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## G = SL2(𝔽3)⋊3Q8order 192 = 26·3

### The semidirect product of SL2(𝔽3) and Q8 acting through Inn(SL2(𝔽3))

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C2×Q8 — SL2(𝔽3)⋊3Q8
 Chief series C1 — C2 — Q8 — C2×Q8 — C2×SL2(𝔽3) — C4×SL2(𝔽3) — SL2(𝔽3)⋊3Q8
 Lower central Q8 — C2×Q8 — SL2(𝔽3)⋊3Q8
 Upper central C1 — C22 — C4⋊C4

Generators and relations for SL2(𝔽3)⋊3Q8
G = < a,b,c,d,e | a4=c3=d4=1, b2=a2, e2=d2, bab-1=dad-1=a-1, cac-1=b, ae=ea, cbc-1=ab, bd=db, be=eb, dcd-1=a-1c, ce=ec, ede-1=d-1 >

Subgroups: 183 in 65 conjugacy classes, 23 normal (15 characteristic)
C1, C2, C3, C4, C4, C22, C6, C2×C4, C2×C4, C2×C4, Q8, Q8, C12, C2×C6, C42, C4⋊C4, C4⋊C4, C2×Q8, C2×Q8, SL2(𝔽3), C2×C12, C4×Q8, C4×Q8, C4×Q8, C42.C2, C4⋊Q8, C3×C4⋊C4, C2×SL2(𝔽3), Q83Q8, C4×SL2(𝔽3), C4×SL2(𝔽3), SL2(𝔽3)⋊3Q8
Quotients: C1, C2, C3, C22, C6, Q8, A4, C2×C6, C3×Q8, C2×A4, C4.A4, C22×A4, Q8×A4, C2×C4.A4, D4.A4, SL2(𝔽3)⋊3Q8

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

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

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

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

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 3 3 4 4 6 type + + - + + - - image C1 C2 C3 C6 Q8 C3×Q8 C4.A4 A4 C2×A4 D4.A4 D4.A4 Q8×A4 kernel SL2(𝔽3)⋊3Q8 C4×SL2(𝔽3) Q8⋊3Q8 C4×Q8 SL2(𝔽3) Q8 C4 C4⋊C4 C2×C4 C2 C2 C2 # reps 1 3 2 6 1 2 12 1 3 1 2 1

Matrix representation of SL2(𝔽3)⋊3Q8 in GL4(𝔽13) generated by

 0 12 0 0 1 0 0 0 0 0 1 0 0 0 0 1
,
 3 4 0 0 4 10 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 10 9 0 0 0 0 1 0 0 0 0 1
,
 2 7 0 0 7 11 0 0 0 0 0 1 0 0 12 0
,
 1 0 0 0 0 1 0 0 0 0 5 0 0 0 0 8
G:=sub<GL(4,GF(13))| [0,1,0,0,12,0,0,0,0,0,1,0,0,0,0,1],[3,4,0,0,4,10,0,0,0,0,1,0,0,0,0,1],[1,10,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[2,7,0,0,7,11,0,0,0,0,0,12,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,5,0,0,0,0,8] >;

SL2(𝔽3)⋊3Q8 in GAP, Magma, Sage, TeX

{\rm SL}_2({\mathbb F}_3)\rtimes_3Q_8
% in TeX

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

G:=SmallGroup(192,1006);
// by ID

G=gap.SmallGroup(192,1006);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,2,-2,84,197,764,438,172,775,285,124]);
// Polycyclic

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

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