metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C15⋊Q8, C5⋊1Dic6, C6.7D10, C10.7D6, C3⋊1Dic10, Dic3.D5, C30.7C22, Dic5.1S3, Dic15.2C2, C2.7(S3×D5), (C5×Dic3).1C2, (C3×Dic5).1C2, SmallGroup(120,14)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C15⋊Q8
G = < a,b,c | a15=b4=1, c2=b2, bab-1=a11, cac-1=a4, cbc-1=b-1 >
Character table of C15⋊Q8
| class | 1 | 2 | 3 | 4A | 4B | 4C | 5A | 5B | 6 | 10A | 10B | 12A | 12B | 15A | 15B | 20A | 20B | 20C | 20D | 30A | 30B | |
| size | 1 | 1 | 2 | 6 | 10 | 30 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 4 | 4 | 6 | 6 | 6 | 6 | 4 | 4 | |
| ρ1 | 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 | 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 | 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 | linear of order 2 |
| ρ5 | 2 | 2 | -1 | 0 | -2 | 0 | 2 | 2 | -1 | 2 | 2 | 1 | 1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | orthogonal lifted from D6 |
| ρ6 | 2 | 2 | 2 | 2 | 0 | 0 | -1+√5/2 | -1-√5/2 | 2 | -1+√5/2 | -1-√5/2 | 0 | 0 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | orthogonal lifted from D5 |
| ρ7 | 2 | 2 | -1 | 0 | 2 | 0 | 2 | 2 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | orthogonal lifted from S3 |
| ρ8 | 2 | 2 | 2 | -2 | 0 | 0 | -1-√5/2 | -1+√5/2 | 2 | -1-√5/2 | -1+√5/2 | 0 | 0 | -1+√5/2 | -1-√5/2 | 1-√5/2 | 1+√5/2 | 1+√5/2 | 1-√5/2 | -1+√5/2 | -1-√5/2 | orthogonal lifted from D10 |
| ρ9 | 2 | 2 | 2 | -2 | 0 | 0 | -1+√5/2 | -1-√5/2 | 2 | -1+√5/2 | -1-√5/2 | 0 | 0 | -1-√5/2 | -1+√5/2 | 1+√5/2 | 1-√5/2 | 1-√5/2 | 1+√5/2 | -1-√5/2 | -1+√5/2 | orthogonal lifted from D10 |
| ρ10 | 2 | 2 | 2 | 2 | 0 | 0 | -1-√5/2 | -1+√5/2 | 2 | -1-√5/2 | -1+√5/2 | 0 | 0 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | orthogonal lifted from D5 |
| ρ11 | 2 | -2 | 2 | 0 | 0 | 0 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | symplectic lifted from Q8, Schur index 2 |
| ρ12 | 2 | -2 | -1 | 0 | 0 | 0 | 2 | 2 | 1 | -2 | -2 | -√3 | √3 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | symplectic lifted from Dic6, Schur index 2 |
| ρ13 | 2 | -2 | -1 | 0 | 0 | 0 | 2 | 2 | 1 | -2 | -2 | √3 | -√3 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | symplectic lifted from Dic6, Schur index 2 |
| ρ14 | 2 | -2 | 2 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | -2 | 1-√5/2 | 1+√5/2 | 0 | 0 | -1-√5/2 | -1+√5/2 | ζ4ζ53-ζ4ζ52 | -ζ43ζ54+ζ43ζ5 | ζ43ζ54-ζ43ζ5 | -ζ4ζ53+ζ4ζ52 | 1+√5/2 | 1-√5/2 | symplectic lifted from Dic10, Schur index 2 |
| ρ15 | 2 | -2 | 2 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | -2 | 1-√5/2 | 1+√5/2 | 0 | 0 | -1-√5/2 | -1+√5/2 | -ζ4ζ53+ζ4ζ52 | ζ43ζ54-ζ43ζ5 | -ζ43ζ54+ζ43ζ5 | ζ4ζ53-ζ4ζ52 | 1+√5/2 | 1-√5/2 | symplectic lifted from Dic10, Schur index 2 |
| ρ16 | 2 | -2 | 2 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | -2 | 1+√5/2 | 1-√5/2 | 0 | 0 | -1+√5/2 | -1-√5/2 | -ζ43ζ54+ζ43ζ5 | -ζ4ζ53+ζ4ζ52 | ζ4ζ53-ζ4ζ52 | ζ43ζ54-ζ43ζ5 | 1-√5/2 | 1+√5/2 | symplectic lifted from Dic10, Schur index 2 |
