direct product, non-abelian, soluble, monomial
Aliases: C5×A4⋊C8, A4⋊C40, C20.10S4, (C5×A4)⋊5C8, (C2×A4).C20, C4.4(C5×S4), (A4×C20).6C2, (C4×A4).2C10, (C10×A4).5C4, C10.7(A4⋊C4), C23.(C5×Dic3), (C22×C20).1S3, (C22×C10).2Dic3, C22⋊(C5×C3⋊C8), (C2×C10)⋊2(C3⋊C8), C2.1(C5×A4⋊C4), (C22×C4).1(C5×S3), SmallGroup(480,255)
Series: Derived ►Chief ►Lower central ►Upper central
| A4 — C5×A4⋊C8 |
Generators and relations for C5×A4⋊C8
G = < a,b,c,d,e | a5=b2=c2=d3=e8=1, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=ebe-1=bc=cb, dcd-1=b, ce=ec, ede-1=d-1 >
Smallest permutation representation of C5×A4⋊C8
►On 120 points
Generators in S120
(1 94 119 78 103)(2 95 120 79 104)(3 96 113 80 97)(4 89 114 73 98)(5 90 115 74 99)(6 91 116 75 100)(7 92 117 76 101)(8 93 118 77 102)(9 34 58 83 108)(10 35 59 84 109)(11 36 60 85 110)(12 37 61 86 111)(13 38 62 87 112)(14 39 63 88 105)(15 40 64 81 106)(16 33 57 82 107)(17 42 66 27 52)(18 43 67 28 53)(19 44 68 29 54)(20 45 69 30 55)(21 46 70 31 56)(22 47 71 32 49)(23 48 72 25 50)(24 41 65 26 51)
(2 6)(4 8)(9 13)(10 14)(11 15)(12 16)(18 22)(20 24)(26 30)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(43 47)(49 53)(51 55)(57 61)(58 62)(59 63)(60 64)(65 69)(67 71)(73 77)(75 79)(81 85)(82 86)(83 87)(84 88)(89 93)(91 95)(98 102)(100 104)(105 109)(106 110)(107 111)(108 112)(114 118)(116 120)
(1 5)(2 6)(3 7)(4 8)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(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)(113 117)(114 118)(115 119)(116 120)
(1 35 43)(2 44 36)(3 37 45)(4 46 38)(5 39 47)(6 48 40)(7 33 41)(8 42 34)(9 102 17)(10 18 103)(11 104 19)(12 20 97)(13 98 21)(14 22 99)(15 100 23)(16 24 101)(25 81 116)(26 117 82)(27 83 118)(28 119 84)(29 85 120)(30 113 86)(31 87 114)(32 115 88)(49 74 105)(50 106 75)(51 76 107)(52 108 77)(53 78 109)(54 110 79)(55 80 111)(56 112 73)(57 65 92)(58 93 66)(59 67 94)(60 95 68)(61 69 96)(62 89 70)(63 71 90)(64 91 72)
(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)| (1,94,119,78,103)(2,95,120,79,104)(3,96,113,80,97)(4,89,114,73,98)(5,90,115,74,99)(6,91,116,75,100)(7,92,117,76,101)(8,93,118,77,102)(9,34,58,83,108)(10,35,59,84,109)(11,36,60,85,110)(12,37,61,86,111)(13,38,62,87,112)(14,39,63,88,105)(15,40,64,81,106)(16,33,57,82,107)(17,42,66,27,52)(18,43,67,28,53)(19,44,68,29,54)(20,45,69,30,55)(21,46,70,31,56)(22,47,71,32,49)(23,48,72,25,50)(24,41,65,26,51), (2,6)(4,8)(9,13)(10,14)(11,15)(12,16)(18,22)(20,24)(26,30)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(43,47)(49,53)(51,55)(57,61)(58,62)(59,63)(60,64)(65,69)(67,71)(73,77)(75,79)(81,85)(82,86)(83,87)(84,88)(89,93)(91,95)(98,102)(100,104)(105,109)(106,110)(107,111)(108,112)(114,118)(116,120), (1,5)(2,6)(3,7)(4,8)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