metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic3.F5, D30.2C4, C15⋊2M4(2), Dic5.10D6, C5⋊C8⋊2S3, C15⋊C8⋊4C2, C5⋊1(C8⋊S3), C3⋊1(C4.F5), C2.7(S3×F5), C6.7(C2×F5), C10.7(C4×S3), C30.7(C2×C4), D30.C2.3C2, (C5×Dic3).2C4, (C3×Dic5).10C22, (C3×C5⋊C8)⋊4C2, SmallGroup(240,101)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic3.F5
G = < a,b,c,d | a6=c5=1, b2=d4=a3, bab-1=a-1, ac=ca, ad=da, bc=cb, dbd-1=a3b, dcd-1=c3 >
Character table of Dic3.F5
| class | 1 | 2A | 2B | 3 | 4A | 4B | 4C | 5 | 6 | 8A | 8B | 8C | 8D | 10 | 12A | 12B | 15 | 20A | 20B | 24A | 24B | 24C | 24D | 30 | |
| size | 1 | 1 | 30 | 2 | 5 | 5 | 6 | 4 | 2 | 10 | 10 | 30 | 30 | 4 | 10 | 10 | 8 | 12 | 12 | 10 | 10 | 10 | 10 | 8 | |
| ρ1 | 1 | 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 | 1 | linear of order 2 |
| ρ3 | 1 | 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 |
| ρ4 | 1 | 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 |
| ρ5 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | i | -i | -i | i | 1 | -1 | -1 | 1 | -1 | -1 | -i | i | i | -i | 1 | linear of order 4 |
| ρ6 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | i | -i | i | -i | 1 | -1 | -1 | 1 | 1 | 1 | -i | i | i | -i | 1 | linear of order 4 |
| ρ7 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | -i | i | -i | i | 1 | -1 | -1 | 1 | 1 | 1 | i | -i | -i | i | 1 | linear of order 4 |
| ρ8 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -i | i | i | -i | 1 | -1 | -1 | 1 | -1 | -1 | i | -i | -i | i | 1 | linear of order 4 |
| ρ9 | 2 | 2 | 0 | -1 | 2 | 2 | 0 | 2 | -1 | 2 | 2 | 0 | 0 | 2 | -1 | -1 | -1 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ10 | 2 | 2 | 0 | -1 | 2 | 2 | 0 | 2 | -1 | -2 | -2 | 0 | 0 | 2 | -1 | -1 | -1 | 0 | 0 | 1 | 1 | 1 | 1 | -1 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | 0 | -1 | -2 | -2 | 0 | 2 | -1 | 2i | -2i | 0 | 0 | 2 | 1 | 1 | -1 | 0 | 0 | i | -i | -i | i | -1 | complex lifted from C4×S3 |
| ρ12 | 2 | -2 | 0 | 2 | 2i | -2i | 0 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | 2i | -2i | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | complex lifted from M4(2) |
| ρ13 | 2 | -2 | 0 | 2 | -2i | 2i | 0 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | -2i | 2i | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | complex lifted from M4(2) |
| ρ14 | 2 | 2 | 0 | -1 | -2 | -2 | 0 | 2 | -1 | -2i | 2i | 0 | 0 | 2 | 1 | 1 | -1 | 0 | 0 | -i | i | i | -i | -1 | complex lifted from C4×S3 |
| ρ15 | 2 | -2 | 0 | -1 | -2i | 2i | 0 | 2 | 1 | 0 | 0 | 0 | 0 | -2 | i | -i | -1 | 0 | 0 | 2ζ85ζ3+ζ85 | 2ζ83ζ3+ζ83 | 2ζ87ζ3+ζ87 | 2ζ8ζ3+ζ8 | 1 | complex lifted from C8⋊S3 |
