metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C7⋊2Q16, Q8.D7, C4.4D14, C14.10D4, C28.4C22, Dic14.2C2, C7⋊C8.C2, (C7×Q8).1C2, C2.7(C7⋊D4), SmallGroup(112,17)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7⋊Q16
G = < a,b,c | a7=b8=1, c2=b4, bab-1=a-1, ac=ca, cbc-1=b-1 >
Character table of C7⋊Q16
| class | 1 | 2 | 4A | 4B | 4C | 7A | 7B | 7C | 8A | 8B | 14A | 14B | 14C | 28A | 28B | 28C | 28D | 28E | 28F | 28G | 28H | 28I | |
| size | 1 | 1 | 2 | 4 | 28 | 2 | 2 | 2 | 14 | 14 | 2 | 2 | 2 | 4 | 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 | trivial |
| ρ2 | 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 | 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 | linear of order 2 |
| ρ5 | 2 | 2 | -2 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | -2 | orthogonal lifted from D4 |
| ρ6 | 2 | 2 | 2 | -2 | 0 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | 0 | 0 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ75+ζ72 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ74-ζ73 | -ζ75-ζ72 | -ζ76-ζ7 | ζ74+ζ73 | -ζ76-ζ7 | ζ76+ζ7 | orthogonal lifted from D14 |
| ρ7 | 2 | 2 | 2 | -2 | 0 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | 0 | 0 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ76+ζ7 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ75-ζ72 | -ζ76-ζ7 | -ζ74-ζ73 | ζ75+ζ72 | -ζ74-ζ73 | ζ74+ζ73 | orthogonal lifted from D14 |
| ρ8 | 2 | 2 | 2 | 2 | 0 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | 0 | 0 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ74+ζ73 | ζ74+ζ73 | ζ76+ζ7 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ75+ζ72 | ζ75+ζ72 | orthogonal lifted from D7 |
| ρ9 | 2 | 2 | 2 | 2 | 0 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | 0 | 0 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ76+ζ7 | ζ76+ζ7 | ζ75+ζ72 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ74+ζ73 | ζ74+ζ73 | orthogonal lifted from D7 |
| ρ10 | 2 | 2 | 2 | -2 | 0 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | 0 | 0 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ74+ζ73 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ76-ζ7 | -ζ74-ζ73 | -ζ75-ζ72 | ζ76+ζ7 | -ζ75-ζ72 | ζ75+ζ72 | orthogonal lifted from D14 |
| ρ11 | 2 | 2 | 2 | 2 | 0 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | 0 | 0 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ75+ζ72 | ζ75+ζ72 | ζ74+ζ73 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ76+ζ7 | ζ76+ζ7 | orthogonal lifted from D7 |
| ρ12 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | 2 | √2 | -√2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q16, Schur index 2 |
| ρ13 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | 2 | -√2 | √2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q16, Schur index 2 |
| ρ14 | 2 | 2 | -2 | 0 | 0 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | 0 | 0 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | -ζ75-ζ72 | -ζ75+ζ72 | -ζ74+ζ73 | ζ74-ζ73 | ζ75-ζ72 | ζ76-ζ7 | -ζ74-ζ73 | -ζ76+ζ7 | -ζ76-ζ7 | complex lifted from C7⋊D4 |
