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## G = A4×C5⋊2C8order 480 = 25·3·5

### Direct product of A4 and C5⋊2C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — A4×C5⋊2C8
 Chief series C1 — C5 — C2×C10 — C22×C10 — C22×C20 — A4×C20 — A4×C5⋊2C8
 Lower central C2×C10 — A4×C5⋊2C8
 Upper central C1 — C4

Generators and relations for A4×C52C8
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=d-1 >

Smallest permutation representation of A4×C52C8
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)(2 110 47 24 54)(3 55 17 48 111)(4 112 41 18 56)(5 49 19 42 105)(6 106 43 20 50)(7 51 21 44 107)(8 108 45 22 52)(9 90 65 73 98)(10 99 74 66 91)(11 92 67 75 100)(12 101 76 68 93)(13 94 69 77 102)(14 103 78 70 95)(15 96 71 79 104)(16 97 80 72 89)(25 116 87 59 33)(26 34 60 88 117)(27 118 81 61 35)(28 36 62 82 119)(29 120 83 63 37)(30 38 64 84 113)(31 114 85 57 39)(32 40 58 86 115)
(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)(2,110,47,24,54)(3,55,17,48,111)(4,112,41,18,56)(5,49,19,42,105)(6,106,43,20,50)(7,51,21,44,107)(8,108,45,22,52)(9,90,65,73,98)(10,99,74,66,91)(11,92,67,75,100)(12,101,76,68,93)(13,94,69,77,102)(14,103,78,70,95)(15,96,71,79,104)(16,97,80,72,89)(25,116,87,59,33)(26,34,60,88,117)(27,118,81,61,35)(28,36,62,82,119)(29,120,83,63,37)(30,38,64,84,113)(31,114,85,57,39)(32,40,58,86,115), (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)(2,110,47,24,54)(3,55,17,48,111)(4,112,41,18,56)(5,49,19,42,105)(6,106,43,20,50)(7,51,21,44,107)(8,108,45,22,52)(9,90,65,73,98)(10,99,74,66,91)(11,92,67,75,100)(12,101,76,68,93)(13,94,69,77,102)(14,103,78,70,95)(15,96,71,79,104)(16,97,80,72,89)(25,116,87,59,33)(26,34,60,88,117)(27,118,81,61,35)(28,36,62,82,119)(29,120,83,63,37)(30,38,64,84,113)(31,114,85,57,39)(32,40,58,86,115), (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),(2,110,47,24,54),(3,55,17,48,111),(4,112,41,18,56),(5,49,19,42,105),(6,106,43,20,50),(7,51,21,44,107),(8,108,45,22,52),(9,90,65,73,98),(10,99,74,66,91),(11,92,67,75,100),(12,101,76,68,93),(13,94,69,77,102),(14,103,78,70,95),(15,96,71,79,104),(16,97,80,72,89),(25,116,87,59,33),(26,34,60,88,117),(27,118,81,61,35),(28,36,62,82,119),(29,120,83,63,37),(30,38,64,84,113),(31,114,85,57,39),(32,40,58,86,115)], [(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)]])

64 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 5A 5B 6A 6B 8A 8B 8C 8D 8E 8F 8G 8H 10A 10B 10C 10D 10E 10F 12A 12B 12C 12D 15A 15B 15C 15D 20A 20B 20C 20D 20E 20F 20G 20H 24A ··· 24H 30A 30B 30C 30D 60A ··· 60H order 1 2 2 2 3 3 4 4 4 4 5 5 6 6 8 8 8 8 8 8 8 8 10 10 10 10 10 10 12 12 12 12 15 15 15 15 20 20 20 20 20 20 20 20 24 ··· 24 30 30 30 30 60 ··· 60 size 1 1 3 3 4 4 1 1 3 3 2 2 4 4 5 5 5 5 15 15 15 15 2 2 6 6 6 6 4 4 4 4 8 8 8 8 2 2 2 2 6 6 6 6 20 ··· 20 8 8 8 8 8 ··· 8

64 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 6 6 6 type + + + - + + + - image C1 C2 C3 C4 C6 C8 C12 C24 D5 Dic5 C3×D5 C5⋊2C8 C3×Dic5 C3×C5⋊2C8 A4 C2×A4 C4×A4 C8×A4 D5×A4 A4×Dic5 A4×C5⋊2C8 kernel A4×C5⋊2C8 A4×C20 C22×C5⋊2C8 C10×A4 C22×C20 C5×A4 C22×C10 C2×C10 C4×A4 C2×A4 C22×C4 A4 C23 C22 C5⋊2C8 C20 C10 C5 C4 C2 C1 # reps 1 1 2 2 2 4 4 8 2 2 4 4 4 8 1 1 2 4 2 2 4

Matrix representation of A4×C52C8 in GL5(𝔽241)

 1 0 0 0 0 0 1 0 0 0 0 0 240 0 0 0 0 0 1 16 0 0 0 0 240
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 226 0 0 0 240 0 0 0 0 0 240
,
 225 0 0 0 0 0 225 0 0 0 0 0 226 1 0 0 0 16 0 0 0 0 239 0 15
,
 0 240 0 0 0 1 51 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 84 158 0 0 0 212 157 0 0 0 0 0 211 0 0 0 0 0 211 0 0 0 0 0 211

G:=sub<GL(5,GF(241))| [1,0,0,0,0,0,1,0,0,0,0,0,240,0,0,0,0,0,1,0,0,0,0,16,240],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,240,0,0,0,226,0,240],[225,0,0,0,0,0,225,0,0,0,0,0,226,16,239,0,0,1,0,0,0,0,0,0,15],[0,1,0,0,0,240,51,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[84,212,0,0,0,158,157,0,0,0,0,0,211,0,0,0,0,0,211,0,0,0,0,0,211] >;

A4×C52C8 in GAP, Magma, Sage, TeX

A_4\times C_5\rtimes_2C_8
% in TeX

G:=Group("A4xC5:2C8");
// GroupNames label

G:=SmallGroup(480,265);
// by ID

G=gap.SmallGroup(480,265);
# by ID

G:=PCGroup([7,-2,-3,-2,-2,-2,2,-5,42,58,1271,516,18822]);
// 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^-1>;
// generators/relations

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