metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C15⋊8M4(2), Dic5.18D6, Dic5.3Dic3, C15⋊C8⋊6C2, (C2×C6).3F5, C22.(C3⋊F5), (C2×C30).2C4, C6.13(C2×F5), C30.13(C2×C4), C5⋊2(C4.Dic3), C3⋊2(C22.F5), (C2×Dic5).6S3, (C3×Dic5).5C4, C10.6(C2×Dic3), (C2×C10).2Dic3, (C6×Dic5).10C2, (C3×Dic5).24C22, C2.6(C2×C3⋊F5), SmallGroup(240,123)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C15⋊8M4(2)
G = < a,b,c | a15=b8=c2=1, bab-1=a2, ac=ca, cbc=b5 >
Character table of C15⋊8M4(2)
| class | 1 | 2A | 2B | 3 | 4A | 4B | 4C | 5 | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 12A | 12B | 12C | 12D | 15A | 15B | 30A | 30B | 30C | 30D | 30E | 30F | |
| size | 1 | 1 | 2 | 2 | 5 | 5 | 10 | 4 | 2 | 2 | 2 | 30 | 30 | 30 | 30 | 4 | 4 | 4 | 10 | 10 | 10 | 10 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | |
| 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 | 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 | -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 | 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 | -1 | -1 | 1 | -1 | -1 | 1 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -i | -i | i | i | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ6 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | -i | i | i | -i | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | linear of order 4 |
| ρ7 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | i | -i | -i | i | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | linear of order 4 |
| ρ8 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | i | i | -i | -i | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ9 | 2 | 2 | -2 | -1 | 2 | 2 | -2 | 2 | 1 | 1 | -1 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | 2 | -1 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ11 | 2 | 2 | 2 | -1 | -2 | -2 | -2 | 2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | symplectic lifted from Dic3, Schur index 2 |
| ρ12 | 2 | 2 | -2 | -1 | -2 | -2 | 2 | 2 | 1 | 1 | -1 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | symplectic lifted from Dic3, Schur index 2 |
| ρ13 | 2 | -2 | 0 | 2 | -2i | 2i | 0 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2i | -2i | 0 | 0 | 2 | 2 | 0 | 0 | -2 | 0 | 0 | -2 | complex lifted from M4(2) |
| ρ14 | 2 | -2 | 0 | 2 | 2i | -2i | 0 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2i | 2i | 0 | 0 | 2 | 2 | 0 | 0 | -2 | 0 | 0 | -2 | complex lifted from M4(2) |
| ρ15 | 2 | -2 | 0 | -1 | -2i | 2i | 0 | 2 | √-3 | -√-3 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -i | i | √3 | -√3 | -1 | -1 | √-3 | √-3 | 1 | -√-3 | -√-3 | 1 | complex lifted from C4.Dic3 |
| ρ16 | 2 | -2 | 0 | -1 | -2i | 2i | 0 | 2 | -√-3 | √-3 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -i | i | -√3 | √3 | -1 | -1 | -√-3 | -√-3 | 1 | √-3 | √-3 | 1 | complex lifted from C4.Dic3 |
| ρ17 | 2 | -2 | 0 | -1 | 2i | -2i | 0 | 2 | √-3 | -√-3 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | i | -i | -√3 | √3 | -1 | -1 | √-3 | √-3 | 1 | -√-3 | -√-3 | 1 | complex lifted from C4.Dic3 |
| ρ18 | 2 | -2 | 0 | -1 | 2i | -2i | 0 | 2 | -√-3 | √-3 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | i | -i | √3 | -√3 | -1 | -1 | -√-3 | -√-3 | 1 | √-3 | √-3 | 1 | complex lifted from C4.Dic3 |
