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## G = Q8⋊SL2(𝔽3)  order 192 = 26·3

### The semidirect product of Q8 and SL2(𝔽3) acting via SL2(𝔽3)/Q8=C3

Aliases: Q8⋊SL2(𝔽3), Q822C3, (C2×Q8).A4, C2.2(Q8⋊A4), C2.2(C23⋊A4), C22.7(C22⋊A4), SmallGroup(192,1022)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — Q82 — Q8⋊SL2(𝔽3)
 Chief series C1 — C2 — C22 — C2×Q8 — Q82 — Q8⋊SL2(𝔽3)
 Lower central Q82 — Q8⋊SL2(𝔽3)
 Upper central C1 — C22

Generators and relations for Q8⋊SL2(𝔽3)
G = < a,b,c,d,e | a4=c4=e3=1, b2=a2, d2=c2, bab-1=a-1, ac=ca, ad=da, eae-1=b, bc=cb, bd=db, ebe-1=ab, dcd-1=c-1, ece-1=d, ede-1=cd >

Subgroups: 254 in 62 conjugacy classes, 14 normal (5 characteristic)
C1, C2, C2, C3, C4, C22, C6, C2×C4, Q8, Q8, C2×C6, C42, C4⋊C4, C2×Q8, C2×Q8, SL2(𝔽3), C4×Q8, C4⋊Q8, C2×SL2(𝔽3), Q82, Q8⋊SL2(𝔽3)
Quotients: C1, C3, A4, SL2(𝔽3), C22⋊A4, Q8⋊A4, C23⋊A4, Q8⋊SL2(𝔽3)

Character table of Q8⋊SL2(𝔽3)

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 4E 4F 4G 6A 6B 6C 6D 6E 6F size 1 1 1 1 16 16 6 6 6 6 12 12 12 16 16 16 16 16 16 ρ1 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 ζ3 ζ32 1 1 1 1 1 1 1 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 linear of order 3 ρ3 1 1 1 1 ζ32 ζ3 1 1 1 1 1 1 1 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 linear of order 3 ρ4 2 2 -2 -2 -1 -1 2 -2 0 0 0 0 0 1 1 -1 -1 1 1 symplectic lifted from SL2(𝔽3), Schur index 2 ρ5 2 -2 2 -2 -1 -1 0 0 2 -2 0 0 0 -1 1 1 1 -1 1 symplectic lifted from SL2(𝔽3), Schur index 2 ρ6 2 2 -2 -2 ζ65 ζ6 2 -2 0 0 0 0 0 ζ32 ζ32 ζ6 ζ65 ζ3 ζ3 complex lifted from SL2(𝔽3) ρ7 2 2 -2 -2 ζ6 ζ65 2 -2 0 0 0 0 0 ζ3 ζ3 ζ65 ζ6 ζ32 ζ32 complex lifted from SL2(𝔽3) ρ8 2 -2 2 -2 ζ65 ζ6 0 0 2 -2 0 0 0 ζ6 ζ32 ζ32 ζ3 ζ65 ζ3 complex lifted from SL2(𝔽3) ρ9 2 -2 2 -2 ζ6 ζ65 0 0 2 -2 0 0 0 ζ65 ζ3 ζ3 ζ32 ζ6 ζ32 complex lifted from SL2(𝔽3) ρ10 3 3 3 3 0 0 -1 -1 -1 -1 3 -1 -1 0 0 0 0 0 0 orthogonal lifted from A4 ρ11 3 3 3 3 0 0 3 3 -1 -1 -1 -1 -1 0 0 0 0 0 0 orthogonal lifted from A4 ρ12 3 3 3 3 0 0 -1 -1 -1 -1 -1 3 -1 0 0 0 0 0 0 orthogonal lifted from A4 ρ13 3 3 3 3 0 0 -1 -1 3 3 -1 -1 -1 0 0 0 0 0 0 orthogonal lifted from A4 ρ14 3 3 3 3 0 0 -1 -1 -1 -1 -1 -1 3 0 0 0 0 0 0 orthogonal lifted from A4 ρ15 4 -4 -4 4 1 1 0 0 0 0 0 0 0 -1 1 -1 -1 -1 1 orthogonal lifted from C23⋊A4 ρ16 4 -4 -4 4 ζ32 ζ3 0 0 0 0 0 0 0 ζ65 ζ3 ζ65 ζ6 ζ6 ζ32 complex lifted from C23⋊A4 ρ17 4 -4 -4 4 ζ3 ζ32 0 0 0 0 0 0 0 ζ6 ζ32 ζ6 ζ65 ζ65 ζ3 complex lifted from C23⋊A4 ρ18 6 6 -6 -6 0 0 -2 2 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q8⋊A4, Schur index 2 ρ19 6 -6 6 -6 0 0 0 0 -2 2 0 0 0 0 0 0 0 0 0 symplectic lifted from Q8⋊A4, Schur index 2