| ρ15 | 2 | 2 | -2 | 0 | 0 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | 0 | 0 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | -ζ74-ζ73 | ζ74-ζ73 | ζ76-ζ7 | -ζ76+ζ7 | -ζ74+ζ73 | ζ75-ζ72 | -ζ76-ζ7 | -ζ75+ζ72 | -ζ75-ζ72 | complex lifted from C7⋊D4 |
| ρ16 | 2 | 2 | -2 | 0 | 0 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | 0 | 0 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | -ζ75-ζ72 | ζ75-ζ72 | ζ74-ζ73 | -ζ74+ζ73 | -ζ75+ζ72 | -ζ76+ζ7 | -ζ74-ζ73 | ζ76-ζ7 | -ζ76-ζ7 | complex lifted from C7⋊D4 |
| ρ17 | 2 | 2 | -2 | 0 | 0 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | 0 | 0 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | -ζ76-ζ7 | -ζ76+ζ7 | ζ75-ζ72 | -ζ75+ζ72 | ζ76-ζ7 | -ζ74+ζ73 | -ζ75-ζ72 | ζ74-ζ73 | -ζ74-ζ73 | complex lifted from C7⋊D4 |
| ρ18 | 2 | 2 | -2 | 0 | 0 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | 0 | 0 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | -ζ74-ζ73 | -ζ74+ζ73 | -ζ76+ζ7 | ζ76-ζ7 | ζ74-ζ73 | -ζ75+ζ72 | -ζ76-ζ7 | ζ75-ζ72 | -ζ75-ζ72 | complex lifted from C7⋊D4 |
| ρ19 | 2 | 2 | -2 | 0 | 0 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | 0 | 0 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | -ζ76-ζ7 | ζ76-ζ7 | -ζ75+ζ72 | ζ75-ζ72 | -ζ76+ζ7 | ζ74-ζ73 | -ζ75-ζ72 | -ζ74+ζ73 | -ζ74-ζ73 | complex lifted from C7⋊D4 |
| ρ20 | 4 | -4 | 0 | 0 | 0 | 2ζ74+2ζ73 | 2ζ76+2ζ7 | 2ζ75+2ζ72 | 0 | 0 | -2ζ76-2ζ7 | -2ζ75-2ζ72 | -2ζ74-2ζ73 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ21 | 4 | -4 | 0 | 0 | 0 | 2ζ76+2ζ7 | 2ζ75+2ζ72 | 2ζ74+2ζ73 | 0 | 0 | -2ζ75-2ζ72 | -2ζ74-2ζ73 | -2ζ76-2ζ7 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ22 | 4 | -4 | 0 | 0 | 0 | 2ζ75+2ζ72 | 2ζ74+2ζ73 | 2ζ76+2ζ7 | 0 | 0 | -2ζ74-2ζ73 | -2ζ76-2ζ7 | -2ζ75-2ζ72 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
►Regular action on 112 points
Generators in S112
(1 102 105 12 23 61 36)(2 37 62 24 13 106 103)(3 104 107 14 17 63 38)(4 39 64 18 15 108 97)(5 98 109 16 19 57 40)(6 33 58 20 9 110 99)(7 100 111 10 21 59 34)(8 35 60 22 11 112 101)(25 74 43 52 70 91 88)(26 81 92 71 53 44 75)(27 76 45 54 72 93 82)(28 83 94 65 55 46 77)(29 78 47 56 66 95 84)(30 85 96 67 49 48 79)(31 80 41 50 68 89 86)(32 87 90 69 51 42 73)
(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)
(1 25 5 29)(2 32 6 28)(3 31 7 27)(4 30 8 26)(9 55 13 51)(10 54 14 50)(11 53 15 49)(12 52 16 56)(17 68 21 72)(18 67 22 71)(19 66 23 70)(20 65 24 69)(33 83 37 87)(34 82 38 86)(35 81 39 85)(36 88 40 84)(41 111 45 107)(42 110 46 106)(43 109 47 105)(44 108 48 112)(57 95 61 91)(58 94 62 90)(59 93 63 89)(60 92 64 96)(73 99 77 103)(74 98 78 102)(75 97 79 101)(76 104 80 100)
G:=sub<Sym(112)| (1,102,105,12,23,61,36)(2,37,62,24,13,106,103)(3,104,107,14,17,63,38)(4,39,64,18,15,108,97)(5,98,109,16,19,57,40)(6,33,58,20,9,110,99)(7,100,111,10,21,59,34)(8,35,60,22,11,112,101)(25,74,43,52,70,91,88)(26,81,92,71,53,44,75)(27,76,45,54,72,93,82)(28,83,94,65,55,46,77)(29,78,47,56,66,95,84)(30,85,96,67,49,48,79)(31,80,41,50,68,89,86)(32,87,90,69,51,42,73), (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), (1,25,5,29)(2,32,6,28)(3,31,7,27)(4,30,8,26)(9,55,13,51)(10,54,14,50)(11,53,15,49)(12,52,16,56)(17,68,21,72)(18,67,22,71)(19,66,23,70)(20,65,24,69)(33,83,37,87)(34,82,38,86)(35,81,39,85)(36,88,40,84)(41,111,45,107)(42,110,46,106)(43,109,47,105)(44,108,48,112)(57,95,61,91)(58,94,62,90)(59,93,63,89)(60,92,64,96)(73,99,77,103)(74,98,78,102)(75,97,79,101)(76,104,80,100)>;
