direct product, non-abelian, soluble
Aliases: C5×U2(𝔽3), C20.11S4, SL2(𝔽3)⋊2C20, C4.5(C5×S4), Q8.(C5×Dic3), C4.A4.2C10, C10.9(A4⋊C4), (C5×Q8).3Dic3, (C5×SL2(𝔽3))⋊8C4, C2.3(C5×A4⋊C4), C4○D4.1(C5×S3), (C5×C4.A4).5C2, (C5×C4○D4).3S3, SmallGroup(480,257)
Series: Derived ►Chief ►Lower central ►Upper central
| SL2(𝔽3) — C5×U2(𝔽3) |
Generators and relations for C5×U2(𝔽3)
G = < a,b,c,d,e,f | a5=b4=e3=1, c2=d2=b2, f2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd-1=b2c, ece-1=b2cd, fcf-1=cd, ede-1=c, fdf-1=b2d, fef-1=e-1 >
Smallest permutation representation of C5×U2(𝔽3)
►On 120 points
Generators in S120
(1 98 74 50 26)(2 99 75 51 27)(3 100 76 52 28)(4 101 77 53 29)(5 102 78 54 30)(6 103 79 55 31)(7 104 80 56 32)(8 97 73 49 25)(9 105 81 57 33)(10 106 82 58 34)(11 107 83 59 35)(12 108 84 60 36)(13 109 85 61 37)(14 110 86 62 38)(15 111 87 63 39)(16 112 88 64 40)(17 113 89 65 41)(18 114 90 66 42)(19 115 91 67 43)(20 116 92 68 44)(21 117 93 69 45)(22 118 94 70 46)(23 119 95 71 47)(24 120 96 72 48)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 64)(65 67 69 71)(66 68 70 72)(73 75 77 79)(74 76 78 80)(81 83 85 87)(82 84 86 88)(89 91 93 95)(90 92 94 96)(97 99 101 103)(98 100 102 104)(105 107 109 111)(106 108 110 112)(113 115 117 119)(114 116 118 120)
(1 2 5 6)(3 4 7 8)(9 15 13 11)(10 18 14 22)(12 20 16 24)(17 19 21 23)(25 28 29 32)(26 27 30 31)(33 39 37 35)(34 42 38 46)(36 44 40 48)(41 43 45 47)(49 52 53 56)(50 51 54 55)(57 63 61 59)(58 66 62 70)(60 68 64 72)(65 67 69 71)(73 76 77 80)(74 75 78 79)(81 87 85 83)(82 90 86 94)(84 92 88 96)(89 91 93 95)(97 100 101 104)(98 99 102 103)(105 111 109 107)(106 114 110 118)(108 116 112 120)(113 115 117 119)
(1 7 5 3)(2 4 6 8)(9 19 13 23)(10 24 14 20)(11 21 15 17)(12 18 16 22)(25 27 29 31)(26 32 30 28)(33 43 37 47)(34 48 38 44)(35 45 39 41)(36 42 40 46)(49 51 53 55)(50 56 54 52)(57 67 61 71)(58 72 62 68)(59 69 63 65)(60 66 64 70)(73 75 77 79)(74 80 78 76)(81 91 85 95)(82 96 86 92)(83 93 87 89)(84 90 88 94)(97 99 101 103)(98 104 102 100)(105 115 109 119)(106 120 110 116)(107 117 111 113)(108 114 112 118)
(1 18 15)(2 16 19)(3 20 9)(4 10 21)(5 22 11)(6 12 23)(7 24 13)(8 14 17)(25 38 41)(26 42 39)(27 40 43)(28 44 33)(29 34 45)(30 46 35)(31 36 47)(32 48 37)(49 62 65)(50 66 63)(51 64 67)(52 68 57)(53 58 69)(54 70 59)(55 60 71)(56 72 61)(73 86 89)(74 90 87)(75 88 91)(76 92 81)(77 82 93)(78 94 83)(79 84 95)(80 96 85)(97 110 113)(98 114 111)(99 112 115)(100 116 105)(101 106 117)(102 118 107)(103 108 119)(104 120 109)
