direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: C3×C4⋊C4, C4⋊C12, C12⋊3C4, C6.3Q8, C6.13D4, C2.(C3×Q8), (C2×C4).1C6, C2.2(C3×D4), C6.11(C2×C4), C2.2(C2×C12), (C2×C12).7C2, C22.3(C2×C6), (C2×C6).14C22, SmallGroup(48,22)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C4⋊C4
G = < a,b,c | a3=b4=c4=1, ab=ba, ac=ca, cbc-1=b-1 >
Character table of C3×C4⋊C4
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 6D | 6E | 6F | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | 12J | 12K | 12L | |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |
| 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 | ζ3 | ζ32 | 1 | -1 | -1 | -1 | 1 | -1 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ32 | ζ65 | ζ65 | ζ65 | ζ6 | ζ6 | ζ6 | ζ6 | ζ3 | ζ3 | ζ32 | ζ65 | linear of order 6 |
| ρ6 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | -1 | 1 | -1 | -1 | -1 | 1 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ6 | ζ3 | ζ65 | ζ65 | ζ6 | ζ6 | ζ32 | ζ32 | ζ65 | ζ65 | ζ6 | ζ3 | linear of order 6 |
| ρ7 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | -1 | 1 | -1 | -1 | -1 | 1 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ65 | ζ32 | ζ6 | ζ6 | ζ65 | ζ65 | ζ3 | ζ3 | ζ6 | ζ6 | ζ65 | ζ32 | linear of order 6 |
| ρ8 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | linear of order 3 |
| ρ9 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | -1 | -1 | -1 | 1 | -1 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ3 | ζ6 | ζ6 | ζ6 | ζ65 | ζ65 | ζ65 | ζ65 | ζ32 | ζ32 | ζ3 | ζ6 | linear of order 6 |
| ρ10 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | linear of order 3 |
| ρ11 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | -1 | -1 | 1 | 1 | -1 | -1 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ65 | ζ6 | ζ32 | ζ32 | ζ3 | ζ3 | ζ65 | ζ65 | ζ6 | ζ6 | ζ65 | ζ6 | linear of order 6 |
| ρ12 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | -1 | -1 | 1 | 1 | -1 | -1 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ6 | ζ65 | ζ3 | ζ3 | ζ32 | ζ32 | ζ6 | ζ6 | ζ65 | ζ65 | ζ6 | ζ65 | linear of order 6 |
| ρ13 | 1 | -1 | 1 | -1 | 1 | 1 | -i | -1 | i | -i | i | 1 | 1 | -1 | -1 | -1 | 1 | -1 | -i | -1 | i | -i | i | -i | 1 | -1 | i | -i | i | 1 | linear of order 4 |
| ρ14 | 1 | -1 | 1 | -1 | 1 | 1 | i | 1 | i | -i | -i | -1 | 1 | -1 | -1 | -1 | 1 | -1 | i | 1 | i | -i | i | -i | -1 | 1 | -i | i | -i | -1 | linear of order 4 |
| ρ15 | 1 | -1 | 1 | -1 | 1 | 1 | -i | 1 | -i | i | i | -1 | 1 | -1 | -1 | -1 | 1 | -1 | -i | 1 | -i | i | -i | i | -1 | 1 | i | -i | i | -1 | linear of order 4 |
