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## G = D8×C3⋊S3order 288 = 25·32

### Direct product of D8 and C3⋊S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — D8×C3⋊S3
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C4×C3⋊S3 — D4×C3⋊S3 — D8×C3⋊S3
 Lower central C32 — C3×C6 — C3×C12 — D8×C3⋊S3
 Upper central C1 — C2 — C4 — D8

Generators and relations for D8×C3⋊S3
G = < a,b,c,d,e | a8=b2=c3=d3=e2=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=c-1, ede=d-1 >

Subgroups: 1236 in 228 conjugacy classes, 57 normal (17 characteristic)
C1, C2, C2 [×6], C3 [×4], C4, C4, C22 [×9], S3 [×16], C6 [×4], C6 [×8], C8, C8, C2×C4, D4 [×2], D4 [×4], C23 [×2], C32, Dic3 [×4], C12 [×4], D6 [×28], C2×C6 [×8], C2×C8, D8, D8 [×3], C2×D4 [×2], C3⋊S3 [×2], C3⋊S3 [×2], C3×C6, C3×C6 [×2], C3⋊C8 [×4], C24 [×4], C4×S3 [×4], D12 [×8], C3⋊D4 [×8], C3×D4 [×8], C22×S3 [×8], C2×D8, C3⋊Dic3, C3×C12, C2×C3⋊S3, C2×C3⋊S3 [×6], C62 [×2], S3×C8 [×4], D24 [×4], D4⋊S3 [×8], C3×D8 [×4], S3×D4 [×8], C324C8, C3×C24, C4×C3⋊S3, C12⋊S3 [×2], C327D4 [×2], D4×C32 [×2], C22×C3⋊S3 [×2], S3×D8 [×4], C8×C3⋊S3, C325D8, C327D8 [×2], C32×D8, D4×C3⋊S3 [×2], D8×C3⋊S3
Quotients: C1, C2 [×7], C22 [×7], S3 [×4], D4 [×2], C23, D6 [×12], D8 [×2], C2×D4, C3⋊S3, C22×S3 [×4], C2×D8, C2×C3⋊S3 [×3], S3×D4 [×4], C22×C3⋊S3, S3×D8 [×4], D4×C3⋊S3, D8×C3⋊S3

Smallest permutation representation of D8×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 8)(3 7)(4 6)(9 13)(10 12)(14 16)(17 23)(18 22)(19 21)(25 29)(26 28)(30 32)(33 35)(36 40)(37 39)(42 48)(43 47)(44 46)(49 51)(52 56)(53 55)(57 61)(58 60)(62 64)(65 69)(66 68)(70 72)
(1 41 20)(2 42 21)(3 43 22)(4 44 23)(5 45 24)(6 46 17)(7 47 18)(8 48 19)(9 25 61)(10 26 62)(11 27 63)(12 28 64)(13 29 57)(14 30 58)(15 31 59)(16 32 60)(33 70 53)(34 71 54)(35 72 55)(36 65 56)(37 66 49)(38 67 50)(39 68 51)(40 69 52)
(1 15 54)(2 16 55)(3 9 56)(4 10 49)(5 11 50)(6 12 51)(7 13 52)(8 14 53)(17 64 68)(18 57 69)(19 58 70)(20 59 71)(21 60 72)(22 61 65)(23 62 66)(24 63 67)(25 36 43)(26 37 44)(27 38 45)(28 39 46)(29 40 47)(30 33 48)(31 34 41)(32 35 42)
(9 56)(10 49)(11 50)(12 51)(13 52)(14 53)(15 54)(16 55)(17 46)(18 47)(19 48)(20 41)(21 42)(22 43)(23 44)(24 45)(25 65)(26 66)(27 67)(28 68)(29 69)(30 70)(31 71)(32 72)(33 58)(34 59)(35 60)(36 61)(37 62)(38 63)(39 64)(40 57)

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

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

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

42 conjugacy classes

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

42 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 D8 S3×D4 S3×D8 kernel D8×C3⋊S3 C8×C3⋊S3 C32⋊5D8 C32⋊7D8 C32×D8 D4×C3⋊S3 C3×D8 C3⋊Dic3 C2×C3⋊S3 C24 C3×D4 C3⋊S3 C6 C3 # reps 1 1 1 2 1 2 4 1 1 4 8 4 4 8

Matrix representation of D8×C3⋊S3 in GL6(𝔽73)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 48 0 0 0 0 38 41
,
 1 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 0 1 0 0 0 0 0 48 72
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 72 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 70 0 0 0 0 1 71 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 0 1
,
 72 3 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 72 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,38,0,0,0,0,48,41],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,48,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,1,0,0,0,0,70,71,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,0,0,0,0,0,3,1,0,0,0,0,0,0,1,72,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

D8×C3⋊S3 in GAP, Magma, Sage, TeX

D_8\times C_3\rtimes S_3
% in TeX

G:=Group("D8xC3:S3");
// GroupNames label

G:=SmallGroup(288,767);
// by ID

G=gap.SmallGroup(288,767);
# by ID

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

G:=Group<a,b,c,d,e|a^8=b^2=c^3=d^3=e^2=1,b*a*b=a^-1,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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