Smallest permutation representation of Q8⋊SL2(𝔽3)
On 64 points
Generators in S64
(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 14 3 16)(2 13 4 15)(5 58 7 60)(6 57 8 59)(9 17 11 19)(10 20 12 18)(21 29 23 31)(22 32 24 30)(25 33 27 35)(26 36 28 34)(37 45 39 47)(38 48 40 46)(41 49 43 51)(42 52 44 50)(53 61 55 63)(54 64 56 62)
(1 24 9 25)(2 21 10 26)(3 22 11 27)(4 23 12 28)(5 50 63 47)(6 51 64 48)(7 52 61 45)(8 49 62 46)(13 29 20 36)(14 30 17 33)(15 31 18 34)(16 32 19 35)(37 58 42 53)(38 59 43 54)(39 60 44 55)(40 57 41 56)
(1 40 9 41)(2 37 10 42)(3 38 11 43)(4 39 12 44)(5 31 63 34)(6 32 64 35)(7 29 61 36)(8 30 62 33)(13 45 20 52)(14 46 17 49)(15 47 18 50)(16 48 19 51)(21 53 26 58)(22 54 27 59)(23 55 28 60)(24 56 25 57)
(2 13 14)(4 15 16)(5 51 28)(6 44 34)(7 49 26)(8 42 36)(10 20 17)(12 18 19)(21 61 46)(22 54 38)(23 63 48)(24 56 40)(25 57 41)(27 59 43)(29 62 37)(30 53 45)(31 64 39)(32 55 47)(33 58 52)(35 60 50)

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

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

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

Matrix representation of Q8⋊SL2(𝔽3) in GL4(𝔽13) generated by

 12 11 0 0 1 1 0 0 0 0 1 0 0 0 0 1
,
 7 8 0 0 10 6 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 0 12 0 0 1 0
,
 1 0 0 0 0 1 0 0 0 0 9 10 0 0 10 4
,
 1 0 0 0 9 9 0 0 0 0 3 0 0 0 12 9
G:=sub<GL(4,GF(13))| [12,1,0,0,11,1,0,0,0,0,1,0,0,0,0,1],[7,10,0,0,8,6,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,12,0],[1,0,0,0,0,1,0,0,0,0,9,10,0,0,10,4],[1,9,0,0,0,9,0,0,0,0,3,12,0,0,0,9] >;

Q8⋊SL2(𝔽3) in GAP, Magma, Sage, TeX

Q_8\rtimes {\rm SL}_2({\mathbb F}_3)
% in TeX

G:=Group("Q8:SL(2,3)");
// GroupNames label

G:=SmallGroup(192,1022);
// by ID

G=gap.SmallGroup(192,1022);
# by ID

G:=PCGroup([7,-3,-2,2,-2,2,-2,-2,85,456,191,723,352,675,136,1264,235,102]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=c^4=e^3=1,b^2=a^2,d^2=c^2,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,e*a*e^-1=b,b*c=c*b,b*d=d*b,e*b*e^-1=a*b,d*c*d^-1=c^-1,e*c*e^-1=d,e*d*e^-1=c*d>;
// generators/relations

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