G = C5⋊U2(𝔽3) order 480 = 25·3·5
The semidirect product of C5 and U2(𝔽3) acting via U2(𝔽3)/SL2(𝔽3)=C4
Aliases: C5⋊U2(𝔽3), Dic5.4S4, SL2(𝔽3)⋊1F5, Q8.(C3⋊F5), (C5×Q8).Dic3, C2.2(A4⋊F5), C10.1(A4⋊C4), Q8⋊2D5.2S3, Dic5.A4.2C2, (C5×SL2(𝔽3))⋊1C4, SmallGroup(480,961)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — Q8 — C5×SL2(𝔽3) — C5⋊U2(𝔽3) |
| C5×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, bab-1=a-1, ac=ca, ad=da, ae=ea, faf-1=a2, 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 >
30C2
4C3
3C4
5C4
15C22
30C4
30C4
4C6
6D5
4C15
15C2×C4
15D4
30C8
30C2×C4
20C12
3D10
3C20
6F5
6F5
4C30
15M4(2)
15C42
20C3⋊C8
3D20
15C4≀C2
Character table of C5⋊U2(𝔽3)
| class | 1 | 2A | 2B | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5 | 6 | 8A | 8B | 10 | 12A | 12B | 15A | 15B | 20 | 30A | 30B | |
| size | 1 | 1 | 30 | 8 | 5 | 5 | 6 | 30 | 30 | 30 | 30 | 4 | 8 | 60 | 60 | 4 | 40 | 40 | 16 | 16 | 24 | 16 | 16 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -i | i | -i | i | 1 | 1 | -i | i | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ4 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | i | -i | i | -i | 1 | 1 | i | -i | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ5 | 2 | 2 | 2 | -1 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 2 | -1 | 0 | 0 | 2 | -1 | -1 | -1 | -1 | 2 | -1 | -1 | orthogonal lifted from S3 |
| ρ6 | 2 | 2 | -2 | -1 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | -1 | 0 | 0 | 2 | 1 | 1 | -1 | -1 | 2 | -1 | -1 | symplectic lifted from Dic3, Schur index 2 |
| ρ7 | 2 | -2 | 0 | -1 | -2i | 2i | 0 | 1-i | 1+i | -1+i | -1-i | 2 | 1 | 0 | 0 | -2 | -i | i | -1 | -1 | 0 | 1 | 1 | complex lifted from U2(𝔽3) |
| ρ8 | 2 | -2 | 0 | -1 | 2i | -2i | 0 | -1-i | -1+i | 1+i | 1-i | 2 | 1 | 0 | 0 | -2 | i | -i | -1 | -1 | 0 | 1 | 1 | complex lifted from U2(𝔽3) |
| ρ9 | 2 | -2 | 0 | -1 | 2i | -2i | 0 | 1+i | 1-i | -1-i | -1+i | 2 | 1 | 0 | 0 | -2 | i | -i | -1 | -1 | 0 | 1 | 1 | complex lifted from U2(𝔽3) |
| ρ10 | 2 | -2 | 0 | -1 | -2i | 2i | 0 | -1+i | -1-i | 1-i | 1+i | 2 | 1 | 0 | 0 | -2 | -i | i | -1 | -1 | 0 | 1 | 1 | complex lifted from U2(𝔽3) |
| ρ11 | 3 | 3 | -1 | 0 | 3 | 3 | -1 | 1 | 1 | 1 | 1 | 3 | 0 | -1 | -1 | 3 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | orthogonal lifted from S4 |
| ρ12 | 3 | 3 | -1 | 0 | 3 | 3 | -1 | -1 | -1 | -1 | -1 | 3 | 0 | 1 | 1 | 3 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | orthogonal lifted from S4 |
| ρ13 | 3 | 3 | 1 | 0 | -3 | -3 | -1 | -i | i | -i | i | 3 | 0 | i | -i | 3 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | complex lifted from A4⋊C4 |
| ρ14 | 3 | 3 | 1 | 0 | -3 | -3 | -1 | i | -i | i | -i | 3 | 0 | -i | i | 3 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | complex lifted from A4⋊C4 |
| ρ15 | 4 | 4 | 0 | 4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | -1 | 4 | 0 | 0 | -1 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ16 | 4 | -4 | 0 | 1 | -4i | 4i | 0 | 0 | 0 | 0 | 0 | 4 | -1 | 0 | 0 | -4 | i | -i | 1 | 1 | 0 | -1 | -1 | complex lifted from U2(𝔽3) |
| ρ17 | 4 | -4 | 0 | 1 | 4i | -4i | 0 | 0 | 0 | 0 | 0 | 4 | -1 | 0 | 0 | -4 | -i | i | 1 | 1 | 0 | -1 | -1 | complex lifted from U2(𝔽3) |
