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## G = C12×D7order 168 = 23·3·7

### Direct product of C12 and D7

Aliases: C12×D7, C844C2, C286C6, Dic75C6, D14.2C6, C6.14D14, C42.14C22, C215(C2×C4), C74(C2×C12), C2.1(C6×D7), (C6×D7).2C2, C14.10(C2×C6), (C3×Dic7)⋊5C2, SmallGroup(168,25)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — C12×D7
 Chief series C1 — C7 — C14 — C42 — C6×D7 — C12×D7
 Lower central C7 — C12×D7
 Upper central C1 — C12

Generators and relations for C12×D7
G = < a,b,c | a12=b7=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of C12×D7
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 18 47 55 82 62 36)(2 19 48 56 83 63 25)(3 20 37 57 84 64 26)(4 21 38 58 73 65 27)(5 22 39 59 74 66 28)(6 23 40 60 75 67 29)(7 24 41 49 76 68 30)(8 13 42 50 77 69 31)(9 14 43 51 78 70 32)(10 15 44 52 79 71 33)(11 16 45 53 80 72 34)(12 17 46 54 81 61 35)
(1 36)(2 25)(3 26)(4 27)(5 28)(6 29)(7 30)(8 31)(9 32)(10 33)(11 34)(12 35)(13 69)(14 70)(15 71)(16 72)(17 61)(18 62)(19 63)(20 64)(21 65)(22 66)(23 67)(24 68)(37 84)(38 73)(39 74)(40 75)(41 76)(42 77)(43 78)(44 79)(45 80)(46 81)(47 82)(48 83)

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

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

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

C12×D7 is a maximal subgroup of   C28.32D6  D6.D14  D125D7  D14.D6

60 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F 7A 7B 7C 12A 12B 12C 12D 12E 12F 12G 12H 14A 14B 14C 21A ··· 21F 28A ··· 28F 42A ··· 42F 84A ··· 84L order 1 2 2 2 3 3 4 4 4 4 6 6 6 6 6 6 7 7 7 12 12 12 12 12 12 12 12 14 14 14 21 ··· 21 28 ··· 28 42 ··· 42 84 ··· 84 size 1 1 7 7 1 1 1 1 7 7 1 1 7 7 7 7 2 2 2 1 1 1 1 7 7 7 7 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + image C1 C2 C2 C2 C3 C4 C6 C6 C6 C12 D7 D14 C3×D7 C4×D7 C6×D7 C12×D7 kernel C12×D7 C3×Dic7 C84 C6×D7 C4×D7 C3×D7 Dic7 C28 D14 D7 C12 C6 C4 C3 C2 C1 # reps 1 1 1 1 2 4 2 2 2 8 3 3 6 6 6 12

Matrix representation of C12×D7 in GL2(𝔽13) generated by

 7 0 0 7
,
 5 7 5 2
,
 11 2 5 2
G:=sub<GL(2,GF(13))| [7,0,0,7],[5,5,7,2],[11,5,2,2] >;

C12×D7 in GAP, Magma, Sage, TeX

C_{12}\times D_7
% in TeX

G:=Group("C12xD7");
// GroupNames label

G:=SmallGroup(168,25);
// by ID

G=gap.SmallGroup(168,25);
# by ID

G:=PCGroup([5,-2,-2,-3,-2,-7,66,3604]);
// Polycyclic

G:=Group<a,b,c|a^12=b^7=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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