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## G = SD16×C3⋊S3order 288 = 25·32

### Direct product of SD16 and C3⋊S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — SD16×C3⋊S3
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C4×C3⋊S3 — D4×C3⋊S3 — SD16×C3⋊S3
 Lower central C32 — C3×C6 — C3×C12 — SD16×C3⋊S3
 Upper central C1 — C2 — C4 — SD16

Generators and relations for SD16×C3⋊S3
G = < a,b,c,d,e | a8=b2=c3=d3=e2=1, bab=a3, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=c-1, ede=d-1 >

Subgroups: 948 in 204 conjugacy classes, 57 normal (27 characteristic)
C1, C2, C2 [×4], C3 [×4], C4, C4 [×3], C22 [×5], S3 [×12], C6 [×4], C6 [×4], C8, C8, C2×C4 [×2], D4, D4 [×2], Q8, Q8 [×2], C23, C32, Dic3 [×8], C12 [×4], C12 [×4], D6 [×16], C2×C6 [×4], C2×C8, SD16, SD16 [×3], C2×D4, C2×Q8, C3⋊S3 [×2], C3⋊S3, C3×C6, C3×C6, C3⋊C8 [×4], C24 [×4], Dic6 [×8], C4×S3 [×8], D12 [×4], C3⋊D4 [×4], C3×D4 [×4], C3×Q8 [×4], C22×S3 [×4], C2×SD16, C3⋊Dic3, C3⋊Dic3, C3×C12, C3×C12, C2×C3⋊S3, C2×C3⋊S3 [×3], C62, S3×C8 [×4], C24⋊C2 [×4], D4.S3 [×4], Q82S3 [×4], C3×SD16 [×4], S3×D4 [×4], S3×Q8 [×4], C324C8, C3×C24, C324Q8, C324Q8, C4×C3⋊S3, C4×C3⋊S3, C12⋊S3, C327D4, D4×C32, Q8×C32, C22×C3⋊S3, S3×SD16 [×4], C8×C3⋊S3, C242S3, C329SD16, C3211SD16, C32×SD16, D4×C3⋊S3, Q8×C3⋊S3, SD16×C3⋊S3
Quotients: C1, C2 [×7], C22 [×7], S3 [×4], D4 [×2], C23, D6 [×12], SD16 [×2], C2×D4, C3⋊S3, C22×S3 [×4], C2×SD16, C2×C3⋊S3 [×3], S3×D4 [×4], C22×C3⋊S3, S3×SD16 [×4], D4×C3⋊S3, SD16×C3⋊S3

Smallest permutation representation of SD16×C3⋊S3
On 72 points
Generators in S72
(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)(65 66 67 68 69 70 71 72)
(2 4)(3 7)(6 8)(9 13)(10 16)(12 14)(17 23)(19 21)(20 24)(25 31)(27 29)(28 32)(33 37)(34 40)(36 38)(42 44)(43 47)(46 48)(49 51)(50 54)(53 55)(57 61)(58 64)(60 62)(65 67)(66 70)(69 71)
(1 15 41)(2 16 42)(3 9 43)(4 10 44)(5 11 45)(6 12 46)(7 13 47)(8 14 48)(17 67 62)(18 68 63)(19 69 64)(20 70 57)(21 71 58)(22 72 59)(23 65 60)(24 66 61)(25 51 38)(26 52 39)(27 53 40)(28 54 33)(29 55 34)(30 56 35)(31 49 36)(32 50 37)
(1 56 68)(2 49 69)(3 50 70)(4 51 71)(5 52 72)(6 53 65)(7 54 66)(8 55 67)(9 37 57)(10 38 58)(11 39 59)(12 40 60)(13 33 61)(14 34 62)(15 35 63)(16 36 64)(17 48 29)(18 41 30)(19 42 31)(20 43 32)(21 44 25)(22 45 26)(23 46 27)(24 47 28)
(9 43)(10 44)(11 45)(12 46)(13 47)(14 48)(15 41)(16 42)(17 34)(18 35)(19 36)(20 37)(21 38)(22 39)(23 40)(24 33)(25 58)(26 59)(27 60)(28 61)(29 62)(30 63)(31 64)(32 57)(49 69)(50 70)(51 71)(52 72)(53 65)(54 66)(55 67)(56 68)

G:=sub<Sym(72)| (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)(65,66,67,68,69,70,71,72), (2,4)(3,7)(6,8)(9,13)(10,16)(12,14)(17,23)(19,21)(20,24)(25,31)(27,29)(28,32)(33,37)(34,40)(36,38)(42,44)(43,47)(46,48)(49,51)(50,54)(53,55)(57,61)(58,64)(60,62)(65,67)(66,70)(69,71), (1,15,41)(2,16,42)(3,9,43)(4,10,44)(5,11,45)(6,12,46)(7,13,47)(8,14,48)(17,67,62)(18,68,63)(19,69,64)(20,70,57)(21,71,58)(22,72,59)(23,65,60)(24,66,61)(25,51,38)(26,52,39)(27,53,40)(28,54,33)(29,55,34)(30,56,35)(31,49,36)(32,50,37), (1,56,68)(2,49,69)(3,50,70)(4,51,71)(5,52,72)(6,53,65)(7,54,66)(8,55,67)(9,37,57)(10,38,58)(11,39,59)(12,40,60)(13,33,61)(14,34,62)(15,35,63)(16,36,64)(17,48,29)(18,41,30)(19,42,31)(20,43,32)(21,44,25)(22,45,26)(23,46,27)(24,47,28), (9,43)(10,44)(11,45)(12,46)(13,47)(14,48)(15,41)(16,42)(17,34)(18,35)(19,36)(20,37)(21,38)(22,39)(23,40)(24,33)(25,58)(26,59)(27,60)(28,61)(29,62)(30,63)(31,64)(32,57)(49,69)(50,70)(51,71)(52,72)(53,65)(54,66)(55,67)(56,68)>;