| ρ18 | 4 | 4 | 0 | -2 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | -1 | -2 | 0 | 0 | -1 | 0 | 0 | 1-√-15/2 | 1+√-15/2 | -1 | 1+√-15/2 | 1-√-15/2 | complex lifted from C3⋊F5 |
| ρ19 | 4 | 4 | 0 | -2 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | -1 | -2 | 0 | 0 | -1 | 0 | 0 | 1+√-15/2 | 1-√-15/2 | -1 | 1-√-15/2 | 1+√-15/2 | complex lifted from C3⋊F5 |
| ρ20 | 8 | -8 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 4 | 0 | 0 | 2 | 0 | 0 | 1 | 1 | 0 | -1 | -1 | orthogonal faithful |
| ρ21 | 8 | -8 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 2 | 0 | 0 | -1-√-15/2 | -1+√-15/2 | 0 | 1-√-15/2 | 1+√-15/2 | complex faithful |
| ρ22 | 8 | -8 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 2 | 0 | 0 | -1+√-15/2 | -1-√-15/2 | 0 | 1+√-15/2 | 1-√-15/2 | complex faithful |
| ρ23 | 12 | 12 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | -3 | 0 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | orthogonal lifted from A4⋊F5 |
►On 120 points
Generators in S120
(1 16 28 18 38)(2 29 39 9 19)(3 40 20 30 10)(4 21 11 33 31)(5 12 32 22 34)(6 25 35 13 23)(7 36 24 26 14)(8 17 15 37 27)(41 51 85 73 103)(42 86 104 52 74)(43 97 75 87 53)(44 76 54 98 88)(45 55 81 77 99)(46 82 100 56 78)(47 101 79 83 49)(48 80 50 102 84)(57 95 117 67 109)(58 118 110 96 68)(59 111 69 119 89)(60 70 90 112 120)(61 91 113 71 105)(62 114 106 92 72)(63 107 65 115 93)(64 66 94 108 116)
(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 4 5 8)(2 3 6 7)(9 30 13 26)(10 23 14 19)(11 32 15 28)(12 17 16 21)(18 33 22 37)(20 35 24 39)(25 36 29 40)(27 38 31 34)(41 108 45 112)(42 44 46 48)(43 110 47 106)(49 114 53 118)(50 104 54 100)(51 116 55 120)(52 98 56 102)(57 65 61 69)(58 83 62 87)(59 67 63 71)(60 85 64 81)(66 77 70 73)(68 79 72 75)(74 88 78 84)(76 82 80 86)(89 117 93 113)(90 103 94 99)(91 119 95 115)(92 97 96 101)(105 111 109 107)
(1 3 5 7)(2 8 6 4)(9 37 13 33)(10 34 14 38)(11 39 15 35)(12 36 16 40)(17 25 21 29)(18 30 22 26)(19 27 23 31)(20 32 24 28)(41 110 45 106)(42 107 46 111)(43 112 47 108)(44 109 48 105)(49 94 53 90)(50 91 54 95)(51 96 55 92)(52 93 56 89)(57 80 61 76)(58 77 62 73)(59 74 63 78)(60 79 64 75)(65 82 69 86)(66 87 70 83)(67 84 71 88)(68 81 72 85)(97 120 101 116)(98 117 102 113)(99 114 103 118)(100 119 104 115)
(1 110 42)(2 43 111)(3 112 44)(4 45 105)(5 106 46)(6 47 107)(7 108 48)(8 41 109)(9 87 89)(10 90 88)(11 81 91)(12 92 82)(13 83 93)(14 94 84)(15 85 95)(16 96 86)(17 51 57)(18 58 52)(19 53 59)(20 60 54)(21 55 61)(22 62 56)(23 49 63)(24 64 50)(25 101 65)(26 66 102)(27 103 67)(28 68 104)(29 97 69)(30 70 98)(31 99 71)(32 72 100)(33 77 113)(34 114 78)(35 79 115)(36 116 80)(37 73 117)(38 118 74)(39 75 119)(40 120 76)
(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)| (1,16,28,18,38)(2,29,39,9,19)(3,40,20,30,10)(4,21,11,33,31)(5,12,32,22,34)(6,25,35,13,23)(7,36,24,26,14)(8,17,15,37,27)(41,51,85,73,103)(42,86,104,52,74)(43,97,75,87,53)(44,76,54,98,88)(45,55,81,77,99)(46,82,100,56,78)(47,101,79,83,49)(48,80,50,102,84)(57,95,117,67,109)(58,118,110,96,68)(59,111,69,119,89)(60,70,90,112,120)(61,91,113,71,105)(62,114,106,92,72)(63,107,65,115,93)(64,66,94,108,116), (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), (1,4,5,8)(2,3,6,7)(9,30,13,