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## G = SL2(𝔽3).D4order 192 = 26·3

### 2nd non-split extension by SL2(𝔽3) of D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — C2×SL2(𝔽3) — SL2(𝔽3).D4
 Chief series C1 — C2 — Q8 — SL2(𝔽3) — C2×SL2(𝔽3) — C2×CSU2(𝔽3) — SL2(𝔽3).D4
 Lower central SL2(𝔽3) — C2×SL2(𝔽3) — SL2(𝔽3).D4
 Upper central C1 — C22 — C2×C4

Generators and relations for SL2(𝔽3).D4
G = < a,b,c,d,e | a4=c3=d4=1, b2=e2=a2, bab-1=dbd-1=ebe-1=a-1, cac-1=b, dad-1=eae-1=a2b, cbc-1=ab, dcd-1=ece-1=c-1, ede-1=d-1 >

Subgroups: 293 in 74 conjugacy classes, 15 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, Dic3, C12, C2×C6, C22⋊C4, C4⋊C4, C2×C8, SD16, Q16, C22×C4, C2×D4, C2×Q8, C2×Q8, C4○D4, SL2(𝔽3), C2×Dic3, C2×C12, C22⋊C8, D4⋊C4, Q8⋊C4, C22⋊Q8, C2×SD16, C2×Q16, C2×C4○D4, Dic3⋊C4, CSU2(𝔽3), C2×SL2(𝔽3), C4.A4, D4.7D4, Q8⋊Dic3, C2×CSU2(𝔽3), C2×C4.A4, SL2(𝔽3).D4
Quotients: C1, C2, C22, S3, D4, D6, C3⋊D4, S4, C2×S4, C4.S4, C4.6S4, A4⋊D4, SL2(𝔽3).D4

Character table of SL2(𝔽3).D4

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 4E 4F 6A 6B 6C 8A 8B 8C 8D 12A 12B 12C 12D size 1 1 1 1 12 8 2 2 6 6 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 0 2 0 0 2 -2 0 0 2 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ6 2 2 2 2 2 -1 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 -1 -2 -2 2 2 0 0 -1 -1 -1 0 0 0 0 1 1 1 1 orthogonal lifted from D6 ρ8 2 -2 -2 2 0 -1 0 0 2 -2 0 0 -1 1 1 0 0 0 0 -√-3 -√-3 √-3 √-3 complex lifted from C3⋊D4 ρ9 2 -2 -2 2 0 -1 0 0 2 -2 0 0 -1 1 1 0 0 0 0 √-3 √-3 -√-3 -√-3 complex lifted from C3⋊D4 ρ10 2 2 -2 -2 0 -1 -2i 2i 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 0 -1 2i -2i 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 0 -1 2i -2i 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 0 -1 -2i 2i 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 -1 0 3 3 -1 -1 -1 -1 0 0 0 1 1 1 1 0 0 0 0 orthogonal lifted from S4 ρ15 3 3 3 3 1 0 -3 -3 -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 -1 0 3 3 -1 -1 1 1 0 0 0 -1 -1 -1 -1 0 0 0 0 orthogonal lifted from S4 ρ17 3 3 3 3 1 0 -3 -3 -1 -1 -1 1 0 0 0 1 -1 1 -1 0 0 0 0 orthogonal lifted from C2×S4 ρ18 4 -4 4 -4 0 -2 0 0 0 0 0 0 2 2 -2 0 0 0 0 0 0 0 0 symplectic lifted from C4.S4, Schur index 2 ρ19 4 -4 4 -4 0 1 0 0 0 0 0 0 -1 -1 1 0 0 0 0 √3 -√3 -√3 √3 symplectic lifted from C4.S4, Schur index 2 ρ20 4 -4 4 -4 0 1 0 0 0 0 0 0 -1 -1 1 0 0 0 0 -√3 √3 √3 -√3 symplectic lifted from C4.S4, Schur index 2 ρ21 4 4 -4 -4 0 1 4i -4i 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 0 1 -4i 4i 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 0 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from A4⋊D4

Smallest permutation representation of SL2(𝔽3).D4
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 5 3 7)(2 8 4 6)(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 38 35 40)(34 37 36 39)(41 46 43 48)(42 45 44 47)(49 54 51 56)(50 53 52 55)(57 62 59 64)(58 61 60 63)
(2 8 5)(4 6 7)(10 14 15)(12 16 13)(18 22 23)(20 24 21)(26 30 31)(28 32 29)(33 38 39)(35 40 37)(41 46 47)(43 48 45)(49 54 55)(51 56 53)(57 62 63)(59 64 61)
(1 60 11 52)(2 61 12 53)(3 58 9 50)(4 63 10 55)(5 59 13 51)(6 62 14 54)(7 57 15 49)(8 64 16 56)(17 44 25 36)(18 45 26 37)(19 42 27 34)(20 47 28 39)(21 41 29 33)(22 48 30 40)(23 43 31 35)(24 46 32 38)
(1 34 3 36)(2 39 4 37)(5 33 7 35)(6 40 8 38)(9 44 11 42)(10 45 12 47)(13 41 15 43)(14 48 16 46)(17 52 19 50)(18 53 20 55)(21 49 23 51)(22 56 24 54)(25 60 27 58)(26 61 28 63)(29 57 31 59)(30 64 32 62)

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

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

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

Matrix representation of SL2(𝔽3).D4 in GL4(𝔽73) generated by

 1 0 0 0 0 1 0 0 0 0 70 31 0 0 35 3
,
 1 0 0 0 0 1 0 0 0 0 41 38 0 0 71 32
,
 72 72 0 0 1 0 0 0 0 0 34 3 0 0 41 38
,
 13 43 0 0 30 60 0 0 0 0 45 42 0 0 70 28
,
 0 72 0 0 72 0 0 0 0 0 26 34 0 0 8 47
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,70,35,0,0,31,3],[1,0,0,0,0,1,0,0,0,0,41,71,0,0,38,32],[72,1,0,0,72,0,0,0,0,0,34,41,0,0,3,38],[13,30,0,0,43,60,0,0,0,0,45,70,0,0,42,28],[0,72,0,0,72,0,0,0,0,0,26,8,0,0,34,47] >;

SL2(𝔽3).D4 in GAP, Magma, Sage, TeX

{\rm SL}_2({\mathbb F}_3).D_4
% in TeX

G:=Group("SL(2,3).D4");
// GroupNames label

G:=SmallGroup(192,984);
// by ID

G=gap.SmallGroup(192,984);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,2,-2,672,85,680,451,1684,655,172,1013,404,285,124]);
// Polycyclic

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

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