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## G = Q8×SL2(𝔽3)  order 192 = 26·3

### Direct product of Q8 and SL2(𝔽3)

Series: Derived Chief Lower central Upper central

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

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, ae=ea, bc=cb, bd=db, be=eb, dcd-1=c-1, ece-1=d, ede-1=cd >

Subgroups: 215 in 77 conjugacy classes, 31 normal (12 characteristic)
C1, C2, C3, C4, C4, C22, C6, C2×C4, C2×C4, Q8, Q8, Q8, C12, C2×C6, C42, C4⋊C4, C2×Q8, C2×Q8, SL2(𝔽3), C2×C12, C3×Q8, C4×Q8, C4×Q8, C4⋊Q8, C2×SL2(𝔽3), C6×Q8, Q82, C4×SL2(𝔽3), Q8×SL2(𝔽3)
Quotients: C1, C2, C3, C22, C6, Q8, A4, C2×C6, SL2(𝔽3), C3×Q8, C2×A4, C2×SL2(𝔽3), C22×A4, C22×SL2(𝔽3), Q8×A4, Q8.A4, Q8×SL2(𝔽3)

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

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

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

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

35 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A ··· 4F 4G 4H 4I 4J 4K 6A ··· 6F 12A ··· 12L order 1 2 2 2 3 3 4 ··· 4 4 4 4 4 4 6 ··· 6 12 ··· 12 size 1 1 1 1 4 4 2 ··· 2 6 6 12 12 12 4 ··· 4 8 ··· 8

35 irreducible representations

 dim 1 1 1 1 2 2 2 2 3 3 4 4 6 type + + - - + + + - image C1 C2 C3 C6 Q8 SL2(𝔽3) SL2(𝔽3) C3×Q8 A4 C2×A4 Q8.A4 Q8.A4 Q8×A4 kernel Q8×SL2(𝔽3) C4×SL2(𝔽3) Q82 C4×Q8 SL2(𝔽3) Q8 Q8 Q8 C2×Q8 C2×C4 C2 C2 C2 # reps 1 3 2 6 1 4 8 2 1 3 1 2 1

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

 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 0 5 0 0 5 0
,
 4 10 0 0 10 9 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 12 0 0 0 0 0 1 0 0 0 0 1
,
 4 1 0 0 10 0 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,0,12,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,0,5,0,0,5,0],[4,10,0,0,10,9,0,0,0,0,1,0,0,0,0,1],[0,12,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[4,10,0,0,1,0,0,0,0,0,1,0,0,0,0,1] >;

Q8×SL2(𝔽3) in GAP, Magma, Sage, TeX

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

G:=Group("Q8xSL(2,3)");
// GroupNames label

G:=SmallGroup(192,1007);
// by ID

G=gap.SmallGroup(192,1007);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,2,-2,84,197,92,438,172,775,285,124]);
// 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,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=c^-1,e*c*e^-1=d,e*d*e^-1=c*d>;
// generators/relations

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