G = C8.(C32⋊C4) order 288 = 25·32
1st non-split extension by C8 of C32⋊C4 acting via C32⋊C4/C3⋊S3=C2
Aliases: (C3×C24).5C4, C8.1(C32⋊C4), C3⋊Dic3.36D4, C32⋊4C8.8C4, C32⋊4(C8.C4), C32⋊M4(2).3C2, (C2×C3⋊S3).6Q8, (C8×C3⋊S3).13C2, C4.11(C2×C32⋊C4), (C3×C6).14(C4⋊C4), (C3×C12).15(C2×C4), C2.7(C4⋊(C32⋊C4)), (C4×C3⋊S3).83C22, SmallGroup(288,419)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8.(C32⋊C4)
G = < a,b,c,d | a8=b3=c3=1, d4=a4, ab=ba, ac=ca, dad-1=a-1, dcd-1=bc=cb, dbd-1=b-1c >
Character table of C8.(C32⋊C4)
| class | 1 | 2A | 2B | 3A | 3B | 4A | 4B | 4C | 6A | 6B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 12C | 12D | 24A | 24B | 24C | 24D | 24E | 24F | 24G | 24H | |
| size | 1 | 1 | 18 | 4 | 4 | 2 | 9 | 9 | 4 | 4 | 2 | 2 | 18 | 18 | 36 | 36 | 36 | 36 | 4 | 4 | 4 | 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 | 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 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | i | i | -i | -i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ6 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | i | -i | -i | i | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ7 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -i | -i | i | i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ8 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -i | i | i | -i | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ9 | 2 | 2 | -2 | 2 | 2 | -2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ11 | 2 | -2 | 0 | 2 | 2 | 0 | 2i | -2i | -2 | -2 | -√2 | √2 | √-2 | -√-2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | -√2 | -√2 | -√2 | -√2 | √2 | √2 | √2 | complex lifted from C8.C4 |
| ρ12 | 2 | -2 | 0 | 2 | 2 | 0 | -2i | 2i | -2 | -2 | √2 | -√2 | √-2 | -√-2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | √2 | √2 | √2 | -√2 | -√2 | -√2 | complex lifted from C8.C4 |
| ρ13 | 2 | -2 | 0 | 2 | 2 | 0 | 2i | -2i | -2 | -2 | √2 | -√2 | -√-2 | √-2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | √2 | √2 | √2 | -√2 | -√2 | -√2 | complex lifted from C8.C4 |
| ρ14 | 2 | -2 | 0 | 2 | 2 | 0 | -2i | 2i | -2 | -2 | -√2 | √2 | -√-2 | √-2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | -√2 | -√2 | -√2 | -√2 | √2 | √2 | √2 | complex lifted from C8.C4 |
| ρ15 | 4 | 4 | 0 | 1 | -2 | 4 | 0 | 0 | -2 | 1 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 1 | 1 | -2 | -2 | -2 | 1 | 1 | -2 | 1 | 1 | orthogonal lifted from C32⋊C4 |
| ρ16 | 4 | 4 | 0 | -2 | 1 | 4 | 0 | 0 | 1 | -2 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -2 | -2 | 1 | 1 | 1 | -2 | -2 | 1 | -2 | -2 | orthogonal lifted from C32⋊C4 |
| ρ17 | 4 | 4 | 0 | -2 | 1 | 4 | 0 | 0 | 1 | -2 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -2 | -2 | -1 | -1 | -1 | 2 | 2 | -1 | 2 | 2 | orthogonal lifted from C2×C32⋊C4 |
