metabelian, soluble, monomial, A-group
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
| C2×C14 — Dic7⋊A4 |
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 |
►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
Export
Subgroup lattice of Dic7⋊A4 in TeX
Character table of Dic7⋊A4 in TeX