Copied to
clipboard

## G = C8.Dic6order 192 = 26·3

### 1st non-split extension by C8 of Dic6 acting via Dic6/C6=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C24 — C8.Dic6
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×C24 — C12.C8 — C8.Dic6
 Lower central C3 — C6 — C12 — C24 — C8.Dic6
 Upper central C1 — C2 — C2×C4 — C2×C8 — C4.Q8

Generators and relations for C8.Dic6
G = < a,b,c | a8=b12=1, c2=ab6, bab-1=a3, cac-1=a5, cbc-1=a-1b-1 >

Character table of C8.Dic6

 class 1 2A 2B 3 4A 4B 4C 4D 6A 6B 6C 8A 8B 8C 8D 8E 12A 12B 12C 12D 12E 12F 16A 16B 16C 16D 24A 24B 24C 24D size 1 1 2 2 2 2 8 8 2 2 2 2 2 4 24 24 4 4 8 8 8 8 12 12 12 12 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 -i i -1 -1 1 1 1 -1 i -i -1 1 -i i i -i 1 -1 -1 1 1 1 -1 -1 linear of order 4 ρ6 1 1 -1 1 -1 1 i -i -1 -1 1 1 1 -1 -i i -1 1 i -i -i i 1 -1 -1 1 1 1 -1 -1 linear of order 4 ρ7 1 1 -1 1 -1 1 -i i -1 -1 1 1 1 -1 -i i -1 1 -i i i -i -1 1 1 -1 1 1 -1 -1 linear of order 4 ρ8 1 1 -1 1 -1 1 i -i -1 -1 1 1 1 -1 i -i -1 1 i -i -i i -1 1 1 -1 1 1 -1 -1 linear of order 4 ρ9 2 2 2 2 2 2 0 0 2 2 2 -2 -2 -2 0 0 2 2 0 0 0 0 0 0 0 0 -2 -2 -2 -2 orthogonal lifted from D4 ρ10 2 2 2 -1 2 2 2 2 -1 -1 -1 2 2 2 0 0 -1 -1 -1 -1 -1 -1 0 0 0 0 -1 -1 -1 -1 orthogonal lifted from S3 ρ11 2 2 2 -1 2 2 -2 -2 -1 -1 -1 2 2 2 0 0 -1 -1 1 1 1 1 0 0 0 0 -1 -1 -1 -1 orthogonal lifted from D6 ρ12 2 2 -2 2 -2 2 0 0 -2 -2 2 -2 -2 2 0 0 -2 2 0 0 0 0 0 0 0 0 -2 -2 2 2 symplectic lifted from Q8, Schur index 2 ρ13 2 2 -2 -1 -2 2 0 0 1 1 -1 -2 -2 2 0 0 1 -1 √3 √3 -√3 -√3 0 0 0 0 1 1 -1 -1 symplectic lifted from Dic6, Schur index 2 ρ14 2 2 -2 -1 -2 2 0 0 1 1 -1 -2 -2 2 0 0 1 -1 -√3 -√3 √3 √3 0 0 0 0 1 1 -1 -1 symplectic lifted from Dic6, Schur index 2 ρ15 2 2 -2 -1 -2 2 -2i 2i 1 1 -1 2 2 -2 0 0 1 -1 i -i -i i 0 0 0 0 -1 -1 1 1 complex lifted from C4×S3 ρ16 2 2 -2 -1 -2 2 2i -2i 1 1 -1 2 2 -2 0 0 1 -1 -i i i -i 0 0 0 0 -1 -1 1 1 complex lifted from C4×S3 ρ17 2 2 2 -1 2 2 0 0 -1 -1 -1 -2 -2 -2 0 0 -1 -1 -√-3 √-3 -√-3 √-3 0 0 0 0 1 1 1 1 complex lifted from C3⋊D4 ρ18 2 2 2 -1 2 2 0 0 -1 -1 -1 -2 -2 -2 0 0 -1 -1 √-3 -√-3 √-3 -√-3 0 0 0 0 1 1 1 1 complex lifted from C3⋊D4 ρ19 2 2 -2 2 2 -2 0 0 -2 -2 2 0 0 0 0 0 2 -2 0 0 0 0 √-2 √-2 -√-2 -√-2 0 0 0 0 complex lifted from SD16 ρ20 2 2 2 2 -2 -2 0 0 2 2 2 0 0 0 0 0 -2 -2 0 0 0 0 √-2 -√-2 √-2 -√-2 0 0 0 0 complex lifted from SD16 ρ21 2 2 -2 2 2 -2 0 0 -2 -2 2 0 0 0 0 0 2 -2 0 0 0 0 -√-2 -√-2 √-2 √-2 0 0 0 0 complex lifted from SD16 ρ22 2 2 2 2 -2 -2 0 0 2 2 2 0 0 0 0 0 -2 -2 0 0 0 0 -√-2 √-2 -√-2 √-2 0 0 0 0 complex lifted from SD16 ρ23 4 4 -4 -2 4 -4 0 0 2 2 -2 0 0 0 0 0 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from Q8⋊2S3 ρ24 4 4 4 -2 -4 -4 0 0 -2 -2 -2 0 0 0 0 0 2 2 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from D4.S3, Schur index 2 ρ25 4 -4 0 4 0 0 0 0 0 0 -4 -2√-2 2√-2 0 0 0 0 0 0 0 0 0 0 0 0 0 -2√-2 2√-2 0 0 complex lifted from C8.Q8 ρ26 4 -4 0 4 0 0 0 0 0 0 -4 2√-2 -2√-2 0 0 0 0 0 0 0 0 0 0 0 0 0 2√-2 -2√-2 0 0 complex lifted from C8.Q8 ρ27 4 -4 0 -2 0 0 0 0 -2√-3 2√-3 2 -2√-2 2√-2 0 0 0 0 0 0 0 0 0 0 0 0 0 √-2 -√-2 √6 -√6 complex faithful ρ28 4 -4 0 -2 0 0 0 0 -2√-3 2√-3 2 2√-2 -2√-2 0 0 0 0 0 0 0 0 0 0 0 0 0 -√-2 √-2 -√6 √6 complex faithful ρ29 4 -4 0 -2 0 0 0 0 2√-3 -2√-3 2 2√-2 -2√-2 0 0 0 0 0 0 0 0 0 0 0 0 0 -√-2 √-2 √6 -√6 complex faithful ρ30 4 -4 0 -2 0 0 0 0 2√-3 -2√-3 2 -2√-2 2√-2 0 0 0 0 0 0 0 0 0 0 0 0 0 √-2 -√-2 -√6 √6 complex faithful