| ρ18 | 4 | 4 | 0 | 1 | -2 | 4 | 0 | 0 | -2 | 1 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 1 | 1 | 2 | 2 | 2 | -1 | -1 | 2 | -1 | -1 | orthogonal lifted from C2×C32⋊C4 |
| ρ19 | 4 | 4 | 0 | -2 | 1 | -4 | 0 | 0 | 1 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 2 | 2 | 3i | 3i | -3i | 0 | 0 | -3i | 0 | 0 | complex lifted from C4⋊(C32⋊C4) |
| ρ20 | 4 | 4 | 0 | 1 | -2 | -4 | 0 | 0 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | 3i | -3i | 0 | 3i | -3i | complex lifted from C4⋊(C32⋊C4) |
| ρ21 | 4 | 4 | 0 | 1 | -2 | -4 | 0 | 0 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | -3i | 3i | 0 | -3i | 3i | complex lifted from C4⋊(C32⋊C4) |
| ρ22 | 4 | 4 | 0 | -2 | 1 | -4 | 0 | 0 | 1 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 2 | 2 | -3i | -3i | 3i | 0 | 0 | 3i | 0 | 0 | complex lifted from C4⋊(C32⋊C4) |
| ρ23 | 4 | -4 | 0 | 1 | -2 | 0 | 0 | 0 | 2 | -1 | 2√2 | -2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -3i | 3i | √2 | -√2 | -√2 | ζ83+2ζ8 | 2ζ87+ζ85 | √2 | ζ87+2ζ85 | 2ζ83+ζ8 | complex faithful |
| ρ24 | 4 | -4 | 0 | -2 | 1 | 0 | 0 | 0 | -1 | 2 | -2√2 | 2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 3i | -3i | 0 | 0 | 2ζ87+ζ85 | 2ζ83+ζ8 | ζ87+2ζ85 | √2 | √2 | ζ83+2ζ8 | -√2 | -√2 | complex faithful |
| ρ25 | 4 | -4 | 0 | 1 | -2 | 0 | 0 | 0 | 2 | -1 | -2√2 | 2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3i | -3i | -√2 | √2 | √2 | 2ζ83+ζ8 | ζ87+2ζ85 | -√2 | 2ζ87+ζ85 | ζ83+2ζ8 | complex faithful |
| ρ26 | 4 | -4 | 0 | -2 | 1 | 0 | 0 | 0 | -1 | 2 | -2√2 | 2√2 | 0 | 0 | 0 | 0 | 0 | 0 | -3i | 3i | 0 | 0 | ζ83+2ζ8 | ζ87+2ζ85 | 2ζ83+ζ8 | √2 | √2 | 2ζ87+ζ85 | -√2 | -√2 | complex faithful |
| ρ27 | 4 | -4 | 0 | -2 | 1 | 0 | 0 | 0 | -1 | 2 | 2√2 | -2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 3i | -3i | 0 | 0 | 2ζ83+ζ8 | 2ζ87+ζ85 | ζ83+2ζ8 | -√2 | -√2 | ζ87+2ζ85 | √2 | √2 | complex faithful |
| ρ28 | 4 | -4 | 0 | 1 | -2 | 0 | 0 | 0 | 2 | -1 | -2√2 | 2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -3i | 3i | -√2 | √2 | √2 | ζ87+2ζ85 | 2ζ83+ζ8 | -√2 | ζ83+2ζ8 | 2ζ87+ζ85 | complex faithful |
| ρ29 | 4 | -4 | 0 | -2 | 1 | 0 | 0 | 0 | -1 | 2 | 2√2 | -2√2 | 0 | 0 | 0 | 0 | 0 | 0 | -3i | 3i | 0 | 0 | ζ87+2ζ85 | ζ83+2ζ8 | 2ζ87+ζ85 | -√2 | -√2 | 2ζ83+ζ8 | √2 | √2 | complex faithful |
| ρ30 | 4 | -4 | 0 | 1 | -2 | 0 | 0 | 0 | 2 | -1 | 2√2 | -2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3i | -3i | √2 | -√2 | -√2 | 2ζ87+ζ85 | ζ83+2ζ8 | √2 | 2ζ83+ζ8 | ζ87+2ζ85 | complex faithful |
►On 48 points
Generators in S48
(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 33 25)(2 34 26)(3 35 27)(4 36 28)(5 37 29)(6 38 30)(7 39 31)(8 40 32)(9 46 20)(10 47 21)(11 48 22)(12 41 23)(13 42 24)(14 43 17)(15 44 18)(16 45 19)
