non-abelian, soluble, monomial
Aliases: A4:Dic7, C14.3S4, C23.D21, C22:Dic21, C7:(A4:C4), (C2xA4).D7, (C7xA4):1C4, C2.1(C7:S4), (A4xC14).1C2, (C2xC14):2Dic3, (C22xC14).2S3, SmallGroup(336,120)
Series: Derived ►Chief ►Lower central ►Upper central
| C7xA4 — A4:Dic7 |
Generators and relations for A4:Dic7
G = < a,b,c,d,e | a2=b2=c3=d14=1, e2=d7, cac-1=eae-1=ab=ba, ad=da, cbc-1=a, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=d-1 >
Quotients: C1, C2, C4, S3, Dic3, D7, S4, Dic7, D21, A4:C4, Dic21, C7:S4, A4:Dic7
Smallest permutation representation of A4:Dic7
►On 84 points
Generators in S84
(1 83)(2 84)(3 71)(4 72)(5 73)(6 74)(7 75)(8 76)(9 77)(10 78)(11 79)(12 80)(13 81)(14 82)(15 58)(16 59)(17 60)(18 61)(19 62)(20 63)(21 64)(22 65)(23 66)(24 67)(25 68)(26 69)(27 70)(28 57)(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)
(1 8)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(15 65)(16 66)(17 67)(18 68)(19 69)(20 70)(21 57)(22 58)(23 59)(24 60)(25 61)(26 62)(27 63)(28 64)(29 52)(30 53)(31 54)(32 55)(33 56)(34 43)(35 44)(36 45)(37 46)(38 47)(39 48)(40 49)(41 50)(42 51)(71 78)(72 79)(73 80)(74 81)(75 82)(76 83)(77 84)
(1 16 29)(2 17 30)(3 18 31)(4 19 32)(5 20 33)(6 21 34)(7 22 35)(8 23 36)(9 24 37)(10 25 38)(11 26 39)(12 27 40)(13 28 41)(14 15 42)(43 81 64)(44 82 65)(45 83 66)(46 84 67)(47 71 68)(48 72 69)(49 73 70)(50 74 57)(51 75 58)(52 76 59)(53 77 60)(54 78 61)(55 79 62)(56 80 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 76 8 83)(2 75 9 82)(3 74 10 81)(4 73 11 80)(5 72 12 79)(6 71 13 78)(7 84 14 77)(15 53 22 46)(16 52 23 45)(17 51 24 44)(18 50 25 43)(19 49 26 56)(20 48 27 55)(21 47 28 54)(29 59 36 66)(30 58 37 65)(31 57 38 64)(32 70 39 63)(33 69 40 62)(34 68 41 61)(35 67 42 60)
G:=sub<Sym(84)| (1,83)(2,84)(3,71)(4,72)(5,73)(6,74)(7,75)(8,76)(9,77)(10,78)(11,79)(12,80)(13,81)(14,82)(15,58)(16,59)(17,60)(18,61)(19,62)(20,63)(21,64)(22,65)(23,66)(24,67)(25,68)(26,69)(27,70)(28,57)(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), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,65)(16,66)(17,67)(18,68)(19,69)(20,70)(21,57)(22,58)(23,59)(24,60)(25,61)(26,62)(27,63)(28,64)(29,52)(30,53)(31,54)(32,55)(33,56)(34,43)(35,44)(36,45)(37,46)(38,47)(39,48)(40,49)(41,50)(42,51)(71,78)(72,79)(73,80)(74,81)(75,82)(76,83)(77,84), (1,16,29)(2,17,30)(3,18,31)(4,19,32)(5,20,33)(6,21,34)(7,22,35)(8,23,36)(9,24,37)(10,25,38)(11,26,39)(12,27,40)(13,28,41)(14,15,42)(43,81,64)(44,82,65)(45,83,66)(46,84,67)(47,71,68)(48,72,69)(49,73,70)(50,74,57)(51,75,58)(52,76,59)(53,77,60)(54,78,61)(55,79,62)(56,80,63), (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), (1,76,8,83)(2,75,9,82)(3,74,10,81)(4,73,11,80)(5,72,12,79)(6,71,13,78)(7,84,14,77)(15,53,22,46)(16,52,23,45)(17,51,24,44)(18,50,25,43)(19,49,26,56)(20,48,27,55)(21,47,28,54)(29,59,36,66)(30,58,37,65)(31,57,38,64)(32,70,39,63)(33,69,40,62)(34,68,41,61)(35,67,42,60)>;