26)(10,23,14,19)(11,32,15,28)(12,17,16,21)(18,33,22,37)(20,35,24,39)(25,36,29,40)(27,38,31,34)(41,108,45,112)(42,44,46,48)(43,110,47,106)(49,114,53,118)(50,104,54,100)(51,116,55,120)(52,98,56,102)(57,65,61,69)(58,83,62,87)(59,67,63,71)(60,85,64,81)(66,77,70,73)(68,79,72,75)(74,88,78,84)(76,82,80,86)(89,117,93,113)(90,103,94,99)(91,119,95,115)(92,97,96,101)(105,111,109,107), (1,3,5,7)(2,8,6,4)(9,37,13,33)(10,34,14,38)(11,39,15,35)(12,36,16,40)(17,25,21,29)(18,30,22,26)(19,27,23,31)(20,32,24,28)(41,110,45,106)(42,107,46,111)(43,112,47,108)(44,109,48,105)(49,94,53,90)(50,91,54,95)(51,96,55,92)(52,93,56,89)(57,80,61,76)(58,77,62,73)(59,74,63,78)(60,79,64,75)(65,82,69,86)(66,87,70,83)(67,84,71,88)(68,81,72,85)(97,120,101,116)(98,117,102,113)(99,114,103,118)(100,119,104,115), (1,110,42)(2,43,111)(3,112,44)(4,45,105)(5,106,46)(6,47,107)(7,108,48)(8,41,109)(9,87,89)(10,90,88)(11,81,91)(12,92,82)(13,83,93)(14,94,84)(15,85,95)(16,96,86)(17,51,57)(18,58,52)(19,53,59)(20,60,54)(21,55,61)(22,62,56)(23,49,63)(24,64,50)(25,101,65)(26,66,102)(27,103,67)(28,68,104)(29,97,69)(30,70,98)(31,99,71)(32,72,100)(33,77,113)(34,114,78)(35,79,115)(36,116,80)(37,73,117)(38,118,74)(39,75,119)(40,120,76), (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( (1,16,28,18,38)(2,29,39,9,19)(3,40,20,30,10)(4,21,11,33,31)(5,12,32,22,34)(6,25,35,13,23)(7,36,24,26,14)(8,17,15,37,27)(41,51,85,73,103)(42,86,104,52,74)(43,97,75,87,53)(44,76,54,98,88)(45,55,81,77,99)(46,82,100,56,78)(47,101,79,83,49)(48,80,50,102,84)(57,95,117,67,109)(58,118,110,96,68)(59,111,69,119,89)(60,70,90,112,120)(61,91,113,71,105)(62,114,106,92,72)(63,107,65,115,93)(64,66,94,108,116), (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), (1,4,5,8)(2,3,6,7)(9,30,13,26)(10,23,14,19)(11,32,15,28)(12,17,16,21)(18,33,22,37)(20,35,24,39)(25,36,29,40)(27,38,31,34)(41,108,45,112)(42,44,46,48)(43,110,47,106)(49,114,53,118)(50,104,54,100)(51,116,55,120)(52,98,56,102)(57,65,61,69)(58,83,62,87)(59,67,63,71)(60,85,64,81)(66,77,70,73)(68,79,72,75)(74,88,78,84)(76,82,80,86)(89,117,93,113)(90,103,94,99)(91,119,95,115)(92,97,96,101)(105,111,109,107), (1,3,5,7)(2,8,6,4)(9,37,13,33)(10,34,14,38)(11,39,15,35)(12,36,16,40)(17,25,21,29)(18,30,22,26)(19,27,23,31)(20,32,24,28)(41,110,45,106)(42,107,46,111)(43,112,47,108)(44,109,48,105)(49,94,53,90)(50,91,54,95)(51,96,55,92)(52,93,56,89)(57,80,61,76)(58,77,62,73)(59,74,63,78)(60,79,64,75)(65,82,69,86)(66,87,70,83)(67,84,71,88)(68,81,72,85)(97,120,101,116)(98,117,102,113)(99,114,103,118)(100,119,104,115), (1,110,42)(2,43,111)(3,112,44)(4,45,105)(5,106,46)(6,47,107)(7,108,48)(8,41,109)(9,87,89)(10,90,88)(11,81,91)(12,92,82)(13,83,93)(14,94,84)(15,85,95)(16,96,86)(17,51,57)(18,58,52)(19,53,59)(20,60,54)(21,55,61)(22,62,56)(23,49,63)(24,64,50)(25,101,65)(26,66,102)(27,103,67)(28,68,104)(29,97,69)(30,70,98)(31,99,71)(32,72,100)(33,77,113)(34,114,78)(35,79,115)(36,116,80)(37,73,117)(38,118,74)(39,75,119)(40,120,76), (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([[(1,16,28,18,38),(2,29,39,9,19),(3,40,20,30,10),(4,21,11,33,31),(5,12,32,22,34),(6,25,35,13,23),(7,36,24,26,14),(8,17,15,37,27),(41,51,85,73,103),(42,86,104,52,74),(43,97,75,87,53),(44,76,54,98,88),(45,55,81,77,99),