direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C2×C7⋊C12, C14⋊C12, C22.F7, Dic7⋊3C6, C7⋊2(C2×C12), (C2×C14).C6, (C2×Dic7)⋊C3, C2.2(C2×F7), C14.4(C2×C6), (C2×C7⋊C3)⋊C4, C7⋊C3⋊2(C2×C4), (C22×C7⋊C3).C2, (C2×C7⋊C3).4C22, SmallGroup(168,10)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C2×C7⋊C12 |
Generators and relations for C2×C7⋊C12
G = < a,b,c | a2=b7=c12=1, ab=ba, ac=ca, cbc-1=b5 >
Character table of C2×C7⋊C12
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 6F | 7 | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 14A | 14B | 14C | |
| size | 1 | 1 | 1 | 1 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 6 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 6 | 6 | 6 | |
| 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 | 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 | 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 | 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 | linear of order 2 |
| ρ5 | 1 | -1 | -1 | 1 | ζ32 | ζ3 | 1 | -1 | -1 | 1 | ζ65 | ζ65 | ζ3 | ζ6 | ζ6 | ζ32 | 1 | ζ3 | ζ6 | ζ65 | ζ65 | ζ32 | ζ32 | ζ3 | ζ6 | 1 | -1 | -1 | linear of order 6 |
| ρ6 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | 1 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | 1 | 1 | 1 | linear of order 3 |
| ρ7 | 1 | -1 | -1 | 1 | ζ32 | ζ3 | -1 | 1 | 1 | -1 | ζ65 | ζ65 | ζ3 | ζ6 | ζ6 | ζ32 | 1 | ζ65 | ζ32 | ζ3 | ζ3 | ζ6 | ζ6 | ζ65 | ζ32 | 1 | -1 | -1 | linear of order 6 |
| ρ8 | 1 | -1 | -1 | 1 | ζ3 | ζ32 | 1 | -1 | -1 | 1 | ζ6 | ζ6 | ζ32 | ζ65 | ζ65 | ζ3 | 1 | ζ32 | ζ65 | ζ6 | ζ6 | ζ3 | ζ3 | ζ32 | ζ65 | 1 | -1 | -1 | linear of order 6 |
| ρ9 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | -1 | -1 | -1 | -1 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | 1 | ζ6 | ζ65 | ζ6 | ζ6 | ζ65 | ζ65 | ζ6 | ζ65 | 1 | 1 | 1 | linear of order 6 |
| ρ10 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | 1 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | 1 | 1 | 1 | linear of order 3 |
| ρ11 | 1 | -1 | -1 | 1 | ζ3 | ζ32 | -1 | 1 | 1 | -1 | ζ6 | ζ6 | ζ32 | ζ65 | ζ65 | ζ3 | 1 | ζ6 | ζ3 | ζ32 | ζ32 | ζ65 | ζ65 | ζ6 | ζ3 | 1 | -1 | -1 | linear of order 6 |
| ρ12 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | -1 | -1 | -1 | -1 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | 1 | ζ65 | ζ6 | ζ65 | ζ65 | ζ6 | ζ6 | ζ65 | ζ6 | 1 | 1 | 1 | linear of order 6 |
| ρ13 | 1 | 1 | -1 | -1 | 1 | 1 | i | i | -i | -i | -1 | 1 | -1 | -1 | 1 | -1 | 1 | i | -i | i | -i | -i | i | -i | i | -1 | 1 | -1 | linear of order 4 |
| ρ14 | 1 | -1 | 1 | -1 | 1 | 1 | -i | i | -i | i | 1 | -1 | -1 | 1 | -1 | -1 | 1 | -i | -i | i | -i | i | -i | i | i | -1 | -1 | 1 | linear of order 4 |
| ρ15 | 1 | -1 | 1 | -1 | 1 | 1 | i | -i | i | -i | 1 | -1 | -1 | 1 | -1 | -1 | 1 | i | i | -i | i | -i | i | -i | -i | -1 | -1 | 1 | linear of order 4 |