Smallest permutation representation of C8.Dic6
On 48 points
Generators in S48
(1 3 5 7 9 11 13 15)(2 12 6 16 10 4 14 8)(17 27 21 31 25 19 29 23)(18 20 22 24 26 28 30 32)(33 43 37 47 41 35 45 39)(34 36 38 40 42 44 46 48)
(1 32 40)(2 35 25 4 41 27 10 43 17 12 33 19)(3 22 42 7 18 46)(5 28 44 13 20 36)(6 47 29 16 45 23 14 39 21 8 37 31)(9 24 48)(11 30 34 15 26 38)
(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)

G:=sub<Sym(48)| (1,3,5,7,9,11,13,15)(2,12,6,16,10,4,14,8)(17,27,21,31,25,19,29,23)(18,20,22,24,26,28,30,32)(33,43,37,47,41,35,45,39)(34,36,38,40,42,44,46,48), (1,32,40)(2,35,25,4,41,27,10,43,17,12,33,19)(3,22,42,7,18,46)(5,28,44,13,20,36)(6,47,29,16,45,23,14,39,21,8,37,31)(9,24,48)(11,30,34,15,26,38), (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)>;

G:=Group( (1,3,5,7,9,11,13,15)(2,12,6,16,10,4,14,8)(17,27,21,31,25,19,29,23)(18,20,22,24,26,28,30,32)(33,43,37,47,41,35,45,39)(34,36,38,40,42,44,46,48), (1,32,40)(2,35,25,4,41,27,10,43,17,12,33,19)(3,22,42,7,18,46)(5,28,44,13,20,36)(6,47,29,16,45,23,14,39,21,8,37,31)(9,24,48)(11,30,34,15,26,38), (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) );

G=PermutationGroup([[(1,3,5,7,9,11,13,15),(2,12,6,16,10,4,14,8),(17,27,21,31,25,19,29,23),(18,20,22,24,26,28,30,32),(33,43,37,47,41,35,45,39),(34,36,38,40,42,44,46,48)], [(1,32,40),(2,35,25,4,41,27,10,43,17,12,33,19),(3,22,42,7,18,46),(5,28,44,13,20,36),(6,47,29,16,45,23,14,39,21,8,37,31),(9,24,48),(11,30,34,15,26,38)], [(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)]])

Matrix representation of C8.Dic6 in GL6(𝔽97)

 96 0 0 0 0 0 0 96 0 0 0 0 0 0 40 57 0 0 0 0 40 40 0 0 0 0 0 0 57 40 0 0 0 0 57 57
,
 0 22 0 0 0 0 75 75 0 0 0 0 0 0 1 0 0 0 0 0 0 96 0 0 0 0 0 0 40 57 0 0 0 0 57 57
,
 82 55 0 0 0 0 70 15 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 40 57 0 0 0 0 40 40 0 0

G:=sub<GL(6,GF(97))| [96,0,0,0,0,0,0,96,0,0,0,0,0,0,40,40,0,0,0,0,57,40,0,0,0,0,0,0,57,57,0,0,0,0,40,57],[0,75,0,0,0,0,22,75,0,0,0,0,0,0,1,0,0,0,0,0,0,96,0,0,0,0,0,0,40,57,0,0,0,0,57,57],[82,70,0,0,0,0,55,15,0,0,0,0,0,0,0,0,40,40,0,0,0,0,57,40,0,0,1,0,0,0,0,0,0,1,0,0] >;

C8.Dic6 in GAP, Magma, Sage, TeX

C_8.{\rm Dic}_6
% in TeX

G:=Group("C8.Dic6");
// GroupNames label

G:=SmallGroup(192,46);
// by ID

G=gap.SmallGroup(192,46);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,365,36,758,346,80,851,102,6278]);
// Polycyclic

G:=Group<a,b,c|a^8=b^12=1,c^2=a*b^6,b*a*b^-1=a^3,c*a*c^-1=a^5,c*b*c^-1=a^-1*b^-1>;
// generators/relations

Export

׿
×
𝔽