(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)| (1,98,74,50,26)(2,99,75,51,27)(3,100,76,52,28)(4,101,77,53,29)(5,102,78,54,30)(6,103,79,55,31)(7,104,80,56,32)(8,97,73,49,25)(9,105,81,57,33)(10,106,82,58,34)(11,107,83,59,35)(12,108,84,60,36)(13,109,85,61,37)(14,110,86,62,38)(15,111,87,63,39)(16,112,88,64,40)(17,113,89,65,41)(18,114,90,66,42)(19,115,91,67,43)(20,116,92,68,44)(21,117,93,69,45)(22,118,94,70,46)(23,119,95,71,47)(24,120,96,72,48), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64)(65,67,69,71)(66,68,70,72)(73,75,77,79)(74,76,78,80)(81,83,85,87)(82,84,86,88)(89,91,93,95)(90,92,94,96)(97,99,101,103)(98,100,102,104)(105,107,109,111)(106,108,110,112)(113,115,117,119)(114,116,118,120), (1,2,5,6)(3,4,7,8)(9,15,13,11)(10,18,14,22)(12,20,16,24)(17,19,21,23)(25,28,29,32)(26,27,30,31)(33,39,37,35)(34,42,38,46)(36,44,40,48)(41,43,45,47)(49,52,53,56)(50,51,54,55)(57,63,61,59)(58,66,62,70)(60,68,64,72)(65,67,69,71)(73,76,77,80)(74,75,78,79)(81,87,85,83)(82,90,86,94)(84,92,88,96)(89,91,93,95)(97,100,101,104)(98,99,102,103)(105,111,109,107)(106,114,110,118)(108,116,112,120)(113,115,117,119), (1,7,5,3)(2,4,6,8)(9,19,13,23)(10,24,14,20)(11,21,15,17)(12,18,16,22)(25,27,29,31)(26,32,30,28)(33,43,37,47)(34,48,38,44)(35,45,39,41)(36,42,40,46)(49,51,53,55)(50,56,54,52)(57,67,61,71)(58,72,62,68)(59,69,63,65)(60,66,64,70)(73,75,77,79)(74,80,78,76)(81,91,85,95)(82,96,86,92)(83,93,87,89)(84,90,88,94)(97,99,101,103)(98,104,102,100)(105,115,109,119)(106,120,110,116)(107,117,111,113)(108,114,112,118), (1,18,15)(2,16,19)(3,20,9)(4,10,21)(5,22,11)(6,12,23)(7,24,13)(8,14,17)(25,38,41)(26,42,39)(27,40,43)(28,44,33)(29,34,45)(30,46,35)(31,36,47)(32,48,37)(49,62,65)(50,66,63)(51,64,67)(52,68,57)(53,58,69)(54,70,59)(55,60,71)(56,72,61)(73,86,89)(74,90,87)(75,88,91)(76,92,81)(77,82,93)(78,94,83)(79,84,95)(80,96,85)(97,110,113)(98,114,111)(99,112,115)(100,116,105)(101,106,117)(102,118,107)(103,108,119)(104,120,109), (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( (1,98,74,50,26)(2,99,75,51,27)(3,100,76,52,28)(4,101,77,53,29)(5,102,78,54,30)(6,103,79,55,31)(7,104,80,56,32)(8,97,73,49,25)(9,105,81,57,33)(10,106,82,58,34)(11,107,83,59,35)(12,108,84,60,36)(13,109,85,61,37)(14,110,86,62,38)(15,111,87,63,39)(16,112,88,64,40)(17,113,89,65,41)(18,114,90,66,42)(19,115,91,67,43)(20,116,92,68,44)(21,117,93,69,45)(22,118,94,70,46)(23,119,95,71,47)(24,120,96,72,48), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64)(65,67,69,71)(66,68,70,72)(73,75,77,79)(74,76,78,80)(81,83,85,87)(82,84,86,88)(89,91,93,95)(90,92,94,96)(97,99,101,103)(98,100,102,104)(105,107,109,111)(106,108,110,112)(113,115,117,119)(114,116,118,120), (1,2,5,6)(3,4,7,8)(9,15,13,11)(10,18,14,22)(12,20,16,24)(17,19,21,23)(25,28,29,32)(26,27,30,31)(33,39,37,35)(34,42,38,46)(36,44,40,48)(41