| ρ16 | 2 | -2 | 0 | -1 | -2i | 2i | 0 | 2 | 1 | 0 | 0 | 0 | 0 | -2 | i | -i | -1 | 0 | 0 | 2ζ8ζ3+ζ8 | 2ζ87ζ3+ζ87 | 2ζ83ζ3+ζ83 | 2ζ85ζ3+ζ85 | 1 | complex lifted from C8⋊S3 |
| ρ17 | 2 | -2 | 0 | -1 | 2i | -2i | 0 | 2 | 1 | 0 | 0 | 0 | 0 | -2 | -i | i | -1 | 0 | 0 | 2ζ83ζ3+ζ83 | 2ζ85ζ3+ζ85 | 2ζ8ζ3+ζ8 | 2ζ87ζ3+ζ87 | 1 | complex lifted from C8⋊S3 |
| ρ18 | 2 | -2 | 0 | -1 | 2i | -2i | 0 | 2 | 1 | 0 | 0 | 0 | 0 | -2 | -i | i | -1 | 0 | 0 | 2ζ87ζ3+ζ87 | 2ζ8ζ3+ζ8 | 2ζ85ζ3+ζ85 | 2ζ83ζ3+ζ83 | 1 | complex lifted from C8⋊S3 |
| ρ19 | 4 | 4 | 0 | 4 | 0 | 0 | 4 | -1 | 4 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | orthogonal lifted from F5 |
| ρ20 | 4 | 4 | 0 | 4 | 0 | 0 | -4 | -1 | 4 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | -1 | orthogonal lifted from C2×F5 |
| ρ21 | 4 | -4 | 0 | 4 | 0 | 0 | 0 | -1 | -4 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | -1 | √-5 | -√-5 | 0 | 0 | 0 | 0 | 1 | complex lifted from C4.F5 |
| ρ22 | 4 | -4 | 0 | 4 | 0 | 0 | 0 | -1 | -4 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | -1 | -√-5 | √-5 | 0 | 0 | 0 | 0 | 1 | complex lifted from C4.F5 |
| ρ23 | 8 | 8 | 0 | -4 | 0 | 0 | 0 | -2 | -4 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | orthogonal lifted from S3×F5 |
| ρ24 | 8 | -8 | 0 | -4 | 0 | 0 | 0 | -2 | 4 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | orthogonal faithful |
►On 120 points
Generators in S120
(1 54 26 5 50 30)(2 55 27 6 51 31)(3 56 28 7 52 32)(4 49 29 8 53 25)(9 64 24 13 60 20)(10 57 17 14 61 21)(11 58 18 15 62 22)(12 59 19 16 63 23)(33 117 41 37 113 45)(34 118 42 38 114 46)(35 119 43 39 115 47)(36 120 44 40 116 48)(65 96 109 69 92 105)(66 89 110 70 93 106)(67 90 111 71 94 107)(68 91 112 72 95 108)(73 100 88 77 104 84)(74 101 81 78 97 85)(75 102 82 79 98 86)(76 103 83 80 99 87)
(1 7 5 3)(2 4 6 8)(9 67 13 71)(10 72 14 68)(11 69 15 65)(12 66 16 70)(17 91 21 95)(18 96 22 92)(19 93 23 89)(20 90 24 94)(25 51 29 55)(26 56 30 52)(27 53 31 49)(28 50 32 54)(33 79 37 75)(34 76 38 80)(35 73 39 77)(36 78 40 74)(41 102 45 98)(42 99 46 103)(43 104 47 100)(44 101 48 97)(57 112 61 108)(58 109 62 105)(59 106 63 110)(60 111 64 107)(81 116 85 120)(82 113 86 117)(83 118 87 114)(84 115 88 119)
(1 24 88 113 96)(2 114 17 89 81)(3 90 115 82 18)(4 83 91 19 116)(5 20 84 117 92)(6 118 21 93 85)(7 94 119 86 22)(8 87 95 23 120)(9 73 41 105 50)(10 106 74 51 42)(11 52 107 43 75)(12 44 53 76 108)(13 77 45 109 54)(14 110 78 55 46)(15 56 111 47 79)(16 48 49 80 112)(25 103 68 59 40)(26 60 104 33 69)(27 34 61 70 97)(28 71 35 98 62)(29 99 72 63 36)(30 64 100 37 65)(31 38 57 66 101)(32 67 39 102 58)