| ρ19 | 4 | 4 | -4 | 4 | 0 | 0 | 0 | -1 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | orthogonal lifted from C2×F5 |
| ρ20 | 4 | 4 | 4 | 4 | 0 | 0 | 0 | -1 | 4 | 4 | 4 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ21 | 4 | -4 | 0 | 4 | 0 | 0 | 0 | -1 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | -√5 | √5 | 1 | 0 | 0 | 0 | 0 | -1 | -1 | -√5 | √5 | 1 | -√5 | √5 | 1 | symplectic lifted from C22.F5, Schur index 2 |
| ρ22 | 4 | -4 | 0 | 4 | 0 | 0 | 0 | -1 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | √5 | -√5 | 1 | 0 | 0 | 0 | 0 | -1 | -1 | √5 | -√5 | 1 | √5 | -√5 | 1 | symplectic lifted from C22.F5, Schur index 2 |
| ρ23 | 4 | 4 | 4 | -2 | 0 | 0 | 0 | -1 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 1+√-15/2 | 1-√-15/2 | 1-√-15/2 | 1+√-15/2 | 1+√-15/2 | 1+√-15/2 | 1-√-15/2 | 1-√-15/2 | complex lifted from C3⋊F5 |
| ρ24 | 4 | -4 | 0 | -2 | 0 | 0 | 0 | -1 | -2√-3 | 2√-3 | 2 | 0 | 0 | 0 | 0 | √5 | -√5 | 1 | 0 | 0 | 0 | 0 | 1+√-15/2 | 1-√-15/2 | -ζ32+ζ53+ζ52 | -ζ32+ζ54+ζ5 | -1-√-15/2 | -ζ3+ζ53+ζ52 | -ζ3+ζ54+ζ5 | -1+√-15/2 | complex faithful |
| ρ25 | 4 | -4 | 0 | -2 | 0 | 0 | 0 | -1 | 2√-3 | -2√-3 | 2 | 0 | 0 | 0 | 0 | -√5 | √5 | 1 | 0 | 0 | 0 | 0 | 1+√-15/2 | 1-√-15/2 | -ζ3+ζ54+ζ5 | -ζ3+ζ53+ζ52 | -1-√-15/2 | -ζ32+ζ54+ζ5 | -ζ32+ζ53+ζ52 | -1+√-15/2 | complex faithful |
| ρ26 | 4 | -4 | 0 | -2 | 0 | 0 | 0 | -1 | -2√-3 | 2√-3 | 2 | 0 | 0 | 0 | 0 | -√5 | √5 | 1 | 0 | 0 | 0 | 0 | 1-√-15/2 | 1+√-15/2 | -ζ32+ζ54+ζ5 | -ζ32+ζ53+ζ52 | -1+√-15/2 | -ζ3+ζ54+ζ5 | -ζ3+ζ53+ζ52 | -1-√-15/2 | complex faithful |
| ρ27 | 4 | -4 | 0 | -2 | 0 | 0 | 0 | -1 | 2√-3 | -2√-3 | 2 | 0 | 0 | 0 | 0 | √5 | -√5 | 1 | 0 | 0 | 0 | 0 | 1-√-15/2 | 1+√-15/2 | -ζ3+ζ53+ζ52 | -ζ3+ζ54+ζ5 | -1+√-15/2 | -ζ32+ζ53+ζ52 | -ζ32+ζ54+ζ5 | -1-√-15/2 | complex faithful |
| ρ28 | 4 | 4 | 4 | -2 | 0 | 0 | 0 | -1 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 1-√-15/2 | 1+√-15/2 | 1+√-15/2 | 1-√-15/2 | 1-√-15/2 | 1-√-15/2 | 1+√-15/2 | 1+√-15/2 | complex lifted from C3⋊F5 |
| ρ29 | 4 | 4 | -4 | -2 | 0 | 0 | 0 | -1 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | 0 | 0 | 0 | 0 | 1+√-15/2 | 1-√-15/2 | -1+√-15/2 | -1-√-15/2 | 1+√-15/2 | -1-√-15/2 | -1+√-15/2 | 1-√-15/2 | complex lifted from C2×C3⋊F5 |
| ρ30 | 4 | 4 | -4 | -2 | 0 | 0 | 0 | -1 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | 0 | 0 | 0 | 0 | 1-√-15/2 | 1+√-15/2 | -1-√-15/2 | -1+√-15/2 | 1-√-15/2 | -1+√-15/2 | -1-√-15/2 | 1+√-15/2 | complex lifted from C2×C3⋊F5 |
►On 120 points
Generators in S120
(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)
(1 91 31 79 20 106 59 75)(2 99 35 81 21 114 48 62)(3 92 39 83 22 107 52 64)(4 100 43 85 23 115 56 66)(5 93 32 87 24 108 60 68)(6 101 36 89 25 116 49 70)(7 94 40 76 26 109 53 72)(8 102 44 78 27 117 57 74)(9 95 33 80 28 110 46 61)(10 103 37 82 29 118 50 63)(11 96 41 84 30 111 54 65)(12 104 45 86 16 119 58 67)(13 97 34 88 17 112 47 69)(14 105 38 90 18 120 51 71)(15 98 42 77 19 113 55 73)
(61 80)(62 81)(63 82)(64 83)(65 84)(66 85)(67 86)(68 87)(69 88)(70 89)(71 90)(72 76)(73 77)(74 78)(75 79)(91 106)(92 107)(93 108)(94 109)(95 110)(96 111)(97 112)(98 113)(99 114)(100 115)(101 116)(102 117)(103 118)(104 119)(105 120)
