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## G = A4×Dic7order 336 = 24·3·7

### Direct product of A4 and Dic7

Aliases: A4×Dic7, C73(C4×A4), (C7×A4)⋊2C4, C2.1(A4×D7), (C2×C14)⋊1C12, C23.(C3×D7), C14.9(C2×A4), (C2×A4).2D7, C22⋊(C3×Dic7), (A4×C14).2C2, (C22×C14).1C6, (C22×Dic7)⋊1C3, SmallGroup(336,133)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — A4×Dic7
 Chief series C1 — C7 — C2×C14 — C22×C14 — A4×C14 — A4×Dic7
 Lower central C2×C14 — A4×Dic7
 Upper central C1 — C2

Generators and relations for A4×Dic7
G = < a,b,c,d,e | a2=b2=c3=d14=1, e2=d7, cac-1=ab=ba, ad=da, ae=ea, cbc-1=a, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Smallest permutation representation of A4×Dic7
On 84 points
Generators in S84
(1 8)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(43 50)(44 51)(45 52)(46 53)(47 54)(48 55)(49 56)(71 78)(72 79)(73 80)(74 81)(75 82)(76 83)(77 84)
(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(21 28)(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(57 64)(58 65)(59 66)(60 67)(61 68)(62 69)(63 70)(71 78)(72 79)(73 80)(74 81)(75 82)(76 83)(77 84)
(1 42 28)(2 29 15)(3 30 16)(4 31 17)(5 32 18)(6 33 19)(7 34 20)(8 35 21)(9 36 22)(10 37 23)(11 38 24)(12 39 25)(13 40 26)(14 41 27)(43 78 64)(44 79 65)(45 80 66)(46 81 67)(47 82 68)(48 83 69)(49 84 70)(50 71 57)(51 72 58)(52 73 59)(53 74 60)(54 75 61)(55 76 62)(56 77 63)
(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 50 8 43)(2 49 9 56)(3 48 10 55)(4 47 11 54)(5 46 12 53)(6 45 13 52)(7 44 14 51)(15 70 22 63)(16 69 23 62)(17 68 24 61)(18 67 25 60)(19 66 26 59)(20 65 27 58)(21 64 28 57)(29 84 36 77)(30 83 37 76)(31 82 38 75)(32 81 39 74)(33 80 40 73)(34 79 41 72)(35 78 42 71)

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

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

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

40 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 6A 6B 7A 7B 7C 12A 12B 12C 12D 14A 14B 14C 14D ··· 14I 21A ··· 21F 42A ··· 42F order 1 2 2 2 3 3 4 4 4 4 6 6 7 7 7 12 12 12 12 14 14 14 14 ··· 14 21 ··· 21 42 ··· 42 size 1 1 3 3 4 4 7 7 21 21 4 4 2 2 2 28 28 28 28 2 2 2 6 ··· 6 8 ··· 8 8 ··· 8

40 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 3 3 3 6 6 type + + + - + + + - image C1 C2 C3 C4 C6 C12 D7 Dic7 C3×D7 C3×Dic7 A4 C2×A4 C4×A4 A4×D7 A4×Dic7 kernel A4×Dic7 A4×C14 C22×Dic7 C7×A4 C22×C14 C2×C14 C2×A4 A4 C23 C22 Dic7 C14 C7 C2 C1 # reps 1 1 2 2 2 4 3 3 6 6 1 1 2 3 3

Matrix representation of A4×Dic7 in GL5(𝔽337)

 1 0 0 0 0 0 1 0 0 0 0 0 336 0 0 0 0 0 1 0 0 0 0 131 336
,
 1 0 0 0 0 0 1 0 0 0 0 0 336 0 0 0 0 0 336 0 0 0 131 206 1
,
 128 0 0 0 0 0 128 0 0 0 0 0 209 128 227 0 0 209 0 0 0 0 0 0 128
,
 227 336 0 0 0 78 304 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 214 162 0 0 0 202 123 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1

G:=sub<GL(5,GF(337))| [1,0,0,0,0,0,1,0,0,0,0,0,336,0,0,0,0,0,1,131,0,0,0,0,336],[1,0,0,0,0,0,1,0,0,0,0,0,336,0,131,0,0,0,336,206,0,0,0,0,1],[128,0,0,0,0,0,128,0,0,0,0,0,209,209,0,0,0,128,0,0,0,0,227,0,128],[227,78,0,0,0,336,304,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[214,202,0,0,0,162,123,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1] >;

A4×Dic7 in GAP, Magma, Sage, TeX

A_4\times {\rm Dic}_7
% in TeX

G:=Group("A4xDic7");
// GroupNames label

G:=SmallGroup(336,133);
// by ID

G=gap.SmallGroup(336,133);
# by ID

G:=PCGroup([6,-2,-3,-2,-2,2,-7,36,441,190,10373]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^3=d^14=1,e^2=d^7,c*a*c^-1=a*b=b*a,a*d=d*a,a*e=e*a,c*b*c^-1=a,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations

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