direct product, metabelian, soluble, monomial, A-group
Aliases: A4×C5⋊2C8, C5⋊2(C8×A4), (C5×A4)⋊4C8, C4.4(D5×A4), (C2×C10)⋊2C24, C20.4(C2×A4), C10.4(C4×A4), (C4×A4).4D5, (A4×C20).5C2, (C10×A4).4C4, C2.1(A4×Dic5), (C22×C20).2C6, (C2×A4).2Dic5, (C22×C10).2C12, C23.2(C3×Dic5), (C22×C5⋊2C8)⋊C3, C22⋊(C3×C5⋊2C8), (C22×C4).2(C3×D5), SmallGroup(480,265)
Series: Derived ►Chief ►Lower central ►Upper central
| C2×C10 — A4×C5⋊2C8 |
Generators and relations for A4×C5⋊2C8
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×C5⋊2C8
►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×C5⋊2C8 ►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×C5⋊2C8 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
Export