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## G = Dic7⋊A4order 336 = 24·3·7

### The semidirect product of Dic7 and A4 acting via A4/C22=C3

Aliases: Dic7⋊A4, C23.F7, C7⋊A4⋊C4, C7⋊(C4×A4), C22⋊(C7⋊C12), (C2×C14)⋊2C12, C14.3(C2×A4), C2.1(D7⋊A4), (C22×C14).4C6, (C22×Dic7)⋊3C3, (C2×C7⋊A4).C2, SmallGroup(336,136)

Series: Derived Chief Lower central Upper central

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

Generators and relations for Dic7⋊A4
G = < a,b,c,d,e | a14=c2=d2=e3=1, b2=a7, bab-1=a-1, ac=ca, ad=da, eae-1=a9, bc=cb, bd=db, be=eb, ece-1=cd=dc, ede-1=c >

Character table of Dic7⋊A4

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 6A 6B 7 12A 12B 12C 12D 14A 14B 14C 14D 14E 14F 14G size 1 1 3 3 28 28 7 7 21 21 28 28 6 28 28 28 28 6 6 6 6 6 6 6 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 linear of order 2 ρ3 1 1 1 1 ζ32 ζ3 -1 -1 -1 -1 ζ3 ζ32 1 ζ65 ζ6 ζ6 ζ65 1 1 1 1 1 1 1 linear of order 6 ρ4 1 1 1 1 ζ32 ζ3 1 1 1 1 ζ3 ζ32 1 ζ3 ζ32 ζ32 ζ3 1 1 1 1 1 1 1 linear of order 3 ρ5 1 1 1 1 ζ3 ζ32 -1 -1 -1 -1 ζ32 ζ3 1 ζ6 ζ65 ζ65 ζ6 1 1 1 1 1 1 1 linear of order 6 ρ6 1 1 1 1 ζ3 ζ32 1 1 1 1 ζ32 ζ3 1 ζ32 ζ3 ζ3 ζ32 1 1 1 1 1 1 1 linear of order 3 ρ7 1 -1 1 -1 1 1 i -i -i i -1 -1 1 i -i i -i -1 -1 1 1 1 -1 -1 linear of order 4 ρ8 1 -1 1 -1 1 1 -i i i -i -1 -1 1 -i i -i i -1 -1 1 1 1 -1 -1 linear of order 4 ρ9 1 -1 1 -1 ζ3 ζ32 i -i -i i ζ6 ζ65 1 ζ4ζ32 ζ43ζ3 ζ4ζ3 ζ43ζ32 -1 -1 1 1 1 -1 -1 linear of order 12 ρ10 1 -1 1 -1 ζ32 ζ3 -i i i -i ζ65 ζ6 1 ζ43ζ3 ζ4ζ32 ζ43ζ32 ζ4ζ3 -1 -1 1 1 1 -1 -1 linear of order 12 ρ11 1 -1 1 -1 ζ3 ζ32 -i i i -i ζ6 ζ65 1 ζ43ζ32 ζ4ζ3 ζ43ζ3 ζ4ζ32 -1 -1 1 1 1 -1 -1 linear of order 12 ρ12 1 -1 1 -1 ζ32 ζ3 i -i -i i ζ65 ζ6 1 ζ4ζ3 ζ43ζ32 ζ4ζ32 ζ43ζ3 -1 -1 1 1 1 -1 -1 linear of order 12 ρ13 3 3 -1 -1 0 0 -3 -3 1 1 0 0 3 0 0 0 0 -1 3 -1 -1 -1 -1 -1 orthogonal lifted from C2×A4 ρ14 3 3 -1 -1 0 0 3 3 -1 -1 0 0 3 0 0 0 0 -1 3 -1 -1 -1 -1 -1 orthogonal lifted from A4 ρ15 3 -3 -1 1 0 0 3i -3i i -i 0 0 3 0 0 0 0 1 -3 -1 -1 -1 1 1 complex lifted from C4×A4 ρ16 3 -3 -1 1 0 0 -3i 3i -i i 0 0 3 0 0 0 0 1 -3 -1 -1 -1 1 1 complex lifted from C4×A4 ρ17 6 6 6 6 0 0 0 0 0 0 0 0 -1 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from F7 ρ18 6 6 -2 -2 0 0 0 0 0 0 0 0 -1 0 0 0 0 2ζ76+2ζ7+1 -1 2ζ74+2ζ73+1 2ζ76+2ζ7+1 2ζ75+2ζ72+1 2ζ75+2ζ72+1 2ζ74+2ζ73+1 orthogonal lifted from D7⋊A4 ρ19 6 6 -2 -2 0 0 0 0 0 0 0 0 -1 0 0 0 0 2ζ75+2ζ72+1 -1 2ζ76+2ζ7+1 2ζ75+2ζ72+1 2ζ74+2ζ73+1 2ζ74+2ζ73+1 2ζ76+2ζ7+1 orthogonal lifted from D7⋊A4 ρ20 6 6 -2 -2 0 0 0 0 0 0 0 0 -1 0 0 0 0 2ζ74+2ζ73+1 -1 2ζ75+2ζ72+1 2ζ74+2ζ73+1 2ζ76+2ζ7+1 2ζ76+2ζ7+1 2ζ75+2ζ72+1 orthogonal lifted from D7⋊A4 ρ21 6 -6 6 -6 0 0 0 0 0 0 0 0 -1 0 0 0 0 1 1 -1 -1 -1 1 1 symplectic lifted from C7⋊C12, Schur index 2 ρ22 6 -6 -2 2 0 0 0 0 0 0 0 0 -1 0 0 0 0 -2ζ75-2ζ72-1 1 2ζ76+2ζ7+1 2ζ75+2ζ72+1 2ζ74+2ζ73+1 -2ζ74-2ζ73-1 -2ζ76-2ζ7-1 symplectic faithful, Schur index 2 ρ23 6 -6 -2 2 0 0 0 0 0 0 0 0 -1 0 0 0 0 -2ζ74-2ζ73-1 1 2ζ75+2ζ72+1 2ζ74+2ζ73+1 2ζ76+2ζ7+1 -2ζ76-2ζ7-1 -2ζ75-2ζ72-1 symplectic faithful, Schur index 2 ρ24 6 -6 -2 2 0 0 0 0 0 0 0 0 -1 0 0 0 0 -2ζ76-2ζ7-1 1 2ζ74+2ζ73+1 2ζ76+2ζ7+1 2ζ75+2ζ72+1 -2ζ75-2ζ72-1 -2ζ74-2ζ73-1 symplectic faithful, Schur index 2