G:=sub<Sym(120)| (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), (1,91,31,79,20,106,59,75)(2,99,35,81,21,114,48,62)(3,92,39,83,22,107,52,64)(4,100,43,85,23,115,56,66)(5,93,32,87,24,108,60,68)(6,101,36,89,25,116,49,70)(7,94,40,76,26,109,53,72)(8,102,44,78,27,117,57,74)(9,95,33,80,28,110,46,61)(10,103,37,82,29,118,50,63)(11,96,41,84,30,111,54,65)(12,104,45,86,16,119,58,67)(13,97,34,88,17,112,47,69)(14,105,38,90,18,120,51,71)(15,98,42,77,19,113,55,73), (61,80)(62,81)(63,82)(64,83)(65,84)(66,85)(67,86)(68,87)(69,88)(70,89)(71,90)(72,76)(73,77)(74,78)(75,79)(91,106)(92,107)(93,108)(94,109)(95,110)(96,111)(97,112)(98,113)(99,114)(100,115)(101,116)(102,117)(103,118)(104,119)(105,120)>;
G:=Group( (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), (1,91,31,79,20,106,59,75)(2,99,35,81,21,114,48,62)(3,92,39,83,22,107,52,64)(4,100,43,85,23,115,56,66)(5,93,32,87,24,108,60,68)(6,101,36,89,25,116,49,70)(7,94,40,76,26,109,53,72)(8,102,44,78,27,117,57,74)(9,95,33,80,28,110,46,61)(10,103,37,82,29,118,50,63)(11,96,41,84,30,111,54,65)(12,104,45,86,16,119,58,67)(13,97,34,88,17,112,47,69)(14,105,38,90,18,120,51,71)(15,98,42,77,19,113,55,73), (61,80)(62,81)(63,82)(64,83)(65,84)(66,85)(67,86)(68,87)(69,88)(70,89)(71,90)(72,76)(73,77)(74,78)(75,79)(91,106)(92,107)(93,108)(94,109)(95,110)(96,111)(97,112)(98,113)(99,114)(100,115)(101,116)(102,117)(103,118)(104,119)(105,120) );
G=PermutationGroup([[(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)], [(1,91,31,79,20,106,59,75),(2,99,35,81,21,114,48,62),(3,92,39,83,22,107,52,64),(4,100,43,85,23,115,56,66),(5,93,32,87,24,108,60,68),(6,101,36,89,25,116,49,70),(7,94,40,76,26,109,53,72),(8,102,44,78,27,117,57,74),(9,95,33,80,28,110,46,61),(10,103,37,82,29,118,50,63),(11,96,41,84,30,111,54,65),(12,104,45,86,16,119,58,67),(13,97,34,88,17,112,47,69),(14,105,38,90,18,120,51,71),(15,98,42,77,19,113,55,73)], [(61,80),(62,81),(63,82),(64,83),(65,84),(66,85),(67,86),(68,87),(69,88),(70,89),(71,90),(72,76),(73,77),(74,78),(75,79),(91,106),(92,107),(93,108),(94,109),(95,110),(96,111),(97,112),(98,113),(99,114),(100,115),(101,116),(102,117),(103,118),(104,119),(105,120)]])
C15⋊8M4(2) is a maximal subgroup of
Dic5.D12 Dic5.4D12 (C2×C60).C4 C5⋊(C12.D4) C5⋊C8.D6 S3×C22.F5 D15⋊2M4(2) C60.59(C2×C4) Dic10.Dic3
C15⋊8M4(2) is a maximal quotient of
C30.11C42 Dic5.13D12 C30.22M4(2)
Matrix representation of C15⋊8M4(2) ►in GL6(𝔽241)
| 240 | 1 | 0 | 0 | 0 | 0 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 51 | 189 | 0 | 0 |
| 0 | 0 | 51 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 1 |
| 0 | 0 | 0 | 0 | 50 | 190 |
| 8 | 22 | 0 | 0 | 0 | 0 |
| 30 | 233 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 237 | 159 | 0 | 0 |
| 0 | 0 | 203 | 4 | 0 | 0 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 |
G:=sub<GL(6,GF(241))| [240,240,0,0,0,0,1,0,0,0,0,0,0,0,51,51,0,0,0,0,189,0,0,0,0,0,0,0,240,50,0,0,0,0,1,190],[8,30,0,0,0,0,22,233,0,0,0,0,0,0,0,0,237,203,0,0,0,0,159,4,0,0,1,0,0,0,0,0,0,1,0,0],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,0,240] >;
C15⋊8M4(2) in GAP, Magma, Sage, TeX
C_{15}\rtimes_8M_4(2) % in TeX
G:=Group("C15:8M4(2)"); // GroupNames label
G:=SmallGroup(240,123);
// by ID
G=gap.SmallGroup(240,123);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-5,24,121,50,964,5189,1745]);
// Polycyclic
G:=Group<a,b,c|a^15=b^8=c^2=1,b*a*b^-1=a^2,a*c=c*a,c*b*c=b^5>;
// generators/relations
Export
Subgroup lattice of C15⋊8M4(2) in TeX
Character table of C15⋊8M4(2) in TeX