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G = S3×C28order 168 = 23·3·7

Direct product of C28 and S3

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: S3×C28, C846C2, D6.C14, C122C14, C14.14D6, Dic32C14, C42.19C22, C31(C2×C28), C216(C2×C4), C2.1(S3×C14), C6.2(C2×C14), (S3×C14).2C2, (C7×Dic3)⋊5C2, SmallGroup(168,30)

Series: Derived Chief Lower central Upper central

C1C3 — S3×C28
C1C3C6C42S3×C14 — S3×C28
C3 — S3×C28
C1C28

Generators and relations for S3×C28
 G = < a,b,c | a28=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

3C2
3C2
3C4
3C22
3C14
3C14
3C2×C4
3C28
3C2×C14
3C2×C28

Smallest permutation representation of S3×C28
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 79 46)(2 80 47)(3 81 48)(4 82 49)(5 83 50)(6 84 51)(7 57 52)(8 58 53)(9 59 54)(10 60 55)(11 61 56)(12 62 29)(13 63 30)(14 64 31)(15 65 32)(16 66 33)(17 67 34)(18 68 35)(19 69 36)(20 70 37)(21 71 38)(22 72 39)(23 73 40)(24 74 41)(25 75 42)(26 76 43)(27 77 44)(28 78 45)
(29 62)(30 63)(31 64)(32 65)(33 66)(34 67)(35 68)(36 69)(37 70)(38 71)(39 72)(40 73)(41 74)(42 75)(43 76)(44 77)(45 78)(46 79)(47 80)(48 81)(49 82)(50 83)(51 84)(52 57)(53 58)(54 59)(55 60)(56 61)

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

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

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

S3×C28 is a maximal subgroup of   D6.Dic7  D285S3  D84⋊C2  D6.D14

84 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D 6 7A···7F12A12B14A···14F14G···14R21A···21F28A···28L28M···28X42A···42F84A···84L
order12223444467···7121214···1414···1421···2128···2828···2842···4284···84
size11332113321···1221···13···32···21···13···32···22···2

84 irreducible representations

dim1111111111222222
type++++++
imageC1C2C2C2C4C7C14C14C14C28S3D6C4×S3S3×C7S3×C14S3×C28
kernelS3×C28C7×Dic3C84S3×C14S3×C7C4×S3Dic3C12D6S3C28C14C7C4C2C1
# reps111146666241126612

Matrix representation of S3×C28 in GL2(𝔽29) generated by

30
03
,
010
2628
,
280
31
G:=sub<GL(2,GF(29))| [3,0,0,3],[0,26,10,28],[28,3,0,1] >;

S3×C28 in GAP, Magma, Sage, TeX

S_3\times C_{28}
% in TeX

G:=Group("S3xC28");
// GroupNames label

G:=SmallGroup(168,30);
// by ID

G=gap.SmallGroup(168,30);
# by ID

G:=PCGroup([5,-2,-2,-7,-2,-3,146,2804]);
// Polycyclic

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

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Subgroup lattice of S3×C28 in TeX

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