(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)| (1,54,26,5,50,30)(2,55,27,6,51,31)(3,56,28,7,52,32)(4,49,29,8,53,25)(9,64,24,13,60,20)(10,57,17,14,61,21)(11,58,18,15,62,22)(12,59,19,16,63,23)(33,117,41,37,113,45)(34,118,42,38,114,46)(35,119,43,39,115,47)(36,120,44,40,116,48)(65,96,109,69,92,105)(66,89,110,70,93,106)(67,90,111,71,94,107)(68,91,112,72,95,108)(73,100,88,77,104,84)(74,101,81,78,97,85)(75,102,82,79,98,86)(76,103,83,80,99,87), (1,7,5,3)(2,4,6,8)(9,67,13,71)(10,72,14,68)(11,69,15,65)(12,66,16,70)(17,91,21,95)(18,96,22,92)(19,93,23,89)(20,90,24,94)(25,51,29,55)(26,56,30,52)(27,53,31,49)(28,50,32,54)(33,79,37,75)(34,76,38,80)(35,73,39,77)(36,78,40,74)(41,102,45,98)(42,99,46,103)(43,104,47,100)(44,101,48,97)(57,112,61,108)(58,109,62,105)(59,106,63,110)(60,111,64,107)(81,116,85,120)(82,113,86,117)(83,118,87,114)(84,115,88,119), (1,24,88,113,96)(2,114,17,89,81)(3,90,115,82,18)(4,83,91,19,116)(5,20,84,117,92)(6,118,21,93,85)(7,94,119,86,22)(8,87,95,23,120)(9,73,41,105,50)(10,106,74,51,42)(11,52,107,43,75)(12,44,53,76,108)(13,77,45,109,54)(14,110,78,55,46)(15,56,111,47,79)(16,48,49,80,112)(25,103,68,59,40)(26,60,104,33,69)(27,34,61,70,97)(28,71,35,98,62)(29,99,72,63,36)(30,64,100,37,65)(31,38,57,66,101)(32,67,39,102,58), (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( (1,54,26,5,50,30)(2,55,27,6,51,31)(3,56,28,7,52,32)(4,49,29,8,53,25)(9,64,24,13,60,20)(10,57,17,14,61,21)(11,58,18,15,62,22)(12,59,19,16,63,23)(33,117,41,37,113,45)(34,118,42,38,114,46)(35,119,43,39,115,47)(36,120,44,40,116,48)(65,96,109,69,92,105)(66,89,110,70,93,106)(67,90,111,71,94,107)(68,91,112,72,95,108)(73,100,88,77,104,84)(74,101,81,78,97,85)(75,102,82,79,98,86)(76,103,83,80,99,87), (1,7,5,3)(2,4,6,8)(9,67,13,71)(10,72,14,68)(11,69,15,65)(12,66,16,70)(17,91,21,95)(18,96,22,92)(19,93,23,89)(20,90,24,94)(25,51,29,55)(26,56,30,52)(27,53,31,49)(28,50,32,54)(33,79,37,75)(34,76,38,80)(35,73,39,77)(36,78,40,74)(41,102,45,98)(42,99,46,103)(43,104,47,100)(44,101,48,97)(57,112,61,108)(58,109,62,105)(59,106,63,110)(60,111,64,107)(81,116,85,120)(82,113,86,117)(83,118,87,114)(84,115,88,119), (1,24,88,113,96)(2,114,17,89,81)(3,90,115,82,18)(4,83,91,19,116)(5,20,84,117,92)(6,118,21,93,85)(7,94,119,86,22)(8,87,95,23,120)(9,73,41,105,50)(10,106,74,51,42)(11,52,107,43,75)(12,44,53,76,108)(13,77,45,109,54)(14,110,78,55,46)(15,56,111,47,79)(16,48,49,80,112)(25,103,68,59,40)(26,60,104,33,69)(27,34,61,70,97)(28,71,35,98,62)(29,99,72,63,36)(30,64,100,37,65)(31,38,57,66,101)(32,67,39,102,58), (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([[(1,54,26,5,50,30),(2,55,27,6,51,31),(3,56,28,7,52,32),(4,49,29,8,53,25),(9,64,24,13,60,20),(10,57,17,14,61,21),(11,58,18,15,62,22),(12,59,19,16,63,23),(33,117,41