| ρ17 | 2 | -2 | 2 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | -2 | 1+√5/2 | 1-√5/2 | 0 | 0 | -1+√5/2 | -1-√5/2 | ζ43ζ54-ζ43ζ5 | ζ4ζ53-ζ4ζ52 | -ζ4ζ53+ζ4ζ52 | -ζ43ζ54+ζ43ζ5 | 1-√5/2 | 1+√5/2 | symplectic lifted from Dic10, Schur index 2 |
| ρ18 | 4 | 4 | -2 | 0 | 0 | 0 | -1-√5 | -1+√5 | -2 | -1-√5 | -1+√5 | 0 | 0 | 1-√5/2 | 1+√5/2 | 0 | 0 | 0 | 0 | 1-√5/2 | 1+√5/2 | orthogonal lifted from S3×D5 |
| ρ19 | 4 | 4 | -2 | 0 | 0 | 0 | -1+√5 | -1-√5 | -2 | -1+√5 | -1-√5 | 0 | 0 | 1+√5/2 | 1-√5/2 | 0 | 0 | 0 | 0 | 1+√5/2 | 1-√5/2 | orthogonal lifted from S3×D5 |
| ρ20 | 4 | -4 | -2 | 0 | 0 | 0 | -1-√5 | -1+√5 | 2 | 1+√5 | 1-√5 | 0 | 0 | 1-√5/2 | 1+√5/2 | 0 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | symplectic faithful, Schur index 2 |
| ρ21 | 4 | -4 | -2 | 0 | 0 | 0 | -1+√5 | -1-√5 | 2 | 1-√5 | 1+√5 | 0 | 0 | 1+√5/2 | 1-√5/2 | 0 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | symplectic faithful, Schur index 2 |
►Regular action 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 46 20 39)(2 57 21 35)(3 53 22 31)(4 49 23 42)(5 60 24 38)(6 56 25 34)(7 52 26 45)(8 48 27 41)(9 59 28 37)(10 55 29 33)(11 51 30 44)(12 47 16 40)(13 58 17 36)(14 54 18 32)(15 50 19 43)(61 96 83 110)(62 92 84 106)(63 103 85 117)(64 99 86 113)(65 95 87 109)(66 91 88 120)(67 102 89 116)(68 98 90 112)(69 94 76 108)(70 105 77 119)(71 101 78 115)(72 97 79 111)(73 93 80 107)(74 104 81 118)(75 100 82 114)
(1 83 20 61)(2 87 21 65)(3 76 22 69)(4 80 23 73)(5 84 24 62)(6 88 25 66)(7 77 26 70)(8 81 27 74)(9 85 28 63)(10 89 29 67)(11 78 30 71)(12 82 16 75)(13 86 17 64)(14 90 18 68)(15 79 19 72)(31 108 53 94)(32 112 54 98)(33 116 55 102)(34 120 56 91)(35 109 57 95)(36 113 58 99)(37 117 59 103)(38 106 60 92)(39 110 46 96)(40 114 47 100)(41 118 48 104)(42 107 49 93)(43 111 50 97)(44 115 51 101)(45 119 52 105)
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), (1,46,20,39)(2,57,21,35)(3,53,22,31)(4,49,23,42)(5,60,24,38)(6,56,25,34)(7,52,26,45)(8,48,27,41)(9,59,28,37)(10,55,29,33)(11,51,30,44)(12,47,16,40)(13,58,17,36)(14,54,18,32)(15,50,19,43)(61,96,83,110)(62,92,84,106)(63,103,85,117)(64,99,86,113)(65,95,87,109)(66,91,88,120)(67,102,89,116)(68,98,90,112)(69,94,76,108)(70,105,77,119)(71,101,78,115)(72,97,79,111)(73,93,80,107)(74,104,81,118)(75,100,82,114), (1,83,20,61)(2,87,21,65)(3,76,22,69)(4,80,23,73)(5,84,24,62)(6,88,25,66)(7,77,26,70)(8,81,27,74)(9,85,28,63)(10,89,29,67)(11,78,30,71)(12,82,16,75)(13,86,17,64)(14,90,18,68)(15,79,19,72)(31,108,53,94)(32,112,54,98)(33,116,55,102)(34,120,56,91)(35,109,57,95)(36,113,58,99)(37,117,59,103)(38,106,60,92)(39,110,46,96)(40,114,47,100)(41,118,48,104)(42,107,49,93)(43,111,50,97)(44,115,51,101)(45,119,52,105)>;
