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## G = Dic7⋊C12order 336 = 24·3·7

### The semidirect product of Dic7 and C12 acting via C12/C2=C6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — Dic7⋊C12
 Chief series C1 — C7 — C14 — C2×C14 — C22×C7⋊C3 — C2×C7⋊C12 — Dic7⋊C12
 Lower central C7 — C14 — Dic7⋊C12
 Upper central C1 — C22 — C2×C4

Generators and relations for Dic7⋊C12
G = < a,b,c | a14=c12=1, b2=a7, bab-1=a-1, cac-1=a9, cbc-1=a7b >

Smallest permutation representation of Dic7⋊C12
On 112 points
Generators in S112
(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)(85 86 87 88 89 90 91 92 93 94 95 96 97 98)(99 100 101 102 103 104 105 106 107 108 109 110 111 112)
(1 64 8 57)(2 63 9 70)(3 62 10 69)(4 61 11 68)(5 60 12 67)(6 59 13 66)(7 58 14 65)(15 81 22 74)(16 80 23 73)(17 79 24 72)(18 78 25 71)(19 77 26 84)(20 76 27 83)(21 75 28 82)(29 94 36 87)(30 93 37 86)(31 92 38 85)(32 91 39 98)(33 90 40 97)(34 89 41 96)(35 88 42 95)(43 105 50 112)(44 104 51 111)(45 103 52 110)(46 102 53 109)(47 101 54 108)(48 100 55 107)(49 99 56 106)
(1 56 18 38)(2 53 27 39 12 51 19 35 10 43 15 33)(3 50 22 40 9 46 20 32 5 44 26 42)(4 47 17 41 6 55 21 29 14 45 23 37)(7 52 16 30 11 54 24 34 13 48 28 36)(8 49 25 31)(57 106 71 85)(58 103 80 86 68 101 72 96 66 107 82 94)(59 100 75 87 65 110 73 93 61 108 79 89)(60 111 84 88 62 105 74 90 70 109 76 98)(63 102 83 91 67 104 77 95 69 112 81 97)(64 99 78 92)

G:=sub<Sym(112)| (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)(85,86,87,88,89,90,91,92,93,94,95,96,97,98)(99,100,101,102,103,104,105,106,107,108,109,110,111,112), (1,64,8,57)(2,63,9,70)(3,62,10,69)(4,61,11,68)(5,60,12,67)(6,59,13,66)(7,58,14,65)(15,81,22,74)(16,80,23,73)(17,79,24,72)(18,78,25,71)(19,77,26,84)(20,76,27,83)(21,75,28,82)(29,94,36,87)(30,93,37,86)(31,92,38,85)(32,91,39,98)(33,90,40,97)(34,89,41,96)(35,88,42,95)(43,105,50,112)(44,104,51,111)(45,103,52,110)(46,102,53,109)(47,101,54,108)(48,100,55,107)(49,99,56,106), (1,56,18,38)(2,53,27,39,12,51,19,35,10,43,15,33)(3,50,22,40,9,46,20,32,5,44,26,42)(4,47,17,41,6,55,21,29,14,45,23,37)(7,52,16,30,11,54,24,34,13,48,28,36)(8,49,25,31)(57,106,71,85)(58,103,80,86,68,101,72,96,66,107,82,94)(59,100,75,87,65,110,73,93,61,108,79,89)(60,111,84,88,62,105,74,90,70,109,76,98)(63,102,83,91,67,104,77,95,69,112,81,97)(64,99,78,92)>;

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)(85,86,87,88,89,90,91,92,93,94,95,96,97,98)(99,100,101,102,103,104,105,106,107,108,109,110,111,112), (1,64,8,57)(2,63,9,70)(3,62,10,69)(4,61,11,68)(5,60,12,67)(6,59,13,66)(7,58,14,65)(15,81,22,74)(16,80,23,73)(17,79,24,72)(18,78,25,71)(19,77,26,84)(20,76,27,83)(21,75,28,82)(29,94,36,87)(30,93,37,86)(31,92,38,85)(32,91,39,98)(33,90,40,97)(34,89,41,96)(35,88,42,95)(43,105,50,112)(44,104,51,111)(45,103,52,110)(46,102,53,109)(47,101,54,108)(48,100,55,107)(49,99,56,106), (1,56,18,38)(2,53,27,39,12,51,19,35,10,43,15,33)(3,50,22,40,9,46,20,32,5,44,26,42)(4,47,17,41,6,55,21,29,14,45,23,37)(7,52,16,30,11,54,24,34,13,48,28,36)(8,49,25,31)(57,106,71,85)(58,103,80,86,68,101,72,96,66,107,82,94)(59,100,75,87,65,110,73,93,61,108,79,89)(60,111,84,88,62,105,74,90,70,109,76,98)(63,102,83,91,67,104,77,95,69,112,81,97)(64,99,78,92) );