| ρ16 | 1 | -1 | 1 | -1 | 1 | 1 | i | -1 | -i | i | -i | 1 | 1 | -1 | -1 | -1 | 1 | -1 | i | -1 | -i | i | -i | i | 1 | -1 | -i | i | -i | 1 | linear of order 4 |
| ρ17 | 1 | -1 | 1 | -1 | ζ3 | ζ32 | -i | -1 | i | -i | i | 1 | ζ3 | ζ65 | ζ65 | ζ6 | ζ32 | ζ6 | ζ43ζ32 | ζ65 | ζ4ζ3 | ζ43ζ3 | ζ4ζ32 | ζ43ζ32 | ζ32 | ζ6 | ζ4ζ3 | ζ43ζ3 | ζ4ζ32 | ζ3 | linear of order 12 |
| ρ18 | 1 | -1 | 1 | -1 | ζ32 | ζ3 | i | -1 | -i | i | -i | 1 | ζ32 | ζ6 | ζ6 | ζ65 | ζ3 | ζ65 | ζ4ζ3 | ζ6 | ζ43ζ32 | ζ4ζ32 | ζ43ζ3 | ζ4ζ3 | ζ3 | ζ65 | ζ43ζ32 | ζ4ζ32 | ζ43ζ3 | ζ32 | linear of order 12 |
| ρ19 | 1 | -1 | 1 | -1 | ζ3 | ζ32 | i | 1 | i | -i | -i | -1 | ζ3 | ζ65 | ζ65 | ζ6 | ζ32 | ζ6 | ζ4ζ32 | ζ3 | ζ4ζ3 | ζ43ζ3 | ζ4ζ32 | ζ43ζ32 | ζ6 | ζ32 | ζ43ζ3 | ζ4ζ3 | ζ43ζ32 | ζ65 | linear of order 12 |
| ρ20 | 1 | -1 | 1 | -1 | ζ32 | ζ3 | i | 1 | i | -i | -i | -1 | ζ32 | ζ6 | ζ6 | ζ65 | ζ3 | ζ65 | ζ4ζ3 | ζ32 | ζ4ζ32 | ζ43ζ32 | ζ4ζ3 | ζ43ζ3 | ζ65 | ζ3 | ζ43ζ32 | ζ4ζ32 | ζ43ζ3 | ζ6 | linear of order 12 |
| ρ21 | 1 | -1 | 1 | -1 | ζ3 | ζ32 | -i | 1 | -i | i | i | -1 | ζ3 | ζ65 | ζ65 | ζ6 | ζ32 | ζ6 | ζ43ζ32 | ζ3 | ζ43ζ3 | ζ4ζ3 | ζ43ζ32 | ζ4ζ32 | ζ6 | ζ32 | ζ4ζ3 | ζ43ζ3 | ζ4ζ32 | ζ65 | linear of order 12 |
| ρ22 | 1 | -1 | 1 | -1 | ζ3 | ζ32 | i | -1 | -i | i | -i | 1 | ζ3 | ζ65 | ζ65 | ζ6 | ζ32 | ζ6 | ζ4ζ32 | ζ65 | ζ43ζ3 | ζ4ζ3 | ζ43ζ32 | ζ4ζ32 | ζ32 | ζ6 | ζ43ζ3 | ζ4ζ3 | ζ43ζ32 | ζ3 | linear of order 12 |
| ρ23 | 1 | -1 | 1 | -1 | ζ32 | ζ3 | -i | -1 | i | -i | i | 1 | ζ32 | ζ6 | ζ6 | ζ65 | ζ3 | ζ65 | ζ43ζ3 | ζ6 | ζ4ζ32 | ζ43ζ32 | ζ4ζ3 | ζ43ζ3 | ζ3 | ζ65 | ζ4ζ32 | ζ43ζ32 | ζ4ζ3 | ζ32 | linear of order 12 |
| ρ24 | 1 | -1 | 1 | -1 | ζ32 | ζ3 | -i | 1 | -i | i | i | -1 | ζ32 | ζ6 | ζ6 | ζ65 | ζ3 | ζ65 | ζ43ζ3 | ζ32 | ζ43ζ32 | ζ4ζ32 | ζ43ζ3 | ζ4ζ3 | ζ65 | ζ3 | ζ4ζ32 | ζ43ζ32 | ζ4ζ3 | ζ6 | linear of order 12 |
| ρ25 | 2 | -2 | -2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ26 | 2 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ27 | 2 | -2 | -2 | 2 | -1-√-3 | -1+√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 1+√-3 | 1+√-3 | -1-√-3 | -1+√-3 | 1-√-3 | 1-√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3×D4 |
| ρ28 | 2 | 2 | -2 | -2 | -1-√-3 | -1+√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 1+√-3 | -1-√-3 | 1+√-3 | 1-√-3 | 1-√-3 | -1+√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3×Q8 |
| ρ29 | 2 | 2 | -2 | -2 | -1+√-3 | -1-√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 1-√-3 | -1+√-3 | 1-√-3 | 1+√-3 | 1+√-3 | -1-√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3×Q8 |
| ρ30 | 2 | -2 | -2 | 2 | -1+√-3 | -1-√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 1-√-3 | 1-√-3 | -1+√-3 | -1-√-3 | 1+√-3 | 1+√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3×D4 |