(9 20 46)(10 21 47)(11 22 48)(12 23 41)(13 24 42)(14 17 43)(15 18 44)(16 19 45)
(1 11 3 9 5 15 7 13)(2 10 4 16 6 14 8 12)(17 32 41 34 21 28 45 38)(18 31 42 33 22 27 46 37)(19 30 43 40 23 26 47 36)(20 29 44 39 24 25 48 35)
G:=sub<Sym(48)| (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), (1,33,25)(2,34,26)(3,35,27)(4,36,28)(5,37,29)(6,38,30)(7,39,31)(8,40,32)(9,46,20)(10,47,21)(11,48,22)(12,41,23)(13,42,24)(14,43,17)(15,44,18)(16,45,19), (9,20,46)(10,21,47)(11,22,48)(12,23,41)(13,24,42)(14,17,43)(15,18,44)(16,19,45), (1,11,3,9,5,15,7,13)(2,10,4,16,6,14,8,12)(17,32,41,34,21,28,45,38)(18,31,42,33,22,27,46,37)(19,30,43,40,23,26,47,36)(20,29,44,39,24,25,48,35)>;
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), (1,33,25)(2,34,26)(3,35,27)(4,36,28)(5,37,29)(6,38,30)(7,39,31)(8,40,32)(9,46,20)(10,47,21)(11,48,22)(12,41,23)(13,42,24)(14,43,17)(15,44,18)(16,45,19), (9,20,46)(10,21,47)(11,22,48)(12,23,41)(13,24,42)(14,17,43)(15,18,44)(16,19,45), (1,11,3,9,5,15,7,13)(2,10,4,16,6,14,8,12)(17,32,41,34,21,28,45,38)(18,31,42,33,22,27,46,37)(19,30,43,40,23,26,47,36)(20,29,44,39,24,25,48,35) );
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)], [(1,33,25),(2,34,26),(3,35,27),(4,36,28),(5,37,29),(6,38,30),(7,39,31),(8,40,32),(9,46,20),(10,47,21),(11,48,22),(12,41,23),(13,42,24),(14,43,17),(15,44,18),(16,45,19)], [(9,20,46),(10,21,47),(11,22,48),(12,23,41),(13,24,42),(14,17,43),(15,18,44),(16,19,45)], [(1,11,3,9,5,15,7,13),(2,10,4,16,6,14,8,12),(17,32,41,34,21,28,45,38),(18,31,42,33,22,27,46,37),(19,30,43,40,23,26,47,36),(20,29,44,39,24,25,48,35)]])
Matrix representation of C8.(C32⋊C4) ►in GL4(𝔽73) generated by
| 63 | 0 | 4 | 4 |
| 0 | 63 | 4 | 4 |
| 0 | 0 | 51 | 0 |
| 0 | 0 | 0 | 51 |
| 72 | 72 | 72 | 0 |
| 1 | 0 | 0 | 1 |
| 0 | 0 | 0 | 72 |
| 0 | 0 | 1 | 72 |
| 1 | 0 | 1 | 0 |
| 0 | 1 | 1 | 0 |
| 0 | 0 | 72 | 1 |
| 0 | 0 | 72 | 0 |
| 27 | 0 | 33 | 34 |
| 27 | 0 | 33 | 33 |
| 19 | 46 | 46 | 46 |
| 46 | 27 | 0 | 0 |
G:=sub<GL(4,GF(73))| [63,0,0,0,0,63,0,0,4,4,51,0,4,4,0,51],[72,1,0,0,72,0,0,0,72,0,0,1,0,1,72,72],[1,0,0,0,0,1,0,0,1,1,72,72,0,0,1,0],[27,27,19,46,0,0,46,27,33,33,46,0,34,33,46,0] >;
C8.(C32⋊C4) in GAP, Magma, Sage, TeX
C_8.(C_3^2\rtimes C_4)
% in TeX
G:=Group("C8.(C3^2:C4)"); // GroupNames label
G:=SmallGroup(288,419);
// by ID
G=gap.SmallGroup(288,419);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,141,176,100,675,80,9413,691,12550,2372]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^3=c^3=1,d^4=a^4,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,d*c*d^-1=b*c=c*b,d*b*d^-1=b^-1*c>;
// generators/relations
Export
Subgroup lattice of C8.(C32⋊C4) in TeX
Character table of C8.(C32⋊C4) in TeX