,37,113,45),(34,118,42,38,114,46),(35,119,43,39,115,47),(36,120,44,40,116,48),(65,96,109,69,92,105),(66,89,110,70,93,106),(67,90,111,71,94,107),(68,91,112,72,95,108),(73,100,88,77,104,84),(74,101,81,78,97,85),(75,102,82,79,98,86),(76,103,83,80,99,87)], [(1,7,5,3),(2,4,6,8),(9,67,13,71),(10,72,14,68),(11,69,15,65),(12,66,16,70),(17,91,21,95),(18,96,22,92),(19,93,23,89),(20,90,24,94),(25,51,29,55),(26,56,30,52),(27,53,31,49),(28,50,32,54),(33,79,37,75),(34,76,38,80),(35,73,39,77),(36,78,40,74),(41,102,45,98),(42,99,46,103),(43,104,47,100),(44,101,48,97),(57,112,61,108),(58,109,62,105),(59,106,63,110),(60,111,64,107),(81,116,85,120),(82,113,86,117),(83,118,87,114),(84,115,88,119)], [(1,24,88,113,96),(2,114,17,89,81),(3,90,115,82,18),(4,83,91,19,116),(5,20,84,117,92),(6,118,21,93,85),(7,94,119,86,22),(8,87,95,23,120),(9,73,41,105,50),(10,106,74,51,42),(11,52,107,43,75),(12,44,53,76,108),(13,77,45,109,54),(14,110,78,55,46),(15,56,111,47,79),(16,48,49,80,112),(25,103,68,59,40),(26,60,104,33,69),(27,34,61,70,97),(28,71,35,98,62),(29,99,72,63,36),(30,64,100,37,65),(31,38,57,66,101),(32,67,39,102,58)], [(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)]])
Dic3.F5 is a maximal subgroup of
S3×C4.F5 D60.C4 Dic6.F5 C5⋊C8⋊D6 C5⋊C8.D6 D15⋊C8⋊C2 D15⋊2M4(2)
Dic3.F5 is a maximal quotient of C30.M4(2) D30⋊C8 C30.4M4(2)
Matrix representation of Dic3.F5 ►in GL6(𝔽241)
| 240 | 1 | 0 | 0 | 0 | 0 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 0 | 34 | 34 |
| 0 | 0 | 207 | 224 | 207 | 0 |
| 0 | 0 | 0 | 207 | 224 | 207 |
| 0 | 0 | 34 | 34 | 0 | 17 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 240 | 240 | 240 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 200 | 159 | 208 | 192 |
| 0 | 0 | 49 | 33 | 82 | 41 |
| 0 | 0 | 49 | 8 | 208 | 16 |
| 0 | 0 | 200 | 8 | 233 | 41 |
G:=sub<GL(6,GF(241))| [240,240,0,0,0,0,1,0,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,17,207,0,34,0,0,0,224,207,34,0,0,34,207,224,0,0,0,34,0,207,17],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,1,0,0,0,0,240,0,1,0,0,0,240,0,0,1,0,0,240,0,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,200,49,49,200,0,0,159,33,8,8,0,0,208,82,208,233,0,0,192,41,16,41] >;
Dic3.F5 in GAP, Magma, Sage, TeX
{\rm Dic}_3.F_5 % in TeX
G:=Group("Dic3.F5"); // GroupNames label
G:=SmallGroup(240,101);
// by ID
G=gap.SmallGroup(240,101);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-5,24,121,55,50,490,3461,1745]);
// Polycyclic
G:=Group<a,b,c,d|a^6=c^5=1,b^2=d^4=a^3,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^3*b,d*c*d^-1=c^3>;
// generators/relations
Export
Subgroup lattice of Dic3.F5 in TeX
Character table of Dic3.F5 in TeX