Aliases: C14.A4, C7⋊SL2(𝔽3), Q8⋊(C7⋊C3), C2.(C7⋊A4), (C7×Q8)⋊2C3, SmallGroup(168,23)
Series: Derived ►Chief ►Lower central ►Upper central
| C7×Q8 — C14.A4 |
Generators and relations for C14.A4
G = < a,b,c,d | a14=d3=1, b2=c2=a7, ab=ba, ac=ca, dad-1=a11, cbc-1=a7b, dbd-1=a7bc, dcd-1=b >
Character table of C14.A4
| class | 1 | 2 | 3A | 3B | 4 | 6A | 6B | 7A | 7B | 14A | 14B | 28A | 28B | 28C | 28D | 28E | 28F | |
| size | 1 | 1 | 28 | 28 | 6 | 28 | 28 | 3 | 3 | 3 | 3 | 6 | 6 | 6 | 6 | 6 | 6 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | ζ32 | ζ3 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ3 | 1 | 1 | ζ3 | ζ32 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ4 | 2 | -2 | -1 | -1 | 0 | 1 | 1 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from SL2(𝔽3), Schur index 2 |
| ρ5 | 2 | -2 | ζ6 | ζ65 | 0 | ζ3 | ζ32 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from SL2(𝔽3) |
| ρ6 | 2 | -2 | ζ65 | ζ6 | 0 | ζ32 | ζ3 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from SL2(𝔽3) |
| ρ7 | 3 | 3 | 0 | 0 | -1 | 0 | 0 | 3 | 3 | 3 | 3 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from A4 |
| ρ8 | 3 | 3 | 0 | 0 | 3 | 0 | 0 | -1-√-7/2 | -1+√-7/2 | -1-√-7/2 | -1+√-7/2 | -1+√-7/2 | -1-√-7/2 | -1-√-7/2 | -1+√-7/2 | -1-√-7/2 | -1+√-7/2 | complex lifted from C7⋊C3 |
| ρ9 | 3 | 3 | 0 | 0 | 3 | 0 | 0 | -1+√-7/2 | -1-√-7/2 | -1+√-7/2 | -1-√-7/2 | -1-√-7/2 | -1+√-7/2 | -1+√-7/2 | -1-√-7/2 | -1+√-7/2 | -1-√-7/2 | complex lifted from C7⋊C3 |
| ρ10 | 3 | 3 | 0 | 0 | -1 | 0 | 0 | -1+√-7/2 | -1-√-7/2 | -1+√-7/2 | -1-√-7/2 | ζ76-ζ75-ζ73 | -ζ74-ζ72+ζ7 | -ζ74+ζ72-ζ7 | -ζ76-ζ75+ζ73 | ζ74-ζ72-ζ7 | -ζ76+ζ75-ζ73 | complex lifted from C7⋊A4 |
| ρ11 | 3 | 3 | 0 | 0 | -1 | 0 | 0 | -1+√-7/2 | -1-√-7/2 | -1+√-7/2 | -1-√-7/2 | -ζ76+ζ75-ζ73 | -ζ74+ζ72-ζ7 | ζ74-ζ72-ζ7 | ζ76-ζ75-ζ73 | -ζ74-ζ72+ζ7 | -ζ76-ζ75+ζ73 | complex lifted from C7⋊A4 |
| ρ12 | 3 | 3 | 0 | 0 | -1 | 0 | 0 | -1+√-7/2 | -1-√-7/2 | -1+√-7/2 | -1-√-7/2 | -ζ76-ζ75+ζ73 | ζ74-ζ72-ζ7 | -ζ74-ζ72+ζ7 | -ζ76+ζ75-ζ73 | -ζ74+ζ72-ζ7 | ζ76-ζ75-ζ73 | complex lifted from C7⋊A4 |
| ρ13 | 3 | 3 | 0 | 0 | -1 | 0 | 0 | -1-√-7/2 | -1+√-7/2 | -1-√-7/2 | -1+√-7/2 | ζ74-ζ72-ζ7 | -ζ76-ζ75+ζ73 | ζ76-ζ75-ζ73 | -ζ74+ζ72-ζ7 | -ζ76+ζ75-ζ73 | -ζ74-ζ72+ζ7 | complex lifted from C7⋊A4 |