(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)(113,117)(114,118)(115,119)(116,120), (1,35,43)(2,44,36)(3,37,45)(4,46,38)(5,39,47)(6,48,40)(7,33,41)(8,42,34)(9,102,17)(10,18,103)(11,104,19)(12,20,97)(13,98,21)(14,22,99)(15,100,23)(16,24,101)(25,81,116)(26,117,82)(27,83,118)(28,119,84)(29,85,120)(30,113,86)(31,87,114)(32,115,88)(49,74,105)(50,106,75)(51,76,107)(52,108,77)(53,78,109)(54,110,79)(55,80,111)(56,112,73)(57,65,92)(58,93,66)(59,67,94)(60,95,68)(61,69,96)(62,89,70)(63,71,90)(64,91,72), (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( (1,94,119,78,103)(2,95,120,79,104)(3,96,113,80,97)(4,89,114,73,98)(5,90,115,74,99)(6,91,116,75,100)(7,92,117,76,101)(8,93,118,77,102)(9,34,58,83,108)(10,35,59,84,109)(11,36,60,85,110)(12,37,61,86,111)(13,38,62,87,112)(14,39,63,88,105)(15,40,64,81,106)(16,33,57,82,107)(17,42,66,27,52)(18,43,67,28,53)(19,44,68,29,54)(20,45,69,30,55)(21,46,70,31,56)(22,47,71,32,49)(23,48,72,25,50)(24,41,65,26,51), (2,6)(4,8)(9,13)(10,14)(11,15)(12,16)(18,22)(20,24)(26,30)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(43,47)(49,53)(51,55)(57,61)(58,62)(59,63)(60,64)(65,69)(67,71)(73,77)(75,79)(81,85)(82,86)(83,87)(84,88)(89,93)(91,95)(98,102)(100,104)(105,109)(106,110)(107,111)(108,112)(114,118)(116,120), (1,5)(2,6)(3,7)(4,8)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(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)(113,117)(114,118)(115,119)(116,120), (1,35,43)(2,44,36)(3,37,45)(4,46,38)(5,39,47)(6,48,40)(7,33,41)(8,42,34)(9,102,17)(10,18,103)(11,104,19)(12,20,97)(13,98,21)(14,22,99)(15,100,23)(16,24,101)(25,81,116)(26,117,82)(27,83,118)(28,119,84)(29,85,120)(30,113,86)(31,87,114)(32,115,88)(49,74,105)(50,106,75)(51,76,107)(52,108,77)(53,78,109)(54,110,79)(55,80,111)(56,112,73)(57,65,92)(58,93,66)(59,67,94)(60,95,68)(61,69,96)(62,89,70)(63,71,90)(64,91,72), (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([[(1,94,119,78,103),(2,95,120,79,104),(3,96,113,80,97),(4,89,114,73,98),(5,90,115,74,99),(6,91,116,75,100),(7,92,117,76,101),(8,93,118,77,102),(9,34,58,83,108),(10,35,59,84,109),(11,36,60,85,110),(12,37,61,86,111),(13,38,62,87,112),(14,39,63,88,105),(15,40,64,81,106),(16,33,57,82,107),(17,42,66,27,52),(18,43,67,28,53),(19,44,68,29,54),(20,45,69,30,55),(21,46,70,31,56),(22,47,71,32,49),(23,48,72,25,50),(24,41,65,26,51)], [(2,6),(4,8),(9,13),(10,14),(11,15),(12,16),(18,22),(20,24),(26,30),(28,32),(33,37),(34,38),(35,39),(36,40),(41,45),(43,47),(49,53),(51,55),(57,61),(58,62),(59,63),(60,64),(65,69),(67,71),(73,77),(75,79),(81,85),(82,86),(83,87),(84,88),(89,93),(91,95),(98,102),(100,104),(105,109),(106,110),(107,111),(108,112),(114,118),(116,120)], [(1,5),(2,6),(3,7),(4,8),(17,21),(18,22),(19,23),(20,24),(25,29),(26,30