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## G = C3×C7⋊C12order 252 = 22·32·7

### Direct product of C3 and C7⋊C12

Aliases: C3×C7⋊C12, C212C12, C6.4F7, C42.4C6, Dic7⋊C32, C7⋊C3⋊C12, C7⋊(C3×C12), C2.(C3×F7), C14.(C3×C6), (C3×Dic7)⋊C3, (C3×C7⋊C3)⋊2C4, (C2×C7⋊C3).C6, (C6×C7⋊C3).2C2, SmallGroup(252,16)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — C3×C7⋊C12
 Chief series C1 — C7 — C14 — C42 — C6×C7⋊C3 — C3×C7⋊C12
 Lower central C7 — C3×C7⋊C12
 Upper central C1 — C6

Generators and relations for C3×C7⋊C12
G = < a,b,c | a3=b7=c12=1, ab=ba, ac=ca, cbc-1=b5 >

Smallest permutation representation of C3×C7⋊C12
On 84 points
Generators in S84
(1 9 8)(2 10 5)(3 11 6)(4 12 7)(13 78 37)(14 79 38)(15 80 39)(16 81 40)(17 82 41)(18 83 42)(19 84 43)(20 73 44)(21 74 45)(22 75 46)(23 76 47)(24 77 48)(25 52 63)(26 53 64)(27 54 65)(28 55 66)(29 56 67)(30 57 68)(31 58 69)(32 59 70)(33 60 71)(34 49 72)(35 50 61)(36 51 62)
(1 41 37 59 45 51 55)(2 52 60 42 56 46 38)(3 47 43 53 39 57 49)(4 58 54 48 50 40 44)(5 25 33 83 29 75 79)(6 76 84 26 80 30 34)(7 31 27 77 35 81 73)(8 82 78 32 74 36 28)(9 17 13 70 21 62 66)(10 63 71 18 67 22 14)(11 23 19 64 15 68 72)(12 69 65 24 61 16 20)
(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)

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

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

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

42 conjugacy classes

 class 1 2 3A 3B 3C ··· 3H 4A 4B 6A 6B 6C ··· 6H 7 12A ··· 12P 14 21A 21B 42A 42B order 1 2 3 3 3 ··· 3 4 4 6 6 6 ··· 6 7 12 ··· 12 14 21 21 42 42 size 1 1 1 1 7 ··· 7 7 7 1 1 7 ··· 7 6 7 ··· 7 6 6 6 6 6

42 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 6 6 6 6 type + + + - image C1 C2 C3 C3 C4 C6 C6 C12 C12 F7 C7⋊C12 C3×F7 C3×C7⋊C12 kernel C3×C7⋊C12 C6×C7⋊C3 C7⋊C12 C3×Dic7 C3×C7⋊C3 C2×C7⋊C3 C42 C7⋊C3 C21 C6 C3 C2 C1 # reps 1 1 6 2 2 6 2 12 4 1 1 2 2

Matrix representation of C3×C7⋊C12 in GL7(𝔽337)

 128 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 0 0 0 0 336 0 1 0 0 0 0 336 0 0 1 0 0 0 336 0 0 0 1 0 0 336 0 0 0 0 1 0 336 0 0 0 0 0 1 336
,
 208 0 0 0 0 0 0 0 238 0 0 238 187 99 0 238 238 187 0 99 0 0 88 0 99 238 99 0 0 0 238 99 238 0 187 0 0 238 0 88 99 99 0 238 88 99 0 0 99

G:=sub<GL(7,GF(337))| [128,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,336,336,336,336,336,336],[208,0,0,0,0,0,0,0,238,238,88,0,0,238,0,0,238,0,238,238,88,0,0,187,99,99,0,99,0,238,0,238,238,88,0,0,187,99,99,0,99,0,0,99,0,0,187,99,99] >;

C3×C7⋊C12 in GAP, Magma, Sage, TeX

C_3\times C_7\rtimes C_{12}
% in TeX

G:=Group("C3xC7:C12");
// GroupNames label

G:=SmallGroup(252,16);
// by ID

G=gap.SmallGroup(252,16);
# by ID

G:=PCGroup([5,-2,-3,-3,-2,-7,90,5404,1809]);
// Polycyclic

G:=Group<a,b,c|a^3=b^7=c^12=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^5>;
// generators/relations

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