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## G = Q8.Dic6order 192 = 26·3

### 1st non-split extension by Q8 of Dic6 acting via Dic6/C4=S3

Aliases: Q8.1Dic6, SL2(𝔽3).Q8, (C2×C4).2S4, (C4×Q8).5S3, (C2×Q8).5D6, C2.4(A4⋊Q8), Q8⋊Dic3.1C2, C22.33(C2×S4), C2.4(C4.6S4), C2.4(Q8.D6), (C4×SL2(𝔽3)).2C2, (C2×SL2(𝔽3)).5C22, SmallGroup(192,948)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — C2×SL2(𝔽3) — Q8.Dic6
 Chief series C1 — C2 — Q8 — SL2(𝔽3) — C2×SL2(𝔽3) — Q8⋊Dic3 — Q8.Dic6
 Lower central SL2(𝔽3) — C2×SL2(𝔽3) — Q8.Dic6
 Upper central C1 — C22 — C2×C4

Generators and relations for Q8.Dic6
G = < a,b,c,d | a4=c12=1, b2=a2, d2=a2c6, bab-1=dbd-1=a-1, cac-1=b, dad-1=a2b, cbc-1=ab, dcd-1=a2c-1 >

Character table of Q8.Dic6

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 4G 6A 6B 6C 8A 8B 8C 8D 12A 12B 12C 12D size 1 1 1 1 8 2 2 6 6 12 24 24 8 8 8 12 12 12 12 8 8 8 8 ρ1 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 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 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 linear of order 2 ρ5 2 2 2 2 -1 -2 -2 2 2 -2 0 0 -1 -1 -1 0 0 0 0 1 1 1 1 orthogonal lifted from D6 ρ6 2 2 2 2 -1 2 2 2 2 2 0 0 -1 -1 -1 0 0 0 0 -1 -1 -1 -1 orthogonal lifted from S3 ρ7 2 2 -2 -2 2 0 0 -2 2 0 0 0 -2 -2 2 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ8 2 2 -2 -2 -1 0 0 -2 2 0 0 0 1 1 -1 0 0 0 0 √3 √3 -√3 -√3 symplectic lifted from Dic6, Schur index 2 ρ9 2 2 -2 -2 -1 0 0 -2 2 0 0 0 1 1 -1 0 0 0 0 -√3 -√3 √3 √3 symplectic lifted from Dic6, Schur index 2 ρ10 2 -2 2 -2 -1 -2i 2i 0 0 0 0 0 1 -1 1 -√2 -√-2 √2 √-2 -i i -i i complex lifted from C4.6S4 ρ11 2 -2 2 -2 -1 2i -2i 0 0 0 0 0 1 -1 1 √2 -√-2 -√2 √-2 i -i i -i complex lifted from C4.6S4 ρ12 2 -2 2 -2 -1 -2i 2i 0 0 0 0 0 1 -1 1 √2 √-2 -√2 -√-2 -i i -i i complex lifted from C4.6S4 ρ13 2 -2 2 -2 -1 2i -2i 0 0 0 0 0 1 -1 1 -√2 √-2 √2 -√-2 i -i i -i complex lifted from C4.6S4 ρ14 3 3 3 3 0 3 3 -1 -1 -1 -1 -1 0 0 0 1 1 1 1 0 0 0 0 orthogonal lifted from S4 ρ15 3 3 3 3 0 -3 -3 -1 -1 1 -1 1 0 0 0 1 -1 1 -1 0 0 0 0 orthogonal lifted from C2×S4 ρ16 3 3 3 3 0 -3 -3 -1 -1 1 1 -1 0 0 0 -1 1 -1 1 0 0 0 0 orthogonal lifted from C2×S4 ρ17 3 3 3 3 0 3 3 -1 -1 -1 1 1 0 0 0 -1 -1 -1 -1 0 0 0 0 orthogonal lifted from S4 ρ18 4 -4 -4 4 -2 0 0 0 0 0 0 0 -2 2 2 0 0 0 0 0 0 0 0 symplectic lifted from Q8.D6, Schur index 2 ρ19 4 -4 -4 4 1 0 0 0 0 0 0 0 1 -1 -1 0 0 0 0 √-3 -√-3 -√-3 √-3 complex lifted from Q8.D6 ρ20 4 -4 -4 4 1 0 0 0 0 0 0 0 1 -1 -1 0 0 0 0 -√-3 √-3 √-3 -√-3 complex lifted from Q8.D6 ρ21 4 -4 4 -4 1 4i -4i 0 0 0 0 0 -1 1 -1 0 0 0 0 -i i -i i complex lifted from C4.6S4 ρ22 4 -4 4 -4 1 -4i 4i 0 0 0 0 0 -1 1 -1 0 0 0 0 i -i i -i complex lifted from C4.6S4 ρ23 6 6 -6 -6 0 0 0 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from A4⋊Q8, Schur index 2

Smallest permutation representation of Q8.Dic6
On 64 points
Generators in S64
(1 23 12 33)(2 20 9 30)(3 17 10 39)(4 26 11 36)(5 60 16 41)(6 57 13 50)(7 54 14 47)(8 63 15 44)(18 22 40 32)(19 37 29 27)(21 25 31 35)(24 28 34 38)(42 46 61 53)(43 58 62 51)(45 49 64 56)(48 52 55 59)
(1 19 12 29)(2 28 9 38)(3 25 10 35)(4 22 11 32)(5 56 16 49)(6 53 13 46)(7 62 14 43)(8 59 15 52)(17 21 39 31)(18 36 40 26)(20 24 30 34)(23 27 33 37)(41 45 60 64)(42 57 61 50)(44 48 63 55)(47 51 54 58)
(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 57 58 59 60 61 62 63 64)
(1 13 10 8)(2 5 11 14)(3 15 12 6)(4 7 9 16)(17 59 33 46)(18 51 34 64)(19 57 35 44)(20 49 36 62)(21 55 37 42)(22 47 38 60)(23 53 39 52)(24 45 40 58)(25 63 29 50)(26 43 30 56)(27 61 31 48)(28 41 32 54)

