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## G = SL2(𝔽3)⋊5D4order 192 = 26·3

### 1st semidirect product of SL2(𝔽3) and D4 acting through Inn(SL2(𝔽3))

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C2×Q8 — SL2(𝔽3)⋊5D4
 Chief series C1 — C2 — Q8 — C2×Q8 — C2×SL2(𝔽3) — C22×SL2(𝔽3) — SL2(𝔽3)⋊5D4
 Lower central Q8 — C2×Q8 — SL2(𝔽3)⋊5D4
 Upper central C1 — C22 — C22⋊C4

Generators and relations for SL2(𝔽3)⋊5D4
G = < a,b,c,d,e | a4=c3=d4=e2=1, b2=a2, 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=d-1 >

Subgroups: 315 in 90 conjugacy classes, 23 normal (21 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, C6, C2×C4, C2×C4, D4, Q8, Q8, C23, C23, C12, C2×C6, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2×Q8, C4○D4, SL2(𝔽3), C2×C12, C22×C6, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C4.4D4, C22×Q8, C2×C4○D4, C3×C22⋊C4, C2×SL2(𝔽3), C2×SL2(𝔽3), C4.A4, Q85D4, C4×SL2(𝔽3), C22×SL2(𝔽3), C2×C4.A4, SL2(𝔽3)⋊5D4
Quotients: C1, C2, C3, C22, C6, D4, A4, C2×C6, C3×D4, C2×A4, C4.A4, C22×A4, D4×A4, C2×C4.A4, D4.A4, SL2(𝔽3)⋊5D4

Smallest permutation representation of SL2(𝔽3)⋊5D4
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 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)
(2 6 7)(4 8 5)(10 14 15)(12 16 13)(18 22 23)(20 24 21)(26 30 31)(28 32 29)
(1 23 9 31)(2 22 10 30)(3 21 11 29)(4 24 12 32)(5 17 13 25)(6 20 14 28)(7 19 15 27)(8 18 16 26)
(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)

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

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

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

35 conjugacy classes

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

35 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 3 3 3 4 4 6 type + + + + + + + + - + image C1 C2 C2 C2 C3 C6 C6 C6 D4 C3×D4 C4.A4 A4 C2×A4 C2×A4 D4.A4 D4.A4 D4×A4 kernel SL2(𝔽3)⋊5D4 C4×SL2(𝔽3) C22×SL2(𝔽3) C2×C4.A4 Q8⋊5D4 C4×Q8 C22×Q8 C2×C4○D4 SL2(𝔽3) Q8 C22 C22⋊C4 C2×C4 C23 C2 C2 C2 # reps 1 1 1 1 2 2 2 2 1 2 12 1 2 1 1 2 1

Matrix representation of SL2(𝔽3)⋊5D4 in GL4(𝔽13) generated by

 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 3 4 0 0 4 10
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 10 9
,
 1 11 0 0 1 12 0 0 0 0 11 6 0 0 6 2
,
 1 0 0 0 1 12 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,0,1,0,0,12,0],[1,0,0,0,0,1,0,0,0,0,3,4,0,0,4,10],[1,0,0,0,0,1,0,0,0,0,1,10,0,0,0,9],[1,1,0,0,11,12,0,0,0,0,11,6,0,0,6,2],[1,1,0,0,0,12,0,0,0,0,1,0,0,0,0,1] >;

SL2(𝔽3)⋊5D4 in GAP, Magma, Sage, TeX

{\rm SL}_2({\mathbb F}_3)\rtimes_5D_4
% in TeX

G:=Group("SL(2,3):5D4");
// GroupNames label

G:=SmallGroup(192,1003);
// by ID

G=gap.SmallGroup(192,1003);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,2,-2,197,680,438,172,775,285,124]);
// 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=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=d^-1>;
// generators/relations

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