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## G = SL2(𝔽3)⋊D4order 192 = 26·3

### 2nd semidirect product of SL2(𝔽3) and 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×GL2(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=e2=1, b2=a2, bab-1=ebe=a-1, cac-1=dad-1=b, eae=a2b, cbc-1=ab, dbd-1=a, dcd-1=ac-1, ece=c-1, ede=d-1 >

Subgroups: 397 in 83 conjugacy classes, 15 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, Dic3, C12, D6, C2×C6, C22⋊C4, C4⋊C4, C2×C8, D8, SD16, C22×C4, C2×D4, C2×Q8, C4○D4, SL2(𝔽3), C2×Dic3, C2×C12, C22×S3, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊D4, C2×D8, C2×SD16, C2×C4○D4, D6⋊C4, GL2(𝔽3), C2×SL2(𝔽3), C4.A4, D4⋊D4, Q8⋊Dic3, C2×GL2(𝔽3), C2×C4.A4, SL2(𝔽3)⋊D4
Quotients: C1, C2, C22, S3, D4, D6, C3⋊D4, S4, C2×S4, C4.6S4, C4.3S4, A4⋊D4, SL2(𝔽3)⋊D4

Character table of SL2(𝔽3)⋊D4

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 6A 6B 6C 8A 8B 8C 8D 12A 12B 12C 12D size 1 1 1 1 12 24 8 2 2 6 6 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 2 0 -1 2 2 2 2 0 -1 -1 -1 0 0 0 0 -1 -1 -1 -1 orthogonal lifted from S3 ρ6 2 2 2 2 -2 0 -1 -2 -2 2 2 0 -1 -1 -1 0 0 0 0 1 1 1 1 orthogonal lifted from D6 ρ7 2 2 -2 -2 0 0 2 0 0 -2 2 0 2 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ8 2 2 -2 -2 0 0 -1 0 0 -2 2 0 -1 1 1 0 0 0 0 √-3 √-3 -√-3 -√-3 complex lifted from C3⋊D4 ρ9 2 2 -2 -2 0 0 -1 0 0 -2 2 0 -1 1 1 0 0 0 0 -√-3 -√-3 √-3 √-3 complex lifted from C3⋊D4 ρ10 2 -2 2 -2 0 0 -1 -2i 2i 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 0 -1 -2i 2i 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 0 -1 2i -2i 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 0 -1 2i -2i 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 -1 0 -3 -3 -1 -1 1 0 0 0 -1 1 -1 1 0 0 0 0 orthogonal lifted from C2×S4 ρ15 3 3 3 3 -1 -1 0 3 3 -1 -1 -1 0 0 0 1 1 1 1 0 0 0 0 orthogonal lifted from S4 ρ16 3 3 3 3 1 1 0 -3 -3 -1 -1 -1 0 0 0 1 -1 1 -1 0 0 0 0 orthogonal lifted from C2×S4 ρ17 3 3 3 3 -1 1 0 3 3 -1 -1 1 0 0 0 -1 -1 -1 -1 0 0 0 0 orthogonal lifted from S4 ρ18 4 -4 -4 4 0 0 -2 0 0 0 0 0 2 2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from C4.3S4 ρ19 4 -4 -4 4 0 0 1 0 0 0 0 0 -1 -1 1 0 0 0 0 -√3 √3 √3 -√3 orthogonal lifted from C4.3S4 ρ20 4 -4 -4 4 0 0 1 0 0 0 0 0 -1 -1 1 0 0 0 0 √3 -√3 -√3 √3 orthogonal lifted from C4.3S4 ρ21 4 -4 4 -4 0 0 1 4i -4i 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 0 1 -4i 4i 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 0 2 -2 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 32 points
Generators in S32
(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)
(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)
(2 8 5)(4 6 7)(10 14 15)(12 16 13)(18 22 23)(20 24 21)(26 30 31)(28 32 29)
(1 22 9 32)(2 20 10 26)(3 24 11 30)(4 18 12 28)(5 23 15 29)(6 19 16 25)(7 21 13 31)(8 17 14 27)
(2 7)(4 5)(6 8)(10 13)(12 15)(14 16)(17 25)(18 29)(19 27)(20 31)(21 26)(22 32)(23 28)(24 30)

G:=sub<Sym(32)| (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), (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), (2,8,5)(4,6,7)(10,14,15)(12,16,13)(18,22,23)(20,24,21)(26,30,31)(28,32,29), (1,22,9,32)(2,20,10,26)(3,24,11,30)(4,18,12,28)(5,23,15,29)(6,19,16,25)(7,21,13,31)(8,17,14,27), (2,7)(4,5)(6,8)(10,13)(12,15)(14,16)(17,25)(18,29)(19,27)(20,31)(21,26)(22,32)(23,28)(24,30)>;

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), (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), (2,8,5)(4,6,7)(10,14,15)(12,16,13)(18,22,23)(20,24,21)(26,30,31)(28,32,29), (1,22,9,32)(2,20,10,26)(3,24,11,30)(4,18,12,28)(5,23,15,29)(6,19,16,25)(7,21,13,31)(8,17,14,27), (2,7)(4,5)(6,8)(10,13)(12,15)(14,16)(17,25)(18,29)(19,27)(20,31)(21,26)(22,32)(23,28)(24,30) );

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)], [(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)], [(2,8,5),(4,6,7),(10,14,15),(12,16,13),(18,22,23),(20,24,21),(26,30,31),(28,32,29)], [(1,22,9,32),(2,20,10,26),(3,24,11,30),(4,18,12,28),(5,23,15,29),(6,19,16,25),(7,21,13,31),(8,17,14,27)], [(2,7),(4,5),(6,8),(10,13),(12,15),(14,16),(17,25),(18,29),(19,27),(20,31),(21,26),(22,32),(23,28),(24,30)]])

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

 1 0 0 0 0 1 0 0 0 0 53 40 0 0 21 20
,
 1 0 0 0 0 1 0 0 0 0 41 52 0 0 21 32
,
 1 0 0 0 0 1 0 0 0 0 72 1 0 0 72 0
,
 0 72 0 0 1 0 0 0 0 0 29 61 0 0 58 44
,
 1 0 0 0 0 72 0 0 0 0 72 1 0 0 0 1
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,53,21,0,0,40,20],[1,0,0,0,0,1,0,0,0,0,41,21,0,0,52,32],[1,0,0,0,0,1,0,0,0,0,72,72,0,0,1,0],[0,1,0,0,72,0,0,0,0,0,29,58,0,0,61,44],[1,0,0,0,0,72,0,0,0,0,72,0,0,0,1,1] >;

SL2(𝔽3)⋊D4 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(192,986);
// by ID

G=gap.SmallGroup(192,986);
# by ID

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

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

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