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)(65,66,67,68,69,70,71,72), (2,4)(3,7)(6,8)(9,13)(10,16)(12,14)(17,23)(19,21)(20,24)(25,31)(27,29)(28,32)(33,37)(34,40)(36,38)(42,44)(43,47)(46,48)(49,51)(50,54)(53,55)(57,61)(58,64)(60,62)(65,67)(66,70)(69,71), (1,15,41)(2,16,42)(3,9,43)(4,10,44)(5,11,45)(6,12,46)(7,13,47)(8,14,48)(17,67,62)(18,68,63)(19,69,64)(20,70,57)(21,71,58)(22,72,59)(23,65,60)(24,66,61)(25,51,38)(26,52,39)(27,53,40)(28,54,33)(29,55,34)(30,56,35)(31,49,36)(32,50,37), (1,56,68)(2,49,69)(3,50,70)(4,51,71)(5,52,72)(6,53,65)(7,54,66)(8,55,67)(9,37,57)(10,38,58)(11,39,59)(12,40,60)(13,33,61)(14,34,62)(15,35,63)(16,36,64)(17,48,29)(18,41,30)(19,42,31)(20,43,32)(21,44,25)(22,45,26)(23,46,27)(24,47,28), (9,43)(10,44)(11,45)(12,46)(13,47)(14,48)(15,41)(16,42)(17,34)(18,35)(19,36)(20,37)(21,38)(22,39)(23,40)(24,33)(25,58)(26,59)(27,60)(28,61)(29,62)(30,63)(31,64)(32,57)(49,69)(50,70)(51,71)(52,72)(53,65)(54,66)(55,67)(56,68) );

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),(65,66,67,68,69,70,71,72)], [(2,4),(3,7),(6,8),(9,13),(10,16),(12,14),(17,23),(19,21),(20,24),(25,31),(27,29),(28,32),(33,37),(34,40),(36,38),(42,44),(43,47),(46,48),(49,51),(50,54),(53,55),(57,61),(58,64),(60,62),(65,67),(66,70),(69,71)], [(1,15,41),(2,16,42),(3,9,43),(4,10,44),(5,11,45),(6,12,46),(7,13,47),(8,14,48),(17,67,62),(18,68,63),(19,69,64),(20,70,57),(21,71,58),(22,72,59),(23,65,60),(24,66,61),(25,51,38),(26,52,39),(27,53,40),(28,54,33),(29,55,34),(30,56,35),(31,49,36),(32,50,37)], [(1,56,68),(2,49,69),(3,50,70),(4,51,71),(5,52,72),(6,53,65),(7,54,66),(8,55,67),(9,37,57),(10,38,58),(11,39,59),(12,40,60),(13,33,61),(14,34,62),(15,35,63),(16,36,64),(17,48,29),(18,41,30),(19,42,31),(20,43,32),(21,44,25),(22,45,26),(23,46,27),(24,47,28)], [(9,43),(10,44),(11,45),(12,46),(13,47),(14,48),(15,41),(16,42),(17,34),(18,35),(19,36),(20,37),(21,38),(22,39),(23,40),(24,33),(25,58),(26,59),(27,60),(28,61),(29,62),(30,63),(31,64),(32,57),(49,69),(50,70),(51,71),(52,72),(53,65),(54,66),(55,67),(56,68)])

42 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F 6G 6H 8A 8B 8C 8D 12A 12B 12C 12D 12E 12F 12G 12H 24A ··· 24H order 1 2 2 2 2 2 3 3 3 3 4 4 4 4 6 6 6 6 6 6 6 6 8 8 8 8 12 12 12 12 12 12 12 12 24 ··· 24 size 1 1 4 9 9 36 2 2 2 2 2 4 18 36 2 2 2 2 8 8 8 8 2 2 18 18 4 4 4 4 8 8 8 8 4 ··· 4

42 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 D6 SD16 S3×D4 S3×SD16 kernel SD16×C3⋊S3 C8×C3⋊S3 C24⋊2S3 C32⋊9SD16 C32⋊11SD16 C32×SD16 D4×C3⋊S3 Q8×C3⋊S3 C3×SD16 C3⋊Dic3 C2×C3⋊S3 C24 C3×D4 C3×Q8 C3⋊S3 C6 C3 # reps 1 1 1 1 1 1 1 1 4 1 1 4 4 4 4 4 8

Matrix representation of SD16×C3⋊S3 in GL6(𝔽73)

 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 69 0 0 0 0 18 61 0 0 0 0 0 0 72 0 0 0 0 0 0 72
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 48 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 72 1 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 1 0 0 0 0 72 0
,
 72 1 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 72 0 0 0 0 1 72
,
 1 0 0 0 0 0 1 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 72 0 0 0 0 0 72

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,18,0,0,0,0,69,61,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,48,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[1,1,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,72,72] >;

SD16×C3⋊S3 in GAP, Magma, Sage, TeX

{\rm SD}_{16}\times C_3\rtimes S_3
% in TeX

G:=Group("SD16xC3:S3");
// GroupNames label

G:=SmallGroup(288,770);
// by ID

G=gap.SmallGroup(288,770);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,135,100,346,185,80,2693,9414]);
// Polycyclic

G:=Group<a,b,c,d,e|a^8=b^2=c^3=d^3=e^2=1,b*a*b=a^3,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations

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