G:=sub<Sym(64)| (1,23,12,33)(2,20,9,30)(3,17,10,39)(4,26,11,36)(5,60,16,41)(6,57,13,50)(7,54,14,47)(8,63,15,44)(18,22,40,32)(19,37,29,27)(21,25,31,35)(24,28,34,38)(42,46,61,53)(43,58,62,51)(45,49,64,56)(48,52,55,59), (1,19,12,29)(2,28,9,38)(3,25,10,35)(4,22,11,32)(5,56,16,49)(6,53,13,46)(7,62,14,43)(8,59,15,52)(17,21,39,31)(18,36,40,26)(20,24,30,34)(23,27,33,37)(41,45,60,64)(42,57,61,50)(44,48,63,55)(47,51,54,58), (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)(53,54,55,56,57,58,59,60,61,62,63,64), (1,13,10,8)(2,5,11,14)(3,15,12,6)(4,7,9,16)(17,59,33,46)(18,51,34,64)(19,57,35,44)(20,49,36,62)(21,55,37,42)(22,47,38,60)(23,53,39,52)(24,45,40,58)(25,63,29,50)(26,43,30,56)(27,61,31,48)(28,41,32,54)>;

G:=Group( (1,23,12,33)(2,20,9,30)(3,17,10,39)(4,26,11,36)(5,60,16,41)(6,57,13,50)(7,54,14,47)(8,63,15,44)(18,22,40,32)(19,37,29,27)(21,25,31,35)(24,28,34,38)(42,46,61,53)(43,58,62,51)(45,49,64,56)(48,52,55,59), (1,19,12,29)(2,28,9,38)(3,25,10,35)(4,22,11,32)(5,56,16,49)(6,53,13,46)(7,62,14,43)(8,59,15,52)(17,21,39,31)(18,36,40,26)(20,24,30,34)(23,27,33,37)(41,45,60,64)(42,57,61,50)(44,48,63,55)(47,51,54,58), (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)(53,54,55,56,57,58,59,60,61,62,63,64), (1,13,10,8)(2,5,11,14)(3,15,12,6)(4,7,9,16)(17,59,33,46)(18,51,34,64)(19,57,35,44)(20,49,36,62)(21,55,37,42)(22,47,38,60)(23,53,39,52)(24,45,40,58)(25,63,29,50)(26,43,30,56)(27,61,31,48)(28,41,32,54) );

G=PermutationGroup([[(1,23,12,33),(2,20,9,30),(3,17,10,39),(4,26,11,36),(5,60,16,41),(6,57,13,50),(7,54,14,47),(8,63,15,44),(18,22,40,32),(19,37,29,27),(21,25,31,35),(24,28,34,38),(42,46,61,53),(43,58,62,51),(45,49,64,56),(48,52,55,59)], [(1,19,12,29),(2,28,9,38),(3,25,10,35),(4,22,11,32),(5,56,16,49),(6,53,13,46),(7,62,14,43),(8,59,15,52),(17,21,39,31),(18,36,40,26),(20,24,30,34),(23,27,33,37),(41,45,60,64),(42,57,61,50),(44,48,63,55),(47,51,54,58)], [(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),(53,54,55,56,57,58,59,60,61,62,63,64)], [(1,13,10,8),(2,5,11,14),(3,15,12,6),(4,7,9,16),(17,59,33,46),(18,51,34,64),(19,57,35,44),(20,49,36,62),(21,55,37,42),(22,47,38,60),(23,53,39,52),(24,45,40,58),(25,63,29,50),(26,43,30,56),(27,61,31,48),(28,41,32,54)]])

Matrix representation of Q8.Dic6 in GL4(𝔽73) generated by

 1 0 0 0 0 1 0 0 0 0 60 71 0 0 12 13
,
 1 0 0 0 0 1 0 0 0 0 1 61 0 0 61 72
,
 59 7 0 0 66 66 0 0 0 0 68 14 0 0 46 32
,
 20 71 0 0 18 53 0 0 0 0 11 13 0 0 2 62
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,60,12,0,0,71,13],[1,0,0,0,0,1,0,0,0,0,1,61,0,0,61,72],[59,66,0,0,7,66,0,0,0,0,68,46,0,0,14,32],[20,18,0,0,71,53,0,0,0,0,11,2,0,0,13,62] >;

Q8.Dic6 in GAP, Magma, Sage, TeX

Q_8.{\rm Dic}_6
% in TeX

G:=Group("Q8.Dic6");
// GroupNames label

G:=SmallGroup(192,948);
// by ID

G=gap.SmallGroup(192,948);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,2,-2,28,85,708,451,1684,655,172,1013,404,285,124]);
// Polycyclic

G:=Group<a,b,c,d|a^4=c^12=1,b^2=a^2,d^2=a^2*c^6,b*a*b^-1=d*b*d^-1=a^-1,c*a*c^-1=b,d*a*d^-1=a^2*b,c*b*c^-1=a*b,d*c*d^-1=a^2*c^-1>;
// generators/relations

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