(76,80)(81,85)(82,86)(83,87)(84,88)(89,93)(90,94)(91,95)(92,96)(97,101)(98,102)(99,103)(100,104)(113,117)(114,118)(115,119)(116,120), (1,100,60)(2,101,61)(3,102,62)(4,103,63)(5,104,64)(6,97,57)(7,98,58)(8,99,59)(9,86,51)(10,87,52)(11,88,53)(12,81,54)(13,82,55)(14,83,56)(15,84,49)(16,85,50)(17,94,119)(18,95,120)(19,96,113)(20,89,114)(21,90,115)(22,91,116)(23,92,117)(24,93,118)(25,45,66)(26,46,67)(27,47,68)(28,48,69)(29,41,70)(30,42,71)(31,43,72)(32,44,65)(33,108,74)(34,109,75)(35,110,76)(36,111,77)(37,112,78)(38,105,79)(39,106,80)(40,107,73), (1,53,23,46,109)(2,47,54,110,24)(3,111,48,17,55)(4,18,112,56,41)(5,49,19,42,105)(6,43,50,106,20)(7,107,44,21,51)(8,22,108,52,45)(9,98,73,65,90)(10,66,99,91,74)(11,92,67,75,100)(12,76,93,101,68)(13,102,77,69,94)(14,70,103,95,78)(15,96,71,79,104)(16,80,89,97,72)(25,59,116,33,87)(26,34,60,88,117)(27,81,35,118,61)(28,119,82,62,36)(29,63,120,37,83)(30,38,64,84,113)(31,85,39,114,57)(32,115,86,58,40), (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) );

G=PermutationGroup([[(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24),(41,45),(42,46),(43,47),(44,48),(49,53),(50,54),(51,55),(52,56),(65,69),(66,70),(67,71),(68,72),(73,77),(74,78),(75,79),(76,80),(89,93),(90,94),(91,95),(92,96),(97,101),(98,102),(99,103),(100,104),(105,109),(106,110),(107,111),(108,112)], [(9,13),(10,14),(11,15),(12,16),(25,29),(26,30),(27,31),(28,32),(33,37),(34,38),(35,39),(36,40),(57,61),(58,62),(59,63),(60,64),(65,69),(66,70),(67,71),(68,72),(73,77),(74,78),(75,79),(76,80),(81,85),(82,86),(83,87),(84,88),(89,93),(90,94),(91,95),(92,96),(97,101),(98,102),(99,103),(100,104),(113,117),(114,118),(115,119),(116,120)], [(1,100,60),(2,101,61),(3,102,62),(4,103,63),(5,104,64),(6,97,57),(7,98,58),(8,99,59),(9,86,51),(10,87,52),(11,88,53),(12,81,54),(13,82,55),(14,83,56),(15,84,49),(16,85,50),(17,94,119),(18,95,120),(19,96,113),(20,89,114),(21,90,115),(22,91,116),(23,92,117),(24,93,118),(25,45,66),(26,46,67),(27,47,68),(28,48,69),(29,41,70),(30,42,71),(31,43,72),(32,44,65),(33,108,74),(34,109,75),(35,110,76),(36,111,77),(37,112,78),(38,105,79),(39,106,80),(40,107,73)], [(1,53,23,46,109),(2,47,54,110,24),(3,111,48,17,55),(4,18,112,56,41),(5,49,19,42,105),(6,43,50,106,20),(7,107,44,21,51),(8,22,108,52,45),(9,98,73,65,90),(10,66,99,91,74),(11,92,67,75,100),(12,76,93,101,68),(13,102,77,69,94),(14,70,103,95,78),(15,96,71,79,104),(16,80,89,97,72),(25,59,116,33,87),(26,34,60,88,117),(27,81,35,118,61),(28,119,82,62,36),(29,63,120,37,83),(30,38,64,84,113),(31,85,39,114,57),(32,115,86,58,40)], [(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)]])

40 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 5 6A 6B 8A 8B 8C 8D 8E 8F 8G 8H 10A 10B 10C 12A 12B 12C 12D 15A 15B 24A ··· 24H 30A 30B order 1 2 2 2 3 3 4 4 4 4 5 6 6 8 8 8 8 8 8 8 8 10 10 10 12 12 12 12 15 15 24 ··· 24 30 30 size 1 1 3 3 4 4 5 5 15 15 4 4 4 5 5 5 5 15 15 15 15 4 12 12 20 20 20 20 16 16 20 ··· 20 16 16

40 irreducible representations

 dim 1 1 1 1 1 1 1 1 12 12 3 3 3 3 4 4 4 4 type + + + - + + + - image C1 C2 C3 C4 C6 C8 C12 C24 A4×F5 A4×C5⋊C8 A4 C2×A4 C4×A4 C8×A4 F5 C5⋊C8 C3×F5 C3×C5⋊C8 kernel A4×C5⋊C8 A4×Dic5 C22×C5⋊C8 C10×A4 C22×Dic5 C5×A4 C22×C10 C2×C10 C2 C1 C5⋊C8 Dic5 C10 C5 C2×A4 A4 C23 C22 # reps 1 1 2 2 2 4 4 8 1 1 1 1 2 4 1 1 2 2

Matrix representation of A4×C5⋊C8 in GL7(𝔽241)

 240 0 0 0 0 0 0 0 1 0 0 0 0 0 0 226 240 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 240 0 0 0 0 0 16 0 240 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 0 16 226 239 0 0 0 0 0 0 15 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 240 0 0 0 1 0 0 240 0 0 0 0 1 0 240 0 0 0 0 0 1 240
,
 233 0 0 0 0 0 0 0 233 0 0 0 0 0 0 0 233 0 0 0 0 0 0 0 51 180 32 157 0 0 0 83 96 94 208 0 0 0 145 147 33 240 0 0 0 84 179 190 61

G:=sub<GL(7,GF(241))| [240,0,0,0,0,0,0,0,1,226,0,0,0,0,0,0,240,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,16,0,0,0,0,0,240,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[0,16,0,0,0,0,0,1,226,0,0,0,0,0,0,239,15,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,240,240,240,240],[233,0,0,0,0,0,0,0,233,0,0,0,0,0,0,0,233,0,0,0,0,0,0,0,51,83,145,84,0,0,0,180,96,147,179,0,0,0,32,94,33,190,0,0,0,157,208,240,61] >;

A4×C5⋊C8 in GAP, Magma, Sage, TeX

A_4\times C_5\rtimes C_8
% in TeX

G:=Group("A4xC5:C8");
// GroupNames label

G:=SmallGroup(480,966);
// by ID

G=gap.SmallGroup(480,966);
# by ID

G:=PCGroup([7,-2,-3,-2,-2,-2,2,-5,42,58,1271,516,9414,3156]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^3=d^5=e^8=1,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^3>;
// generators/relations

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