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## G = SL2(𝔽3)⋊6D4order 192 = 26·3

### 2nd 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)⋊6D4
 Chief series C1 — C2 — Q8 — C2×Q8 — C2×SL2(𝔽3) — C2×C4.A4 — SL2(𝔽3)⋊6D4
 Lower central Q8 — C2×Q8 — SL2(𝔽3)⋊6D4
 Upper central C1 — C22 — C4⋊C4

Generators and relations for SL2(𝔽3)⋊6D4
G = < a,b,c,d,e | a4=c3=d4=e2=1, b2=a2, bab-1=eae=a-1, cac-1=b, ad=da, cbc-1=ab, bd=db, be=eb, cd=dc, ece=a-1c, ede=d-1 >

Subgroups: 323 in 87 conjugacy classes, 23 normal (15 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4 [×5], C22, C22 [×6], C6 [×3], C2×C4, C2×C4 [×2], C2×C4 [×6], D4 [×8], Q8, Q8, C23 [×2], C12 [×4], C2×C6, C42, C22⋊C4 [×2], C4⋊C4, C4⋊C4, C22×C4 [×2], C2×D4 [×5], C2×Q8, C4○D4 [×4], SL2(𝔽3), C2×C12 [×3], C4×D4, C4×Q8, C4⋊D4 [×2], C41D4, C2×C4○D4 [×2], C3×C4⋊C4, C2×SL2(𝔽3), C4.A4 [×2], Q86D4, C4×SL2(𝔽3), C2×C4.A4 [×2], SL2(𝔽3)⋊6D4
Quotients: C1, C2 [×3], C3, C22, C6 [×3], D4, A4, C2×C6, C3×D4, C2×A4 [×3], C4.A4 [×2], C22×A4, D4×A4, C2×C4.A4, Q8.A4, SL2(𝔽3)⋊6D4

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

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

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

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

35 conjugacy classes

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

35 irreducible representations

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

Matrix representation of SL2(𝔽3)⋊6D4 in GL4(𝔽13) generated by

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

SL2(𝔽3)⋊6D4 in GAP, Magma, Sage, TeX

{\rm SL}_2({\mathbb F}_3)\rtimes_6D_4
% in TeX

G:=Group("SL(2,3):6D4");
// GroupNames label

G:=SmallGroup(192,1005);
// by ID

G=gap.SmallGroup(192,1005);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,2,-2,672,197,680,438,172,775,285,124]);
// 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=e*a*e=a^-1,c*a*c^-1=b,a*d=d*a,c*b*c^-1=a*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=a^-1*c,e*d*e=d^-1>;
// generators/relations

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