►Regular action on 48 points
Generators in S48
(1 28 23)(2 25 24)(3 26 21)(4 27 22)(5 9 18)(6 10 19)(7 11 20)(8 12 17)(13 45 34)(14 46 35)(15 47 36)(16 48 33)(29 40 44)(30 37 41)(31 38 42)(32 39 43)
(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)
(1 38 8 34)(2 37 5 33)(3 40 6 36)(4 39 7 35)(9 16 25 41)(10 15 26 44)(11 14 27 43)(12 13 28 42)(17 45 23 31)(18 48 24 30)(19 47 21 29)(20 46 22 32)
G:=sub<Sym(48)| (1,28,23)(2,25,24)(3,26,21)(4,27,22)(5,9,18)(6,10,19)(7,11,20)(8,12,17)(13,45,34)(14,46,35)(15,47,36)(16,48,33)(29,40,44)(30,37,41)(31,38,42)(32,39,43), (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), (1,38,8,34)(2,37,5,33)(3,40,6,36)(4,39,7,35)(9,16,25,41)(10,15,26,44)(11,14,27,43)(12,13,28,42)(17,45,23,31)(18,48,24,30)(19,47,21,29)(20,46,22,32)>;
G:=Group( (1,28,23)(2,25,24)(3,26,21)(4,27,22)(5,9,18)(6,10,19)(7,11,20)(8,12,17)(13,45,34)(14,46,35)(15,47,36)(16,48,33)(29,40,44)(30,37,41)(31,38,42)(32,39,43), (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), (1,38,8,34)(2,37,5,33)(3,40,6,36)(4,39,7,35)(9,16,25,41)(10,15,26,44)(11,14,27,43)(12,13,28,42)(17,45,23,31)(18,48,24,30)(19,47,21,29)(20,46,22,32) );
G=PermutationGroup([[(1,28,23),(2,25,24),(3,26,21),(4,27,22),(5,9,18),(6,10,19),(7,11,20),(8,12,17),(13,45,34),(14,46,35),(15,47,36),(16,48,33),(29,40,44),(30,37,41),(31,38,42),(32,39,43)], [(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)], [(1,38,8,34),(2,37,5,33),(3,40,6,36),(4,39,7,35),(9,16,25,41),(10,15,26,44),(11,14,27,43),(12,13,28,42),(17,45,23,31),(18,48,24,30),(19,47,21,29),(20,46,22,32)]])
C3×C4⋊C4 is a maximal subgroup of
C6.Q16 C12.Q8 C6.D8 C6.SD16 Dic6⋊C4 C12⋊Q8 Dic3.Q8 C4.Dic6 C4⋊C4⋊7S3 Dic3⋊5D4 D6.D4 C12⋊D4 D6⋊Q8 C4.D12 C4⋊C4⋊S3 D4×C12 Q8×C12 C4○D4⋊C12 SL2(𝔽3)⋊6D4 SL2(𝔽3)⋊3Q8 Dic7⋊C12 C28⋊C12 C2.SU3(𝔽2) C4⋊(He3⋊C4)
C3×C4⋊C4 is a maximal quotient of
Dic7⋊C12 C28⋊C12
Matrix representation of C3×C4⋊C4 ►in GL3(𝔽13) generated by
| 1 | 0 | 0 |
| 0 | 9 | 0 |
| 0 | 0 | 9 |
| 12 | 0 | 0 |
| 0 | 0 | 12 |
| 0 | 1 | 0 |
| 5 | 0 | 0 |
| 0 | 0 | 5 |
| 0 | 5 | 0 |
G:=sub<GL(3,GF(13))| [1,0,0,0,9,0,0,0,9],[12,0,0,0,0,1,0,12,0],[5,0,0,0,0,5,0,5,0] >;
C3×C4⋊C4 in GAP, Magma, Sage, TeX
C_3\times C_4\rtimes C_4
% in TeX
G:=Group("C3xC4:C4"); // GroupNames label
G:=SmallGroup(48,22);
// by ID
G=gap.SmallGroup(48,22);
# by ID
G:=PCGroup([5,-2,-2,-3,-2,-2,120,141,66]);
// Polycyclic
G:=Group<a,b,c|a^3=b^4=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export
Subgroup lattice of C3×C4⋊C4 in TeX
Character table of C3×C4⋊C4 in TeX