),(27,31),(28,32),(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),(113,117),(114,118),(115,119),(116,120)], [(1,35,43),(2,44,36),(3,37,45),(4,46,38),(5,39,47),(6,48,40),(7,33,41),(8,42,34),(9,102,17),(10,18,103),(11,104,19),(12,20,97),(13,98,21),(14,22,99),(15,100,23),(16,24,101),(25,81,116),(26,117,82),(27,83,118),(28,119,84),(29,85,120),(30,113,86),(31,87,114),(32,115,88),(49,74,105),(50,106,75),(51,76,107),(52,108,77),(53,78,109),(54,110,79),(55,80,111),(56,112,73),(57,65,92),(58,93,66),(59,67,94),(60,95,68),(61,69,96),(62,89,70),(63,71,90),(64,91,72)], [(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)]])
100 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 5A | 5B | 5C | 5D | 6 | 8A | ··· | 8H | 10A | 10B | 10C | 10D | 10E | ··· | 10L | 12A | 12B | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 20I | ··· | 20P | 30A | 30B | 30C | 30D | 40A | ··· | 40AF | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 8 | ··· | 8 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | 30 | 30 | 30 | 40 | ··· | 40 | 60 | ··· | 60 |
| size | 1 | 1 | 3 | 3 | 8 | 1 | 1 | 3 | 3 | 1 | 1 | 1 | 1 | 8 | 6 | ··· | 6 | 1 | 1 | 1 | 1 | 3 | ··· | 3 | 8 | 8 | 8 | 8 | 8 | 8 | 1 | ··· | 1 | 3 | ··· | 3 | 8 | 8 | 8 | 8 | 6 | ··· | 6 | 8 | ··· | 8 |
100 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 |
| type | + | + | + | - | + | |||||||||||||||
| image | C1 | C2 | C4 | C5 | C8 | C10 | C20 | C40 | S3 | Dic3 | C3⋊C8 | C5×S3 | C5×Dic3 | C5×C3⋊C8 | S4 | A4⋊C4 | A4⋊C8 | C5×S4 | C5×A4⋊C4 | C5×A4⋊C8 |
| kernel | C5×A4⋊C8 | A4×C20 | C10×A4 | A4⋊C8 | C5×A4 | C4×A4 | C2×A4 | A4 | C22×C20 | C22×C10 | C2×C10 | C22×C4 | C23 | C22 | C20 | C10 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 4 | 4 | 8 | 16 | 1 | 1 | 2 | 4 | 4 | 8 | 2 | 2 | 4 | 8 | 8 | 16 |
Matrix representation of C5×A4⋊C8 ►in GL3(𝔽241) generated by
| 87 | 0 | 0 |
| 0 | 87 | 0 |
| 0 | 0 | 87 |
| 1 | 0 | 0 |
| 1 | 240 | 0 |
| 1 | 0 | 240 |
| 240 | 0 | 0 |
| 240 | 1 | 0 |
| 0 | 0 | 240 |
| 1 | 0 | 239 |
| 0 | 0 | 240 |
| 0 | 1 | 240 |
| 8 | 0 | 225 |
| 0 | 8 | 233 |
| 0 | 0 | 233 |
G:=sub<GL(3,GF(241))| [87,0,0,0,87,0,0,0,87],[1,1,1,0,240,0,0,0,240],[240,240,0,0,1,0,0,0,240],[1,0,0,0,0,1,239,240,240],[8,0,0,0,8,0,225,233,233] >;
C5×A4⋊C8 in GAP, Magma, Sage, TeX
C_5\times A_4\rtimes C_8
% in TeX
G:=Group("C5xA4:C8"); // GroupNames label
G:=SmallGroup(480,255);
// by ID
G=gap.SmallGroup(480,255);
# by ID
G:=PCGroup([7,-2,-5,-2,-2,-3,-2,2,70,58,2804,10085,285,5886,475]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^2=c^2=d^3=e^8=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=e*b*e^-1=b*c=c*b,d*c*d^-1=b,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations
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