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

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

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

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

Matrix representation of Dic7⋊A4 in GL6(𝔽337)

 1 336 0 0 0 0 229 109 0 0 0 0 0 0 0 304 0 0 0 0 143 194 0 0 93 208 22 106 1 110 9 244 74 180 227 33
,
 46 300 0 0 0 0 39 291 0 0 0 0 0 0 46 300 0 0 0 0 39 291 0 0 134 0 238 0 300 302 150 282 196 18 328 37
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 336 0 0 0 0 0 0 336 0 0 121 237 0 0 336 0 121 237 0 0 0 336
,
 336 0 0 0 0 0 0 336 0 0 0 0 0 0 336 0 0 0 0 0 0 336 0 0 216 100 107 243 1 0 216 100 107 243 0 1
,
 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 336 1 36 129 74 209 335 227 0 0 0 0 128 0 1 0 0 0 128 0

G:=sub<GL(6,GF(337))| [1,229,0,0,93,9,336,109,0,0,208,244,0,0,0,143,22,74,0,0,304,194,106,180,0,0,0,0,1,227,0,0,0,0,110,33],[46,39,0,0,134,150,300,291,0,0,0,282,0,0,46,39,238,196,0,0,300,291,0,18,0,0,0,0,300,328,0,0,0,0,302,37],[1,0,0,0,121,121,0,1,0,0,237,237,0,0,336,0,0,0,0,0,0,336,0,0,0,0,0,0,336,0,0,0,0,0,0,336],[336,0,0,0,216,216,0,336,0,0,100,100,0,0,336,0,107,107,0,0,0,336,243,243,0,0,0,0,1,0,0,0,0,0,0,1],[0,0,0,36,0,1,0,0,0,129,0,0,1,0,0,74,0,0,0,1,0,209,0,0,0,0,336,335,128,128,0,0,1,227,0,0] >;

Dic7⋊A4 in GAP, Magma, Sage, TeX

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

G:=Group("Dic7:A4");
// GroupNames label

G:=SmallGroup(336,136);
// by ID

G=gap.SmallGroup(336,136);
# by ID

G:=PCGroup([6,-2,-3,-2,-2,2,-7,36,297,550,10373,3467]);
// Polycyclic

G:=Group<a,b,c,d,e|a^14=c^2=d^2=e^3=1,b^2=a^7,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,e*a*e^-1=a^9,b*c=c*b,b*d=d*b,b*e=e*b,e*c*e^-1=c*d=d*c,e*d*e^-1=c>;
// generators/relations

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