G:=Group( (1,102,105,12,23,61,36)(2,37,62,24,13,106,103)(3,104,107,14,17,63,38)(4,39,64,18,15,108,97)(5,98,109,16,19,57,40)(6,33,58,20,9,110,99)(7,100,111,10,21,59,34)(8,35,60,22,11,112,101)(25,74,43,52,70,91,88)(26,81,92,71,53,44,75)(27,76,45,54,72,93,82)(28,83,94,65,55,46,77)(29,78,47,56,66,95,84)(30,85,96,67,49,48,79)(31,80,41,50,68,89,86)(32,87,90,69,51,42,73), (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), (1,25,5,29)(2,32,6,28)(3,31,7,27)(4,30,8,26)(9,55,13,51)(10,54,14,50)(11,53,15,49)(12,52,16,56)(17,68,21,72)(18,67,22,71)(19,66,23,70)(20,65,24,69)(33,83,37,87)(34,82,38,86)(35,81,39,85)(36,88,40,84)(41,111,45,107)(42,110,46,106)(43,109,47,105)(44,108,48,112)(57,95,61,91)(58,94,62,90)(59,93,63,89)(60,92,64,96)(73,99,77,103)(74,98,78,102)(75,97,79,101)(76,104,80,100) );
G=PermutationGroup([[(1,102,105,12,23,61,36),(2,37,62,24,13,106,103),(3,104,107,14,17,63,38),(4,39,64,18,15,108,97),(5,98,109,16,19,57,40),(6,33,58,20,9,110,99),(7,100,111,10,21,59,34),(8,35,60,22,11,112,101),(25,74,43,52,70,91,88),(26,81,92,71,53,44,75),(27,76,45,54,72,93,82),(28,83,94,65,55,46,77),(29,78,47,56,66,95,84),(30,85,96,67,49,48,79),(31,80,41,50,68,89,86),(32,87,90,69,51,42,73)], [(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)], [(1,25,5,29),(2,32,6,28),(3,31,7,27),(4,30,8,26),(9,55,13,51),(10,54,14,50),(11,53,15,49),(12,52,16,56),(17,68,21,72),(18,67,22,71),(19,66,23,70),(20,65,24,69),(33,83,37,87),(34,82,38,86),(35,81,39,85),(36,88,40,84),(41,111,45,107),(42,110,46,106),(43,109,47,105),(44,108,48,112),(57,95,61,91),(58,94,62,90),(59,93,63,89),(60,92,64,96),(73,99,77,103),(74,98,78,102),(75,97,79,101),(76,104,80,100)]])
C7⋊Q16 is a maximal subgroup of
SD16⋊D7 SD16⋊3D7 D7×Q16 Q16⋊D7 C28.C23 D4.8D14 D4.9D14 Q8.2F7 C21⋊Q16 C7⋊Dic12 C21⋊7Q16 Q8.D21
C7⋊Q16 is a maximal quotient of
C28.Q8 C14.Q16 Q8⋊Dic7 C21⋊Q16 C7⋊Dic12 C21⋊7Q16
Matrix representation of C7⋊Q16 ►in GL4(𝔽113) generated by
| 112 | 1 | 0 | 0 |
| 32 | 80 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 71 | 35 | 0 | 0 |
| 111 | 42 | 0 | 0 |
| 0 | 0 | 0 | 51 |
| 0 | 0 | 31 | 51 |
| 34 | 101 | 0 | 0 |
| 68 | 79 | 0 | 0 |
| 0 | 0 | 57 | 22 |
| 0 | 0 | 68 | 56 |
G:=sub<GL(4,GF(113))| [112,32,0,0,1,80,0,0,0,0,1,0,0,0,0,1],[71,111,0,0,35,42,0,0,0,0,0,31,0,0,51,51],[34,68,0,0,101,79,0,0,0,0,57,68,0,0,22,56] >;
C7⋊Q16 in GAP, Magma, Sage, TeX
C_7\rtimes Q_{16} % in TeX
G:=Group("C7:Q16"); // GroupNames label
G:=SmallGroup(112,17);
// by ID
G=gap.SmallGroup(112,17);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-7,40,61,46,182,97,42,2404]);
// Polycyclic
G:=Group<a,b,c|a^7=b^8=1,c^2=b^4,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export
Subgroup lattice of C7⋊Q16 in TeX
Character table of C7⋊Q16 in TeX