(46,82,100,56,78),(47,101,79,83,49),(48,80,50,102,84),(57,95,117,67,109),(58,118,110,96,68),(59,111,69,119,89),(60,70,90,112,120),(61,91,113,71,105),(62,114,106,92,72),(63,107,65,115,93),(64,66,94,108,116)], [(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)], [(1,4,5,8),(2,3,6,7),(9,30,13,26),(10,23,14,19),(11,32,15,28),(12,17,16,21),(18,33,22,37),(20,35,24,39),(25,36,29,40),(27,38,31,34),(41,108,45,112),(42,44,46,48),(43,110,47,106),(49,114,53,118),(50,104,54,100),(51,116,55,120),(52,98,56,102),(57,65,61,69),(58,83,62,87),(59,67,63,71),(60,85,64,81),(66,77,70,73),(68,79,72,75),(74,88,78,84),(76,82,80,86),(89,117,93,113),(90,103,94,99),(91,119,95,115),(92,97,96,101),(105,111,109,107)], [(1,3,5,7),(2,8,6,4),(9,37,13,33),(10,34,14,38),(11,39,15,35),(12,36,16,40),(17,25,21,29),(18,30,22,26),(19,27,23,31),(20,32,24,28),(41,110,45,106),(42,107,46,111),(43,112,47,108),(44,109,48,105),(49,94,53,90),(50,91,54,95),(51,96,55,92),(52,93,56,89),(57,80,61,76),(58,77,62,73),(59,74,63,78),(60,79,64,75),(65,82,69,86),(66,87,70,83),(67,84,71,88),(68,81,72,85),(97,120,101,116),(98,117,102,113),(99,114,103,118),(100,119,104,115)], [(1,110,42),(2,43,111),(3,112,44),(4,45,105),(5,106,46),(6,47,107),(7,108,48),(8,41,109),(9,87,89),(10,90,88),(11,81,91),(12,92,82),(13,83,93),(14,94,84),(15,85,95),(16,96,86),(17,51,57),(18,58,52),(19,53,59),(20,60,54),(21,55,61),(22,62,56),(23,49,63),(24,64,50),(25,101,65),(26,66,102),(27,103,67),(28,68,104),(29,97,69),(30,70,98),(31,99,71),(32,72,100),(33,77,113),(34,114,78),(35,79,115),(36,116,80),(37,73,117),(38,118,74),(39,75,119),(40,120,76)], [(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)]])
Matrix representation of C5⋊U2(𝔽3) ►in GL6(𝔽241)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 240 | 240 | 240 | 240 |
| 64 | 0 | 0 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 240 | 240 | 240 | 240 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 64 | 0 | 0 | 0 | 0 | 0 |
| 0 | 177 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 88 | 152 | 0 | 0 | 0 | 0 |
| 153 | 152 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 64 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 240 | 240 | 240 | 240 |
| 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,240,0,0,1,0,0,240,0,0,0,1,0,240,0,0,0,0,1,240],[64,0,0,0,0,0,0,64,0,0,0,0,0,0,1,240,0,0,0,0,0,240,0,0,0,0,0,240,0,1,0,0,0,240,1,0],[0,240,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[64,0,0,0,0,0,0,177,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[88,153,0,0,0,0,152,152,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,64,0,0,0,0,1,0,0,0,0,0,0,0,1,0,240,0,0,0,0,0,240,1,0,0,0,1,240,0,0,0,0,0,240,0] >;
C5⋊U2(𝔽3) in GAP, Magma, Sage, TeX
C_5\rtimes {\rm U}_2({\mathbb F}_3) % in TeX
G:=Group("C5:U(2,3)"); // GroupNames label
G:=SmallGroup(480,961);
// by ID
G=gap.SmallGroup(480,961);
# by ID
G:=PCGroup([7,-2,-2,-3,-5,-2,2,-2,14,1688,170,1011,682,4204,3168,172,2525,1909,285,124]);
// 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,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,f*a*f^-1=a^2,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
Subgroup lattice of C5⋊U2(𝔽3) in TeX
Character table of C5⋊U2(𝔽3) in TeX