,43,45,47)(49,52,53,56)(50,51,54,55)(57,63,61,59)(58,66,62,70)(60,68,64,72)(65,67,69,71)(73,76,77,80)(74,75,78,79)(81,87,85,83)(82,90,86,94)(84,92,88,96)(89,91,93,95)(97,100,101,104)(98,99,102,103)(105,111,109,107)(106,114,110,118)(108,116,112,120)(113,115,117,119), (1,7,5,3)(2,4,6,8)(9,19,13,23)(10,24,14,20)(11,21,15,17)(12,18,16,22)(25,27,29,31)(26,32,30,28)(33,43,37,47)(34,48,38,44)(35,45,39,41)(36,42,40,46)(49,51,53,55)(50,56,54,52)(57,67,61,71)(58,72,62,68)(59,69,63,65)(60,66,64,70)(73,75,77,79)(74,80,78,76)(81,91,85,95)(82,96,86,92)(83,93,87,89)(84,90,88,94)(97,99,101,103)(98,104,102,100)(105,115,109,119)(106,120,110,116)(107,117,111,113)(108,114,112,118), (1,18,15)(2,16,19)(3,20,9)(4,10,21)(5,22,11)(6,12,23)(7,24,13)(8,14,17)(25,38,41)(26,42,39)(27,40,43)(28,44,33)(29,34,45)(30,46,35)(31,36,47)(32,48,37)(49,62,65)(50,66,63)(51,64,67)(52,68,57)(53,58,69)(54,70,59)(55,60,71)(56,72,61)(73,86,89)(74,90,87)(75,88,91)(76,92,81)(77,82,93)(78,94,83)(79,84,95)(80,96,85)(97,110,113)(98,114,111)(99,112,115)(100,116,105)(101,106,117)(102,118,107)(103,108,119)(104,120,109), (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([[(1,98,74,50,26),(2,99,75,51,27),(3,100,76,52,28),(4,101,77,53,29),(5,102,78,54,30),(6,103,79,55,31),(7,104,80,56,32),(8,97,73,49,25),(9,105,81,57,33),(10,106,82,58,34),(11,107,83,59,35),(12,108,84,60,36),(13,109,85,61,37),(14,110,86,62,38),(15,111,87,63,39),(16,112,88,64,40),(17,113,89,65,41),(18,114,90,66,42),(19,115,91,67,43),(20,116,92,68,44),(21,117,93,69,45),(22,118,94,70,46),(23,119,95,71,47),(24,120,96,72,48)], [(1,3,5,7),(2,4,6,8),(9,11,13,15),(10,12,14,16),(17,19,21,23),(18,20,22,24),(25,27,29,31),(26,28,30,32),(33,35,37,39),(34,36,38,40),(41,43,45,47),(42,44,46,48),(49,51,53,55),(50,52,54,56),(57,59,61,63),(58,60,62,64),(65,67,69,71),(66,68,70,72),(73,75,77,79),(74,76,78,80),(81,83,85,87),(82,84,86,88),(89,91,93,95),(90,92,94,96),(97,99,101,103),(98,100,102,104),(105,107,109,111),(106,108,110,112),(113,115,117,119),(114,116,118,120)], [(1,2,5,6),(3,4,7,8),(9,15,13,11),(10,18,14,22),(12,20,16,24),(17,19,21,23),(25,28,29,32),(26,27,30,31),(33,39,37,35),(34,42,38,46),(36,44,40,48),(41,43,45,47),(49,52,53,56),(50,51,54,55),(57,63,61,59),(58,66,62,70),(60,68,64,72),(65,67,69,71),(73,76,77,80),(74,75,78,79),(81,87,85,83),(82,90,86,94),(84,92,88,96),(89,91,93,95),(97,100,101,104),(98,99,102,103),(105,111,109,107),(106,114,110,118),(108,116,112,120),(113,115,117,119)], [(1,7,5,3),(2,4,6,8),(9,19,13,23),(10,24,14,20),(11,21,15,17),(12,18,16,22),(25,27,29,31),(26,32,30,28),(33,43,37,47),(34,48,38,44),(35,45,39,41),(36,42,40,46),(49,51,53,55),(50,56,54,52),(57,67,61,71),