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## G = S3×C7×D4order 336 = 24·3·7

### Direct product of C7, S3 and D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — S3×C7×D4
 Chief series C1 — C3 — C6 — C42 — S3×C14 — S3×C2×C14 — S3×C7×D4
 Lower central C3 — C6 — S3×C7×D4
 Upper central C1 — C14 — C7×D4

Generators and relations for S3×C7×D4
G = < a,b,c,d,e | a7=b3=c2=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 240 in 108 conjugacy classes, 50 normal (26 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, S3, C6, C6, C7, C2×C4, D4, D4, C23, Dic3, C12, D6, D6, D6, C2×C6, C14, C14, C2×D4, C21, C4×S3, D12, C3⋊D4, C3×D4, C22×S3, C28, C28, C2×C14, C2×C14, S3×C7, S3×C7, C42, C42, S3×D4, C2×C28, C7×D4, C7×D4, C22×C14, C7×Dic3, C84, S3×C14, S3×C14, S3×C14, C2×C42, D4×C14, S3×C28, C7×D12, C7×C3⋊D4, D4×C21, S3×C2×C14, S3×C7×D4
Quotients: C1, C2, C22, S3, C7, D4, C23, D6, C14, C2×D4, C22×S3, C2×C14, S3×C7, S3×D4, C7×D4, C22×C14, S3×C14, D4×C14, S3×C2×C14, S3×C7×D4

Smallest permutation representation of S3×C7×D4
On 84 points
Generators in S84
(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 43 50)(2 44 51)(3 45 52)(4 46 53)(5 47 54)(6 48 55)(7 49 56)(8 15 78)(9 16 79)(10 17 80)(11 18 81)(12 19 82)(13 20 83)(14 21 84)(22 29 36)(23 30 37)(24 31 38)(25 32 39)(26 33 40)(27 34 41)(28 35 42)(57 64 71)(58 65 72)(59 66 73)(60 67 74)(61 68 75)(62 69 76)(63 70 77)
(8 15)(9 16)(10 17)(11 18)(12 19)(13 20)(14 21)(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(43 50)(44 51)(45 52)(46 53)(47 54)(48 55)(49 56)(64 71)(65 72)(66 73)(67 74)(68 75)(69 76)(70 77)
(1 78 57 22)(2 79 58 23)(3 80 59 24)(4 81 60 25)(5 82 61 26)(6 83 62 27)(7 84 63 28)(8 64 29 43)(9 65 30 44)(10 66 31 45)(11 67 32 46)(12 68 33 47)(13 69 34 48)(14 70 35 49)(15 71 36 50)(16 72 37 51)(17 73 38 52)(18 74 39 53)(19 75 40 54)(20 76 41 55)(21 77 42 56)
(8 29)(9 30)(10 31)(11 32)(12 33)(13 34)(14 35)(15 36)(16 37)(17 38)(18 39)(19 40)(20 41)(21 42)(22 78)(23 79)(24 80)(25 81)(26 82)(27 83)(28 84)

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

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

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

105 conjugacy classes

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

105 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C7 C14 C14 C14 C14 C14 S3 D4 D6 D6 S3×C7 C7×D4 S3×C14 S3×C14 S3×D4 S3×C7×D4 kernel S3×C7×D4 S3×C28 C7×D12 C7×C3⋊D4 D4×C21 S3×C2×C14 S3×D4 C4×S3 D12 C3⋊D4 C3×D4 C22×S3 C7×D4 S3×C7 C28 C2×C14 D4 S3 C4 C22 C7 C1 # reps 1 1 1 2 1 2 6 6 6 12 6 12 1 2 1 2 6 12 6 12 1 6

Matrix representation of S3×C7×D4 in GL4(𝔽337) generated by

 295 0 0 0 0 295 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 0 336 0 0 1 336
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0
,
 0 336 0 0 1 0 0 0 0 0 336 0 0 0 0 336
,
 1 0 0 0 0 336 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(337))| [295,0,0,0,0,295,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,336,336],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0],[0,1,0,0,336,0,0,0,0,0,336,0,0,0,0,336],[1,0,0,0,0,336,0,0,0,0,1,0,0,0,0,1] >;

S3×C7×D4 in GAP, Magma, Sage, TeX

S_3\times C_7\times D_4
% in TeX

G:=Group("S3xC7xD4");
// GroupNames label

G:=SmallGroup(336,188);
// by ID

G=gap.SmallGroup(336,188);
# by ID

G:=PCGroup([6,-2,-2,-2,-7,-2,-3,548,8069]);
// Polycyclic

G:=Group<a,b,c,d,e|a^7=b^3=c^2=d^4=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

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