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## G = S3×C2×C14order 168 = 23·3·7

### Direct product of C2×C14 and S3

Aliases: S3×C2×C14, C214C23, C424C22, C6⋊(C2×C14), C3⋊(C22×C14), (C2×C6)⋊3C14, (C2×C42)⋊7C2, SmallGroup(168,55)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — S3×C2×C14
 Chief series C1 — C3 — C21 — S3×C7 — S3×C14 — S3×C2×C14
 Lower central C3 — S3×C2×C14
 Upper central C1 — C2×C14

Generators and relations for S3×C2×C14
G = < a,b,c,d | a2=b14=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 108 in 64 conjugacy classes, 42 normal (10 characteristic)
C1, C2, C2, C3, C22, C22, S3, C6, C7, C23, D6, C2×C6, C14, C14, C21, C22×S3, C2×C14, C2×C14, S3×C7, C42, C22×C14, S3×C14, C2×C42, S3×C2×C14
Quotients: C1, C2, C22, S3, C7, C23, D6, C14, C22×S3, C2×C14, S3×C7, C22×C14, S3×C14, S3×C2×C14

Smallest permutation representation of S3×C2×C14
On 84 points
Generators in S84
(1 25)(2 26)(3 27)(4 28)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(11 21)(12 22)(13 23)(14 24)(29 59)(30 60)(31 61)(32 62)(33 63)(34 64)(35 65)(36 66)(37 67)(38 68)(39 69)(40 70)(41 57)(42 58)(43 80)(44 81)(45 82)(46 83)(47 84)(48 71)(49 72)(50 73)(51 74)(52 75)(53 76)(54 77)(55 78)(56 79)
(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 50 61)(2 51 62)(3 52 63)(4 53 64)(5 54 65)(6 55 66)(7 56 67)(8 43 68)(9 44 69)(10 45 70)(11 46 57)(12 47 58)(13 48 59)(14 49 60)(15 77 35)(16 78 36)(17 79 37)(18 80 38)(19 81 39)(20 82 40)(21 83 41)(22 84 42)(23 71 29)(24 72 30)(25 73 31)(26 74 32)(27 75 33)(28 76 34)
(1 25)(2 26)(3 27)(4 28)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(11 21)(12 22)(13 23)(14 24)(29 48)(30 49)(31 50)(32 51)(33 52)(34 53)(35 54)(36 55)(37 56)(38 43)(39 44)(40 45)(41 46)(42 47)(57 83)(58 84)(59 71)(60 72)(61 73)(62 74)(63 75)(64 76)(65 77)(66 78)(67 79)(68 80)(69 81)(70 82)

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

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

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

S3×C2×C14 is a maximal subgroup of   D6⋊Dic7

84 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 6A 6B 6C 7A ··· 7F 14A ··· 14R 14S ··· 14AP 21A ··· 21F 42A ··· 42R order 1 2 2 2 2 2 2 2 3 6 6 6 7 ··· 7 14 ··· 14 14 ··· 14 21 ··· 21 42 ··· 42 size 1 1 1 1 3 3 3 3 2 2 2 2 1 ··· 1 1 ··· 1 3 ··· 3 2 ··· 2 2 ··· 2

84 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + + + image C1 C2 C2 C7 C14 C14 S3 D6 S3×C7 S3×C14 kernel S3×C2×C14 S3×C14 C2×C42 C22×S3 D6 C2×C6 C2×C14 C14 C22 C2 # reps 1 6 1 6 36 6 1 3 6 18

Matrix representation of S3×C2×C14 in GL4(𝔽43) generated by

 42 0 0 0 0 42 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 42 0 0 0 0 16 0 0 0 0 16
,
 1 0 0 0 0 1 0 0 0 0 42 42 0 0 1 0
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 42 42
G:=sub<GL(4,GF(43))| [42,0,0,0,0,42,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,42,0,0,0,0,16,0,0,0,0,16],[1,0,0,0,0,1,0,0,0,0,42,1,0,0,42,0],[1,0,0,0,0,1,0,0,0,0,1,42,0,0,0,42] >;

S3×C2×C14 in GAP, Magma, Sage, TeX

S_3\times C_2\times C_{14}
% in TeX

G:=Group("S3xC2xC14");
// GroupNames label

G:=SmallGroup(168,55);
// by ID

G=gap.SmallGroup(168,55);
# by ID

G:=PCGroup([5,-2,-2,-2,-7,-3,2804]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^14=c^3=d^2=1,a*b=b*a,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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