| ρ14 | 3 | 3 | 0 | 0 | -1 | 0 | 0 | -1-√-7/2 | -1+√-7/2 | -1-√-7/2 | -1+√-7/2 | -ζ74-ζ72+ζ7 | ζ76-ζ75-ζ73 | -ζ76+ζ75-ζ73 | ζ74-ζ72-ζ7 | -ζ76-ζ75+ζ73 | -ζ74+ζ72-ζ7 | complex lifted from C7⋊A4 |
| ρ15 | 3 | 3 | 0 | 0 | -1 | 0 | 0 | -1-√-7/2 | -1+√-7/2 | -1-√-7/2 | -1+√-7/2 | -ζ74+ζ72-ζ7 | -ζ76+ζ75-ζ73 | -ζ76-ζ75+ζ73 | -ζ74-ζ72+ζ7 | ζ76-ζ75-ζ73 | ζ74-ζ72-ζ7 | complex lifted from C7⋊A4 |
| ρ16 | 6 | -6 | 0 | 0 | 0 | 0 | 0 | -1-√-7 | -1+√-7 | 1+√-7 | 1-√-7 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful, Schur index 2 |
| ρ17 | 6 | -6 | 0 | 0 | 0 | 0 | 0 | -1+√-7 | -1-√-7 | 1-√-7 | 1+√-7 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful, Schur index 2 |
►On 56 points
Generators in S56
(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)
(1 41 8 34)(2 42 9 35)(3 29 10 36)(4 30 11 37)(5 31 12 38)(6 32 13 39)(7 33 14 40)(15 51 22 44)(16 52 23 45)(17 53 24 46)(18 54 25 47)(19 55 26 48)(20 56 27 49)(21 43 28 50)
(1 17 8 24)(2 18 9 25)(3 19 10 26)(4 20 11 27)(5 21 12 28)(6 22 13 15)(7 23 14 16)(29 48 36 55)(30 49 37 56)(31 50 38 43)(32 51 39 44)(33 52 40 45)(34 53 41 46)(35 54 42 47)
(2 10 12)(3 5 9)(4 14 6)(7 13 11)(15 49 33)(16 44 30)(17 53 41)(18 48 38)(19 43 35)(20 52 32)(21 47 29)(22 56 40)(23 51 37)(24 46 34)(25 55 31)(26 50 42)(27 45 39)(28 54 36)
G:=sub<Sym(56)| (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), (1,41,8,34)(2,42,9,35)(3,29,10,36)(4,30,11,37)(5,31,12,38)(6,32,13,39)(7,33,14,40)(15,51,22,44)(16,52,23,45)(17,53,24,46)(18,54,25,47)(19,55,26,48)(20,56,27,49)(21,43,28,50), (1,17,8,24)(2,18,9,25)(3,19,10,26)(4,20,11,27)(5,21,12,28)(6,22,13,15)(7,23,14,16)(29,48,36,55)(30,49,37,56)(31,50,38,43)(32,51,39,44)(33,52,40,45)(34,53,41,46)(35,54,42,47), (2,10,12)(3,5,9)(4,14,6)(7,13,11)(15,49,33)(16,44,30)(17,53,41)(18,48,38)(19,43,35)(20,52,32)(21,47,29)(22,56,40)(23,51,37)(24,46,34)(25,55,31)(26,50,42)(27,45,39)(28,54,36)>;
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), (1,41,8,34)(2,42,9,35)(3,29,10,36)(4,30,11,37)(5,31,12,38)(6,32,13,39)(7,33,14,40)(15,51,22,44)(16,52,23,45)(17,53,24,46)(18,54,25,47)(19,55,26,48)(20,56,27,49)(21,43,28,50), (1,17,8,24)(2,18,9,25)(3,19,10,26)(4,20,11,27)(5,21,12,28)(6,22,13,15)(7,23,14,16)(29,48,36,55)(30,49,37,56)(31,50,38,43)(32,51,39,44)(33,52,40,45)(34,53,41,46)(35,54,42,47), (2,10,12)(3,5,9)(4,14,6)(7,13,11)(15,49,33)(16,44,30)(17,53,41)(18,48,38)(19,43,35)(20,52,32)(21,47,29)(22,56,40)(23,51,37)(24,46,34)(25,55,31)(26,50,42)(27,45,39)(28,54,36) );
