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## G = C7×C3⋊D4order 168 = 23·3·7

### Direct product of C7 and C3⋊D4

Aliases: C7×C3⋊D4, C219D4, D62C14, Dic3⋊C14, C14.17D6, C42.22C22, C32(C7×D4), (C2×C14)⋊3S3, (C2×C42)⋊6C2, (C2×C6)⋊2C14, (S3×C14)⋊5C2, C2.5(S3×C14), C6.5(C2×C14), C222(S3×C7), (C7×Dic3)⋊4C2, SmallGroup(168,33)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C7×C3⋊D4
G = < a,b,c,d | a7=b3=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Smallest permutation representation of C7×C3⋊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 62 69)(2 63 70)(3 57 64)(4 58 65)(5 59 66)(6 60 67)(7 61 68)(8 40 32)(9 41 33)(10 42 34)(11 36 35)(12 37 29)(13 38 30)(14 39 31)(15 22 79)(16 23 80)(17 24 81)(18 25 82)(19 26 83)(20 27 84)(21 28 78)(43 75 51)(44 76 52)(45 77 53)(46 71 54)(47 72 55)(48 73 56)(49 74 50)
(1 78 45 35)(2 79 46 29)(3 80 47 30)(4 81 48 31)(5 82 49 32)(6 83 43 33)(7 84 44 34)(8 66 18 50)(9 67 19 51)(10 68 20 52)(11 69 21 53)(12 70 15 54)(13 64 16 55)(14 65 17 56)(22 71 37 63)(23 72 38 57)(24 73 39 58)(25 74 40 59)(26 75 41 60)(27 76 42 61)(28 77 36 62)
(8 25)(9 26)(10 27)(11 28)(12 22)(13 23)(14 24)(15 37)(16 38)(17 39)(18 40)(19 41)(20 42)(21 36)(29 79)(30 80)(31 81)(32 82)(33 83)(34 84)(35 78)(50 74)(51 75)(52 76)(53 77)(54 71)(55 72)(56 73)(57 64)(58 65)(59 66)(60 67)(61 68)(62 69)(63 70)

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

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

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

C7×C3⋊D4 is a maximal subgroup of   C42.C23  Dic3.D14  D6⋊D14  S3×C7×D4

63 conjugacy classes

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

63 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + image C1 C2 C2 C2 C7 C14 C14 C14 S3 D4 D6 C3⋊D4 S3×C7 C7×D4 S3×C14 C7×C3⋊D4 kernel C7×C3⋊D4 C7×Dic3 S3×C14 C2×C42 C3⋊D4 Dic3 D6 C2×C6 C2×C14 C21 C14 C7 C22 C3 C2 C1 # reps 1 1 1 1 6 6 6 6 1 1 1 2 6 6 6 12

Matrix representation of C7×C3⋊D4 in GL2(𝔽43) generated by

 41 0 0 41
,
 41 7 18 1
,
 13 27 16 30
,
 1 36 0 42
G:=sub<GL(2,GF(43))| [41,0,0,41],[41,18,7,1],[13,16,27,30],[1,0,36,42] >;

C7×C3⋊D4 in GAP, Magma, Sage, TeX

C_7\times C_3\rtimes D_4
% in TeX

G:=Group("C7xC3:D4");
// GroupNames label

G:=SmallGroup(168,33);
// by ID

G=gap.SmallGroup(168,33);
# by ID

G:=PCGroup([5,-2,-2,-7,-2,-3,301,2804]);
// Polycyclic

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

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