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## G = C7×Dic3order 84 = 22·3·7

### Direct product of C7 and Dic3

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

 Derived series C1 — C3 — C7×Dic3
 Chief series C1 — C3 — C6 — C42 — C7×Dic3
 Lower central C3 — C7×Dic3
 Upper central C1 — C14

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

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

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

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

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

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

42 conjugacy classes

 class 1 2 3 4A 4B 6 7A ··· 7F 14A ··· 14F 21A ··· 21F 28A ··· 28L 42A ··· 42F order 1 2 3 4 4 6 7 ··· 7 14 ··· 14 21 ··· 21 28 ··· 28 42 ··· 42 size 1 1 2 3 3 2 1 ··· 1 1 ··· 1 2 ··· 2 3 ··· 3 2 ··· 2

42 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + - image C1 C2 C4 C7 C14 C28 S3 Dic3 S3×C7 C7×Dic3 kernel C7×Dic3 C42 C21 Dic3 C6 C3 C14 C7 C2 C1 # reps 1 1 2 6 6 12 1 1 6 6

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

 25 0 0 25
,
 3 19 21 27
,
 0 9 16 0
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

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