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),(85,86,87,88,89,90,91,92,93,94,95,96,97,98),(99,100,101,102,103,104,105,106,107,108,109,110,111,112)], [(1,64,8,57),(2,63,9,70),(3,62,10,69),(4,61,11,68),(5,60,12,67),(6,59,13,66),(7,58,14,65),(15,81,22,74),(16,80,23,73),(17,79,24,72),(18,78,25,71),(19,77,26,84),(20,76,27,83),(21,75,28,82),(29,94,36,87),(30,93,37,86),(31,92,38,85),(32,91,39,98),(33,90,40,97),(34,89,41,96),(35,88,42,95),(43,105,50,112),(44,104,51,111),(45,103,52,110),(46,102,53,109),(47,101,54,108),(48,100,55,107),(49,99,56,106)], [(1,56,18,38),(2,53,27,39,12,51,19,35,10,43,15,33),(3,50,22,40,9,46,20,32,5,44,26,42),(4,47,17,41,6,55,21,29,14,45,23,37),(7,52,16,30,11,54,24,34,13,48,28,36),(8,49,25,31),(57,106,71,85),(58,103,80,86,68,101,72,96,66,107,82,94),(59,100,75,87,65,110,73,93,61,108,79,89),(60,111,84,88,62,105,74,90,70,109,76,98),(63,102,83,91,67,104,77,95,69,112,81,97),(64,99,78,92)]])

38 conjugacy classes

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

38 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 6 6 6 6 6 type + + + + - + + - image C1 C2 C2 C3 C4 C6 C6 C12 D4 Q8 C3×D4 C3×Q8 F7 C2×F7 C4.F7 C4×F7 Dic7⋊C6 kernel Dic7⋊C12 C2×C7⋊C12 C2×C4×C7⋊C3 Dic7⋊C4 C7⋊C12 C2×Dic7 C2×C28 Dic7 C2×C7⋊C3 C2×C7⋊C3 C14 C14 C2×C4 C22 C2 C2 C2 # reps 1 2 1 2 4 4 2 8 1 1 2 2 1 1 2 2 2

Matrix representation of Dic7⋊C12 in GL8(𝔽337)

 336 0 0 0 0 0 0 0 0 336 0 0 0 0 0 0 0 0 1 336 0 0 0 0 0 0 1 0 336 0 0 0 0 0 1 0 0 336 0 0 0 0 1 0 0 0 336 0 0 0 1 0 0 0 0 336 0 0 1 0 0 0 0 0
,
 49 235 0 0 0 0 0 0 235 288 0 0 0 0 0 0 0 0 0 266 71 266 0 181 0 0 266 0 0 266 181 71 0 0 0 266 0 110 71 71 0 0 266 266 181 0 71 0 0 0 266 110 71 0 0 71 0 0 110 0 71 266 71 0
,
 0 208 0 0 0 0 0 0 129 0 0 0 0 0 0 0 0 0 23 180 314 0 0 314 0 0 0 23 0 180 314 314 0 0 0 23 314 23 0 157 0 0 180 0 314 23 314 0 0 0 23 23 157 0 314 0 0 0 23 0 0 23 157 314

G:=sub<GL(8,GF(337))| [336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,1,1,1,1,1,1,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0],[49,235,0,0,0,0,0,0,235,288,0,0,0,0,0,0,0,0,0,266,0,266,266,110,0,0,266,0,266,266,110,0,0,0,71,0,0,181,71,71,0,0,266,266,110,0,0,266,0,0,0,181,71,71,0,71,0,0,181,71,71,0,71,0],[0,129,0,0,0,0,0,0,208,0,0,0,0,0,0,0,0,0,23,0,0,180,23,23,0,0,180,23,23,0,23,0,0,0,314,0,314,314,157,0,0,0,0,180,23,23,0,23,0,0,0,314,0,314,314,157,0,0,314,314,157,0,0,314] >;

Dic7⋊C12 in GAP, Magma, Sage, TeX

{\rm Dic}_7\rtimes C_{12}
% in TeX

G:=Group("Dic7:C12");
// GroupNames label

G:=SmallGroup(336,15);
// by ID

G=gap.SmallGroup(336,15);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-7,144,313,79,10373,1745]);
// Polycyclic

G:=Group<a,b,c|a^14=c^12=1,b^2=a^7,b*a*b^-1=a^-1,c*a*c^-1=a^9,c*b*c^-1=a^7*b>;
// generators/relations

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