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), (1,46,20,39)(2,57,21,35)(3,53,22,31)(4,49,23,42)(5,60,24,38)(6,56,25,34)(7,52,26,45)(8,48,27,41)(9,59,28,37)(10,55,29,33)(11,51,30,44)(12,47,16,40)(13,58,17,36)(14,54,18,32)(15,50,19,43)(61,96,83,110)(62,92,84,106)(63,103,85,117)(64,99,86,113)(65,95,87,109)(66,91,88,120)(67,102,89,116)(68,98,90,112)(69,94,76,108)(70,105,77,119)(71,101,78,115)(72,97,79,111)(73,93,80,107)(74,104,81,118)(75,100,82,114), (1,83,20,61)(2,87,21,65)(3,76,22,69)(4,80,23,73)(5,84,24,62)(6,88,25,66)(7,77,26,70)(8,81,27,74)(9,85,28,63)(10,89,29,67)(11,78,30,71)(12,82,16,75)(13,86,17,64)(14,90,18,68)(15,79,19,72)(31,108,53,94)(32,112,54,98)(33,116,55,102)(34,120,56,91)(35,109,57,95)(36,113,58,99)(37,117,59,103)(38,106,60,92)(39,110,46,96)(40,114,47,100)(41,118,48,104)(42,107,49,93)(43,111,50,97)(44,115,51,101)(45,119,52,105) );
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)], [(1,46,20,39),(2,57,21,35),(3,53,22,31),(4,49,23,42),(5,60,24,38),(6,56,25,34),(7,52,26,45),(8,48,27,41),(9,59,28,37),(10,55,29,33),(11,51,30,44),(12,47,16,40),(13,58,17,36),(14,54,18,32),(15,50,19,43),(61,96,83,110),(62,92,84,106),(63,103,85,117),(64,99,86,113),(65,95,87,109),(66,91,88,120),(67,102,89,116),(68,98,90,112),(69,94,76,108),(70,105,77,119),(71,101,78,115),(72,97,79,111),(73,93,80,107),(74,104,81,118),(75,100,82,114)], [(1,83,20,61),(2,87,21,65),(3,76,22,69),(4,80,23,73),(5,84,24,62),(6,88,25,66),(7,77,26,70),(8,81,27,74),(9,85,28,63),(10,89,29,67),(11,78,30,71),(12,82,16,75),(13,86,17,64),(14,90,18,68),(15,79,19,72),(31,108,53,94),(32,112,54,98),(33,116,55,102),(34,120,56,91),(35,109,57,95),(36,113,58,99),(37,117,59,103),(38,106,60,92),(39,110,46,96),(40,114,47,100),(41,118,48,104),(42,107,49,93),(43,111,50,97),(44,115,51,101),(45,119,52,105)]])
C15⋊Q8 is a maximal subgroup of
D5×Dic6 S3×Dic10 D15⋊Q8 D6.D10 Dic5.D6 C30.C23 Dic3.D10 C45⋊Q8 C15⋊Dic6 C3⋊Dic30 C32⋊3Dic10 CSU2(𝔽3)⋊D5 A4⋊Dic10
C15⋊Q8 is a maximal quotient of
C30.Q8 Dic15⋊5C4 C6.Dic10 C45⋊Q8 C15⋊Dic6 C3⋊Dic30 C32⋊3Dic10 A4⋊Dic10
Matrix representation of C15⋊Q8 ►in GL4(𝔽61) generated by
| 60 | 17 | 0 | 0 |
| 44 | 44 | 0 | 0 |
| 0 | 0 | 60 | 1 |
| 0 | 0 | 60 | 0 |
| 29 | 54 | 0 | 0 |
| 7 | 32 | 0 | 0 |
| 0 | 0 | 11 | 50 |
| 0 | 0 | 0 | 50 |
| 20 | 58 | 0 | 0 |
| 32 | 41 | 0 | 0 |
| 0 | 0 | 23 | 15 |
| 0 | 0 | 46 | 38 |
G:=sub<GL(4,GF(61))| [60,44,0,0,17,44,0,0,0,0,60,60,0,0,1,0],[29,7,0,0,54,32,0,0,0,0,11,0,0,0,50,50],[20,32,0,0,58,41,0,0,0,0,23,46,0,0,15,38] >;
C15⋊Q8 in GAP, Magma, Sage, TeX
C_{15}\rtimes Q_8 % in TeX
G:=Group("C15:Q8"); // GroupNames label
G:=SmallGroup(120,14);
// by ID
G=gap.SmallGroup(120,14);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-5,20,61,26,168,2404]);
// Polycyclic
G:=Group<a,b,c|a^15=b^4=1,c^2=b^2,b*a*b^-1=a^11,c*a*c^-1=a^4,c*b*c^-1=b^-1>;
// generators/relations
Export
Subgroup lattice of C15⋊Q8 in TeX
Character table of C15⋊Q8 in TeX