| ρ16 | 1 | 1 | -1 | -1 | 1 | 1 | -i | -i | i | i | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -i | i | -i | i | i | -i | i | -i | -1 | 1 | -1 | linear of order 4 |
| ρ17 | 1 | -1 | 1 | -1 | ζ3 | ζ32 | i | -i | i | -i | ζ32 | ζ6 | ζ6 | ζ3 | ζ65 | ζ65 | 1 | ζ4ζ32 | ζ4ζ3 | ζ43ζ32 | ζ4ζ32 | ζ43ζ3 | ζ4ζ3 | ζ43ζ32 | ζ43ζ3 | -1 | -1 | 1 | linear of order 12 |
| ρ18 | 1 | 1 | -1 | -1 | ζ32 | ζ3 | -i | -i | i | i | ζ65 | ζ3 | ζ65 | ζ6 | ζ32 | ζ6 | 1 | ζ43ζ3 | ζ4ζ32 | ζ43ζ3 | ζ4ζ3 | ζ4ζ32 | ζ43ζ32 | ζ4ζ3 | ζ43ζ32 | -1 | 1 | -1 | linear of order 12 |
| ρ19 | 1 | -1 | 1 | -1 | ζ32 | ζ3 | -i | i | -i | i | ζ3 | ζ65 | ζ65 | ζ32 | ζ6 | ζ6 | 1 | ζ43ζ3 | ζ43ζ32 | ζ4ζ3 | ζ43ζ3 | ζ4ζ32 | ζ43ζ32 | ζ4ζ3 | ζ4ζ32 | -1 | -1 | 1 | linear of order 12 |
| ρ20 | 1 | 1 | -1 | -1 | ζ3 | ζ32 | i | i | -i | -i | ζ6 | ζ32 | ζ6 | ζ65 | ζ3 | ζ65 | 1 | ζ4ζ32 | ζ43ζ3 | ζ4ζ32 | ζ43ζ32 | ζ43ζ3 | ζ4ζ3 | ζ43ζ32 | ζ4ζ3 | -1 | 1 | -1 | linear of order 12 |
| ρ21 | 1 | 1 | -1 | -1 | ζ32 | ζ3 | i | i | -i | -i | ζ65 | ζ3 | ζ65 | ζ6 | ζ32 | ζ6 | 1 | ζ4ζ3 | ζ43ζ32 | ζ4ζ3 | ζ43ζ3 | ζ43ζ32 | ζ4ζ32 | ζ43ζ3 | ζ4ζ32 | -1 | 1 | -1 | linear of order 12 |
| ρ22 | 1 | -1 | 1 | -1 | ζ3 | ζ32 | -i | i | -i | i | ζ32 | ζ6 | ζ6 | ζ3 | ζ65 | ζ65 | 1 | ζ43ζ32 | ζ43ζ3 | ζ4ζ32 | ζ43ζ32 | ζ4ζ3 | ζ43ζ3 | ζ4ζ32 | ζ4ζ3 | -1 | -1 | 1 | linear of order 12 |
| ρ23 | 1 | -1 | 1 | -1 | ζ32 | ζ3 | i | -i | i | -i | ζ3 | ζ65 | ζ65 | ζ32 | ζ6 | ζ6 | 1 | ζ4ζ3 | ζ4ζ32 | ζ43ζ3 | ζ4ζ3 | ζ43ζ32 | ζ4ζ32 | ζ43ζ3 | ζ43ζ32 | -1 | -1 | 1 | linear of order 12 |
| ρ24 | 1 | 1 | -1 | -1 | ζ3 | ζ32 | -i | -i | i | i | ζ6 | ζ32 | ζ6 | ζ65 | ζ3 | ζ65 | 1 | ζ43ζ32 | ζ4ζ3 | ζ43ζ32 | ζ4ζ32 | ζ4ζ3 | ζ43ζ3 | ζ4ζ32 | ζ43ζ3 | -1 | 1 | -1 | linear of order 12 |
| ρ25 | 6 | -6 | -6 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | orthogonal lifted from C2×F7 |
| ρ26 | 6 | 6 | 6 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | orthogonal lifted from F7 |
| ρ27 | 6 | -6 | 6 | -6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | symplectic lifted from C7⋊C12, Schur index 2 |
| ρ28 | 6 | 6 | -6 | -6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | symplectic lifted from C7⋊C12, Schur index 2 |
►On 56 points
Generators in S56
(1 7)(2 8)(3 5)(4 6)(9 22)(10 23)(11 24)(12 25)(13 26)(14 27)(15 28)(16 29)(17 30)(18 31)(19 32)(20 21)(33 47)(34 48)(35 49)(36 50)(37 51)(38 52)(39 53)(40 54)(41 55)(42 56)(43 45)(44 46)
(1 28 24 53 32 45 49)(2 46 54 29 50 21 25)(3 22 30 47 26 51 55)(4 52 48 23 56 27 31)(5 9 17 33 13 37 41)(6 38 34 10 42 14 18)(7 15 11 39 19 43 35)(8 44 40 16 36 20 12)
(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)
G:=sub<Sym(56)| (1,7)(2,8)(3,5)(4,6)(9,22)(10,23)(11,24)(12,25)(13,26)(14,27)(15,28)(16,29)(17,30)(18,31)(19,32)(20,21)(33,47)(34,48)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56)(43,45)(44,46), (1,28,24,53,32,45,49)(2,46,54,29,50,21,25)(3,22,30,47,26,51,55)(4,52,48,23,56,27,31)(5,9,17,33,13,37,41)(6,38,34,10,42,14,18)(7,15,11,39,19,43,35)(8,44,40,16,36,20,12), (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)>;