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)], [(1,41,8,34),(2,42,9,35),(3,29,10,36),(4,30,11,37),(5,31,12,38),(6,32,13,39),(7,33,14,40),(15,51,22,44),(16,52,23,45),(17,53,24,46),(18,54,25,47),(19,55,26,48),(20,56,27,49),(21,43,28,50)], [(1,17,8,24),(2,18,9,25),(3,19,10,26),(4,20,11,27),(5,21,12,28),(6,22,13,15),(7,23,14,16),(29,48,36,55),(30,49,37,56),(31,50,38,43),(32,51,39,44),(33,52,40,45),(34,53,41,46),(35,54,42,47)], [(2,10,12),(3,5,9),(4,14,6),(7,13,11),(15,49,33),(16,44,30),(17,53,41),(18,48,38),(19,43,35),(20,52,32),(21,47,29),(22,56,40),(23,51,37),(24,46,34),(25,55,31),(26,50,42),(27,45,39),(28,54,36)]])
C14.A4 is a maximal subgroup of
Q8.F7 Q8⋊F7 C28.A4
Matrix representation of C14.A4 ►in GL5(𝔽337)
| 336 | 0 | 0 | 0 | 0 |
| 0 | 336 | 0 | 0 | 0 |
| 0 | 0 | 212 | 211 | 336 |
| 0 | 0 | 336 | 124 | 0 |
| 0 | 0 | 124 | 125 | 125 |
| 238 | 79 | 0 | 0 | 0 |
| 166 | 99 | 0 | 0 | 0 |
| 0 | 0 | 46 | 190 | 277 |
| 0 | 0 | 276 | 283 | 336 |
| 0 | 0 | 151 | 87 | 7 |
| 263 | 2 | 0 | 0 | 0 |
| 126 | 74 | 0 | 0 | 0 |
| 0 | 0 | 219 | 272 | 336 |
| 0 | 0 | 60 | 257 | 61 |
| 0 | 0 | 250 | 64 | 197 |
| 1 | 0 | 0 | 0 | 0 |
| 156 | 208 | 0 | 0 | 0 |
| 0 | 0 | 0 | 124 | 1 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 336 | 0 | 336 |
G:=sub<GL(5,GF(337))| [336,0,0,0,0,0,336,0,0,0,0,0,212,336,124,0,0,211,124,125,0,0,336,0,125],[238,166,0,0,0,79,99,0,0,0,0,0,46,276,151,0,0,190,283,87,0,0,277,336,7],[263,126,0,0,0,2,74,0,0,0,0,0,219,60,250,0,0,272,257,64,0,0,336,61,197],[1,156,0,0,0,0,208,0,0,0,0,0,0,0,336,0,0,124,1,0,0,0,1,0,336] >;
C14.A4 in GAP, Magma, Sage, TeX
C_{14}.A_4 % in TeX
G:=Group("C14.A4"); // GroupNames label
G:=SmallGroup(168,23);
// by ID
G=gap.SmallGroup(168,23);
# by ID
G:=PCGroup([5,-3,-2,2,-7,-2,61,286,137,457,222,483]);
// Polycyclic
G:=Group<a,b,c,d|a^14=d^3=1,b^2=c^2=a^7,a*b=b*a,a*c=c*a,d*a*d^-1=a^11,c*b*c^-1=a^7*b,d*b*d^-1=a^7*b*c,d*c*d^-1=b>;
// generators/relations
Export
Subgroup lattice of C14.A4 in TeX
Character table of C14.A4 in TeX