(58,72,62,68),(59,69,63,65),(60,66,64,70),(73,75,77,79),(74,80,78,76),(81,91,85,95),(82,96,86,92),(83,93,87,89),(84,90,88,94),(97,99,101,103),(98,104,102,100),(105,115,109,119),(106,120,110,116),(107,117,111,113),(108,114,112,118)], [(1,18,15),(2,16,19),(3,20,9),(4,10,21),(5,22,11),(6,12,23),(7,24,13),(8,14,17),(25,38,41),(26,42,39),(27,40,43),(28,44,33),(29,34,45),(30,46,35),(31,36,47),(32,48,37),(49,62,65),(50,66,63),(51,64,67),(52,68,57),(53,58,69),(54,70,59),(55,60,71),(56,72,61),(73,86,89),(74,90,87),(75,88,91),(76,92,81),(77,82,93),(78,94,83),(79,84,95),(80,96,85),(97,110,113),(98,114,111),(99,112,115),(100,116,105),(101,106,117),(102,118,107),(103,108,119),(104,120,109)], [(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)]])
80 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 4C | ··· | 4G | 5A | 5B | 5C | 5D | 6 | 8A | 8B | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 12A | 12B | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 20I | ··· | 20AB | 30A | 30B | 30C | 30D | 40A | ··· | 40H | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 6 | 8 | 8 | 10 | 10 | 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 | 6 | 8 | 1 | 1 | 6 | ··· | 6 | 1 | 1 | 1 | 1 | 8 | 12 | 12 | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 1 | ··· | 1 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 8 | ··· | 8 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 |
| type | + | + | + | - | + | |||||||||||||
| image | C1 | C2 | C4 | C5 | C10 | C20 | S3 | Dic3 | C5×S3 | C5×Dic3 | U2(𝔽3) | C5×U2(𝔽3) | S4 | A4⋊C4 | C5×S4 | C5×A4⋊C4 | U2(𝔽3) | C5×U2(𝔽3) |
| kernel | C5×U2(𝔽3) | C5×C4.A4 | C5×SL2(𝔽3) | U2(𝔽3) | C4.A4 | SL2(𝔽3) | C5×C4○D4 | C5×Q8 | C4○D4 | Q8 | C5 | C1 | C20 | C10 | C4 | C2 | C5 | C1 |
| # reps | 1 | 1 | 2 | 4 | 4 | 8 | 1 | 1 | 4 | 4 | 4 | 16 | 2 | 2 | 8 | 8 | 2 | 8 |
Matrix representation of C5×U2(𝔽3) ►in GL2(𝔽41) generated by
| 16 | 0 |
| 0 | 16 |
| 32 | 0 |
| 0 | 32 |
| 1 | 4 |
| 20 | 40 |
| 32 | 5 |
| 0 | 9 |
| 8 | 16 |
| 39 | 32 |
| 9 | 20 |
| 16 | 32 |
G:=sub<GL(2,GF(41))| [16,0,0,16],[32,0,0,32],[1,20,4,40],[32,0,5,9],[8,39,16,32],[9,16,20,32] >;
C5×U2(𝔽3) in GAP, Magma, Sage, TeX
C_5\times {\rm U}_2({\mathbb F}_3) % in TeX
G:=Group("C5xU(2,3)"); // GroupNames label
G:=SmallGroup(480,257);
// by ID
G=gap.SmallGroup(480,257);
# by ID
G:=PCGroup([7,-2,-5,-2,-3,-2,2,-2,70,520,1123,4204,655,172,2525,404,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^5=b^4=e^3=1,c^2=d^2=b^2,f^2=b,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,d*c*d^-1=b^2*c,e*c*e^-1=b^2*c*d,f*c*f^-1=c*d,e*d*e^-1=c,f*d*f^-1=b^2*d,f*e*f^-1=e^-1>;
// generators/relations
Export