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## G = S3×D28order 336 = 24·3·7

### Direct product of S3 and D28

Series: Derived Chief Lower central Upper central

 Derived series C1 — C42 — S3×D28
 Chief series C1 — C7 — C21 — C42 — C6×D7 — C2×S3×D7 — S3×D28
 Lower central C21 — C42 — S3×D28
 Upper central C1 — C2 — C4

Generators and relations for S3×D28
G = < a,b,c,d | a3=b2=c28=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 804 in 108 conjugacy classes, 36 normal (22 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, S3, C6, C6, C7, C2×C4, D4, C23, Dic3, C12, D6, D6, C2×C6, D7, C14, C14, C2×D4, C21, C4×S3, D12, C3⋊D4, C3×D4, C22×S3, C28, C28, D14, D14, C2×C14, S3×C7, C3×D7, D21, C42, S3×D4, D28, D28, C2×C28, C22×D7, C7×Dic3, C84, S3×D7, C6×D7, S3×C14, D42, C2×D28, C3⋊D28, C3×D28, S3×C28, D84, C2×S3×D7, S3×D28
Quotients: C1, C2, C22, S3, D4, C23, D6, D7, C2×D4, C22×S3, D14, S3×D4, D28, C22×D7, S3×D7, C2×D28, C2×S3×D7, S3×D28

Smallest permutation representation of S3×D28
On 84 points
Generators in S84
(1 52 66)(2 53 67)(3 54 68)(4 55 69)(5 56 70)(6 29 71)(7 30 72)(8 31 73)(9 32 74)(10 33 75)(11 34 76)(12 35 77)(13 36 78)(14 37 79)(15 38 80)(16 39 81)(17 40 82)(18 41 83)(19 42 84)(20 43 57)(21 44 58)(22 45 59)(23 46 60)(24 47 61)(25 48 62)(26 49 63)(27 50 64)(28 51 65)
(29 71)(30 72)(31 73)(32 74)(33 75)(34 76)(35 77)(36 78)(37 79)(38 80)(39 81)(40 82)(41 83)(42 84)(43 57)(44 58)(45 59)(46 60)(47 61)(48 62)(49 63)(50 64)(51 65)(52 66)(53 67)(54 68)(55 69)(56 70)
(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 73 74 75 76 77 78 79 80 81 82 83 84)
(1 7)(2 6)(3 5)(8 28)(9 27)(10 26)(11 25)(12 24)(13 23)(14 22)(15 21)(16 20)(17 19)(29 53)(30 52)(31 51)(32 50)(33 49)(34 48)(35 47)(36 46)(37 45)(38 44)(39 43)(40 42)(54 56)(57 81)(58 80)(59 79)(60 78)(61 77)(62 76)(63 75)(64 74)(65 73)(66 72)(67 71)(68 70)(82 84)

G:=sub<Sym(84)| (1,52,66)(2,53,67)(3,54,68)(4,55,69)(5,56,70)(6,29,71)(7,30,72)(8,31,73)(9,32,74)(10,33,75)(11,34,76)(12,35,77)(13,36,78)(14,37,79)(15,38,80)(16,39,81)(17,40,82)(18,41,83)(19,42,84)(20,43,57)(21,44,58)(22,45,59)(23,46,60)(24,47,61)(25,48,62)(26,49,63)(27,50,64)(28,51,65), (29,71)(30,72)(31,73)(32,74)(33,75)(34,76)(35,77)(36,78)(37,79)(38,80)(39,81)(40,82)(41,83)(42,84)(43,57)(44,58)(45,59)(46,60)(47,61)(48,62)(49,63)(50,64)(51,65)(52,66)(53,67)(54,68)(55,69)(56,70), (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)(57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84), (1,7)(2,6)(3,5)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,20)(17,19)(29,53)(30,52)(31,51)(32,50)(33,49)(34,48)(35,47)(36,46)(37,45)(38,44)(39,43)(40,42)(54,56)(57,81)(58,80)(59,79)(60,78)(61,77)(62,76)(63,75)(64,74)(65,73)(66,72)(67,71)(68,70)(82,84)>;

