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G = C2×C6×D7order 168 = 23·3·7

Direct product of C2×C6 and D7

Aliases: C2×C6×D7, C213C23, C423C22, C143(C2×C6), (C2×C42)⋊5C2, C73(C22×C6), (C2×C14)⋊11C6, SmallGroup(168,54)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — C2×C6×D7
 Chief series C1 — C7 — C21 — C3×D7 — C6×D7 — C2×C6×D7
 Lower central C7 — C2×C6×D7
 Upper central C1 — C2×C6

Generators and relations for C2×C6×D7
G = < a,b,c,d | a2=b6=c7=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 196 in 64 conjugacy classes, 42 normal (10 characteristic)
C1, C2, C2, C3, C22, C22, C6, C6, C7, C23, C2×C6, C2×C6, D7, C14, C21, C22×C6, D14, C2×C14, C3×D7, C42, C22×D7, C6×D7, C2×C42, C2×C6×D7
Quotients: C1, C2, C3, C22, C6, C23, C2×C6, D7, C22×C6, D14, C3×D7, C22×D7, C6×D7, C2×C6×D7

Smallest permutation representation of C2×C6×D7
On 84 points
Generators in S84
(1 69)(2 70)(3 64)(4 65)(5 66)(6 67)(7 68)(8 71)(9 72)(10 73)(11 74)(12 75)(13 76)(14 77)(15 78)(16 79)(17 80)(18 81)(19 82)(20 83)(21 84)(22 43)(23 44)(24 45)(25 46)(26 47)(27 48)(28 49)(29 50)(30 51)(31 52)(32 53)(33 54)(34 55)(35 56)(36 57)(37 58)(38 59)(39 60)(40 61)(41 62)(42 63)
(1 34 20 27 13 41)(2 35 21 28 14 42)(3 29 15 22 8 36)(4 30 16 23 9 37)(5 31 17 24 10 38)(6 32 18 25 11 39)(7 33 19 26 12 40)(43 71 57 64 50 78)(44 72 58 65 51 79)(45 73 59 66 52 80)(46 74 60 67 53 81)(47 75 61 68 54 82)(48 76 62 69 55 83)(49 77 63 70 56 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 26)(2 25)(3 24)(4 23)(5 22)(6 28)(7 27)(8 31)(9 30)(10 29)(11 35)(12 34)(13 33)(14 32)(15 38)(16 37)(17 36)(18 42)(19 41)(20 40)(21 39)(43 66)(44 65)(45 64)(46 70)(47 69)(48 68)(49 67)(50 73)(51 72)(52 71)(53 77)(54 76)(55 75)(56 74)(57 80)(58 79)(59 78)(60 84)(61 83)(62 82)(63 81)

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

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

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

C2×C6×D7 is a maximal subgroup of   D14⋊Dic3

60 conjugacy classes

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

60 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + + + image C1 C2 C2 C3 C6 C6 D7 D14 C3×D7 C6×D7 kernel C2×C6×D7 C6×D7 C2×C42 C22×D7 D14 C2×C14 C2×C6 C6 C22 C2 # reps 1 6 1 2 12 2 3 9 6 18

Matrix representation of C2×C6×D7 in GL3(𝔽43) generated by

 1 0 0 0 42 0 0 0 42
,
 7 0 0 0 42 0 0 0 42
,
 1 0 0 0 0 1 0 42 15
,
 1 0 0 0 0 42 0 42 0
G:=sub<GL(3,GF(43))| [1,0,0,0,42,0,0,0,42],[7,0,0,0,42,0,0,0,42],[1,0,0,0,0,42,0,1,15],[1,0,0,0,0,42,0,42,0] >;

C2×C6×D7 in GAP, Magma, Sage, TeX

C_2\times C_6\times D_7
% in TeX

G:=Group("C2xC6xD7");
// GroupNames label

G:=SmallGroup(168,54);
// by ID

G=gap.SmallGroup(168,54);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-7,3604]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^6=c^7=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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