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G = C7×Dic3order 84 = 22·3·7

Direct product of C7 and Dic3

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

Aliases: C7×Dic3, C3⋊C28, C213C4, C6.C14, C42.3C2, C14.2S3, C2.(S3×C7), SmallGroup(84,3)

Series: Derived Chief Lower central Upper central

C1C3 — C7×Dic3
C1C3C6C42 — C7×Dic3
C3 — C7×Dic3
C1C14

Generators and relations for C7×Dic3
 G = < a,b,c | a7=b6=1, c2=b3, ab=ba, ac=ca, cbc-1=b-1 >

3C4
3C28

Smallest permutation representation of C7×Dic3
Regular action 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 65 56 57 44 77)(2 66 50 58 45 71)(3 67 51 59 46 72)(4 68 52 60 47 73)(5 69 53 61 48 74)(6 70 54 62 49 75)(7 64 55 63 43 76)(8 36 78 23 15 35)(9 37 79 24 16 29)(10 38 80 25 17 30)(11 39 81 26 18 31)(12 40 82 27 19 32)(13 41 83 28 20 33)(14 42 84 22 21 34)
(1 35 57 78)(2 29 58 79)(3 30 59 80)(4 31 60 81)(5 32 61 82)(6 33 62 83)(7 34 63 84)(8 56 23 77)(9 50 24 71)(10 51 25 72)(11 52 26 73)(12 53 27 74)(13 54 28 75)(14 55 22 76)(15 44 36 65)(16 45 37 66)(17 46 38 67)(18 47 39 68)(19 48 40 69)(20 49 41 70)(21 43 42 64)

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

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

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

C7×Dic3 is a maximal subgroup of   D21⋊C4  C3⋊D28  C21⋊Q8  S3×C28

42 conjugacy classes

class 1  2  3 4A4B 6 7A···7F14A···14F21A···21F28A···28L42A···42F
order1234467···714···1421···2128···2842···42
size1123321···11···12···23···32···2

42 irreducible representations

dim1111112222
type+++-
imageC1C2C4C7C14C28S3Dic3S3×C7C7×Dic3
kernelC7×Dic3C42C21Dic3C6C3C14C7C2C1
# reps11266121166

Matrix representation of C7×Dic3 in GL2(𝔽29) generated by

250
025
,
319
2127
,
09
160
G:=sub<GL(2,GF(29))| [25,0,0,25],[3,21,19,27],[0,16,9,0] >;

C7×Dic3 in GAP, Magma, Sage, TeX

C_7\times {\rm Dic}_3
% in TeX

G:=Group("C7xDic3");
// GroupNames label

G:=SmallGroup(84,3);
// by ID

G=gap.SmallGroup(84,3);
# by ID

G:=PCGroup([4,-2,-7,-2,-3,56,899]);
// Polycyclic

G:=Group<a,b,c|a^7=b^6=1,c^2=b^3,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C7×Dic3 in TeX

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