G:=Group( (1,52,66)(2,53,67)(3,54,68)(4,55,69)(5,56,70)(6,29,71)(7,30,72)(8,31,73)(9,32,74)(10,33,75)(11,34,76)(12,35,77)(13,36,78)(14,37,79)(15,38,80)(16,39,81)(17,40,82)(18,41,83)(19,42,84)(20,43,57)(21,44,58)(22,45,59)(23,46,60)(24,47,61)(25,48,62)(26,49,63)(27,50,64)(28,51,65), (29,71)(30,72)(31,73)(32,74)(33,75)(34,76)(35,77)(36,78)(37,79)(38,80)(39,81)(40,82)(41,83)(42,84)(43,57)(44,58)(45,59)(46,60)(47,61)(48,62)(49,63)(50,64)(51,65)(52,66)(53,67)(54,68)(55,69)(56,70), (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)(57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84), (1,7)(2,6)(3,5)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,20)(17,19)(29,53)(30,52)(31,51)(32,50)(33,49)(34,48)(35,47)(36,46)(37,45)(38,44)(39,43)(40,42)(54,56)(57,81)(58,80)(59,79)(60,78)(61,77)(62,76)(63,75)(64,74)(65,73)(66,72)(67,71)(68,70)(82,84) );

G=PermutationGroup([[(1,52,66),(2,53,67),(3,54,68),(4,55,69),(5,56,70),(6,29,71),(7,30,72),(8,31,73),(9,32,74),(10,33,75),(11,34,76),(12,35,77),(13,36,78),(14,37,79),(15,38,80),(16,39,81),(17,40,82),(18,41,83),(19,42,84),(20,43,57),(21,44,58),(22,45,59),(23,46,60),(24,47,61),(25,48,62),(26,49,63),(27,50,64),(28,51,65)], [(29,71),(30,72),(31,73),(32,74),(33,75),(34,76),(35,77),(36,78),(37,79),(38,80),(39,81),(40,82),(41,83),(42,84),(43,57),(44,58),(45,59),(46,60),(47,61),(48,62),(49,63),(50,64),(51,65),(52,66),(53,67),(54,68),(55,69),(56,70)], [(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),(57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84)], [(1,7),(2,6),(3,5),(8,28),(9,27),(10,26),(11,25),(12,24),(13,23),(14,22),(15,21),(16,20),(17,19),(29,53),(30,52),(31,51),(32,50),(33,49),(34,48),(35,47),(36,46),(37,45),(38,44),(39,43),(40,42),(54,56),(57,81),(58,80),(59,79),(60,78),(61,77),(62,76),(63,75),(64,74),(65,73),(66,72),(67,71),(68,70),(82,84)]])

51 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 6A 6B 6C 7A 7B 7C 12 14A 14B 14C 14D ··· 14I 21A 21B 21C 28A ··· 28F 28G ··· 28L 42A 42B 42C 84A ··· 84F order 1 2 2 2 2 2 2 2 3 4 4 6 6 6 7 7 7 12 14 14 14 14 ··· 14 21 21 21 28 ··· 28 28 ··· 28 42 42 42 84 ··· 84 size 1 1 3 3 14 14 42 42 2 2 6 2 28 28 2 2 2 4 2 2 2 6 ··· 6 4 4 4 2 ··· 2 6 ··· 6 4 4 4 4 ··· 4

51 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 S3 D4 D6 D6 D7 D14 D14 D14 D28 S3×D4 S3×D7 C2×S3×D7 S3×D28 kernel S3×D28 C3⋊D28 C3×D28 S3×C28 D84 C2×S3×D7 D28 S3×C7 C28 D14 C4×S3 Dic3 C12 D6 S3 C7 C4 C2 C1 # reps 1 2 1 1 1 2 1 2 1 2 3 3 3 3 12 1 3 3 6

Matrix representation of S3×D28 in GL4(𝔽337) generated by

 1 0 0 0 0 1 0 0 0 0 335 141 0 0 43 1
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 294 336
,
 24 299 0 0 38 319 0 0 0 0 1 0 0 0 0 1
,
 109 143 0 0 228 228 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(337))| [1,0,0,0,0,1,0,0,0,0,335,43,0,0,141,1],[1,0,0,0,0,1,0,0,0,0,1,294,0,0,0,336],[24,38,0,0,299,319,0,0,0,0,1,0,0,0,0,1],[109,228,0,0,143,228,0,0,0,0,1,0,0,0,0,1] >;

S3×D28 in GAP, Magma, Sage, TeX

S_3\times D_{28}
% in TeX

G:=Group("S3xD28");
// GroupNames label

G:=SmallGroup(336,149);
// by ID

G=gap.SmallGroup(336,149);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-7,218,50,490,10373]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^2=c^28=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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