G:=Group( (1,7)(2,8)(3,5)(4,6)(9,22)(10,23)(11,24)(12,25)(13,26)(14,27)(15,28)(16,29)(17,30)(18,31)(19,32)(20,21)(33,47)(34,48)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56)(43,45)(44,46), (1,28,24,53,32,45,49)(2,46,54,29,50,21,25)(3,22,30,47,26,51,55)(4,52,48,23,56,27,31)(5,9,17,33,13,37,41)(6,38,34,10,42,14,18)(7,15,11,39,19,43,35)(8,44,40,16,36,20,12), (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) );
G=PermutationGroup([[(1,7),(2,8),(3,5),(4,6),(9,22),(10,23),(11,24),(12,25),(13,26),(14,27),(15,28),(16,29),(17,30),(18,31),(19,32),(20,21),(33,47),(34,48),(35,49),(36,50),(37,51),(38,52),(39,53),(40,54),(41,55),(42,56),(43,45),(44,46)], [(1,28,24,53,32,45,49),(2,46,54,29,50,21,25),(3,22,30,47,26,51,55),(4,52,48,23,56,27,31),(5,9,17,33,13,37,41),(6,38,34,10,42,14,18),(7,15,11,39,19,43,35),(8,44,40,16,36,20,12)], [(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)]])
C2×C7⋊C12 is a maximal subgroup of
Dic7⋊C12 C28⋊C12 D14⋊C12 C23.2F7 C2×C4×F7 D4⋊2F7
C2×C7⋊C12 is a maximal quotient of C28.C12 C28⋊C12 C23.2F7
Matrix representation of C2×C7⋊C12 ►in GL7(𝔽337)
| 336 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 336 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 336 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 336 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 336 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 336 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 336 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 336 |
| 0 | 1 | 0 | 0 | 0 | 0 | 336 |
| 0 | 0 | 1 | 0 | 0 | 0 | 336 |
| 0 | 0 | 0 | 1 | 0 | 0 | 336 |
| 0 | 0 | 0 | 0 | 1 | 0 | 336 |
| 0 | 0 | 0 | 0 | 0 | 1 | 336 |
| 209 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 45 | 0 | 0 | 45 | 113 | 292 |
| 0 | 45 | 45 | 113 | 0 | 292 | 0 |
| 0 | 158 | 0 | 292 | 45 | 292 | 0 |
| 0 | 0 | 45 | 292 | 45 | 0 | 113 |
| 0 | 0 | 45 | 0 | 158 | 292 | 292 |
| 0 | 45 | 158 | 292 | 0 | 0 | 292 |
G:=sub<GL(7,GF(337))| [336,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,336],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,336,336,336,336,336,336],[209,0,0,0,0,0,0,0,45,45,158,0,0,45,0,0,45,0,45,45,158,0,0,113,292,292,0,292,0,45,0,45,45,158,0,0,113,292,292,0,292,0,0,292,0,0,113,292,292] >;
C2×C7⋊C12 in GAP, Magma, Sage, TeX
C_2\times C_7\rtimes C_{12} % in TeX
G:=Group("C2xC7:C12"); // GroupNames label
G:=SmallGroup(168,10);
// by ID
G=gap.SmallGroup(168,10);
# by ID
G:=PCGroup([5,-2,-2,-3,-2,-7,60,3604,614]);
// Polycyclic
G:=Group<a,b,c|a^2=b^7=c^12=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^5>;
// generators/relations
Export
Subgroup lattice of C2×C7⋊C12 in TeX
Character table of C2×C7⋊C12 in TeX