G:=Group( (1,83)(2,84)(3,71)(4,72)(5,73)(6,74)(7,75)(8,76)(9,77)(10,78)(11,79)(12,80)(13,81)(14,82)(15,58)(16,59)(17,60)(18,61)(19,62)(20,63)(21,64)(22,65)(23,66)(24,67)(25,68)(26,69)(27,70)(28,57)(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), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,65)(16,66)(17,67)(18,68)(19,69)(20,70)(21,57)(22,58)(23,59)(24,60)(25,61)(26,62)(27,63)(28,64)(29,52)(30,53)(31,54)(32,55)(33,56)(34,43)(35,44)(36,45)(37,46)(38,47)(39,48)(40,49)(41,50)(42,51)(71,78)(72,79)(73,80)(74,81)(75,82)(76,83)(77,84), (1,16,29)(2,17,30)(3,18,31)(4,19,32)(5,20,33)(6,21,34)(7,22,35)(8,23,36)(9,24,37)(10,25,38)(11,26,39)(12,27,40)(13,28,41)(14,15,42)(43,81,64)(44,82,65)(45,83,66)(46,84,67)(47,71,68)(48,72,69)(49,73,70)(50,74,57)(51,75,58)(52,76,59)(53,77,60)(54,78,61)(55,79,62)(56,80,63), (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), (1,76,8,83)(2,75,9,82)(3,74,10,81)(4,73,11,80)(5,72,12,79)(6,71,13,78)(7,84,14,77)(15,53,22,46)(16,52,23,45)(17,51,24,44)(18,50,25,43)(19,49,26,56)(20,48,27,55)(21,47,28,54)(29,59,36,66)(30,58,37,65)(31,57,38,64)(32,70,39,63)(33,69,40,62)(34,68,41,61)(35,67,42,60) );
G=PermutationGroup([[(1,83),(2,84),(3,71),(4,72),(5,73),(6,74),(7,75),(8,76),(9,77),(10,78),(11,79),(12,80),(13,81),(14,82),(15,58),(16,59),(17,60),(18,61),(19,62),(20,63),(21,64),(22,65),(23,66),(24,67),(25,68),(26,69),(27,70),(28,57),(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)], [(1,8),(2,9),(3,10),(4,11),(5,12),(6,13),(7,14),(15,65),(16,66),(17,67),(18,68),(19,69),(20,70),(21,57),(22,58),(23,59),(24,60),(25,61),(26,62),(27,63),(28,64),(29,52),(30,53),(31,54),(32,55),(33,56),(34,43),(35,44),(36,45),(37,46),(38,47),(39,48),(40,49),(41,50),(42,51),(71,78),(72,79),(73,80),(74,81),(75,82),(76,83),(77,84)], [(1,16,29),(2,17,30),(3,18,31),(4,19,32),(5,20,33),(6,21,34),(7,22,35),(8,23,36),(9,24,37),(10,25,38),(11,26,39),(12,27,40),(13,28,41),(14,15,42),(43,81,64),(44,82,65),(45,83,66),(46,84,67),(47,71,68),(48,72,69),(49,73,70),(50,74,57),(51,75,58),(52,76,59),(53,77,60),(54,78,61),(55,79,62),(56,80,63)], [(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)], [(1,76,8,83),(2,75,9,82),(3,74,10,81),(4,73,11,80),(5,72,12,79),(6,71,13,78),(7,84,14,77),(15,53,22,46),(16,52,23,45),(17,51,24,44),(18,50,25,43),(19,49,26,56),(20,48,27,55),(21,47,28,54),(29,59,36,66),(30,58,37,65),(31,57,38,64),(32,70,39,63),(33,69,40,62),(34,68,41,61),(35,67,42,60)]])
34 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 6 | 7A | 7B | 7C | 14A | 14B | 14C | 14D | ··· | 14I | 21A | ··· | 21F | 42A | ··· | 42F |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 6 | 7 | 7 | 7 | 14 | 14 | 14 | 14 | ··· | 14 | 21 | ··· | 21 | 42 | ··· | 42 |
| size | 1 | 1 | 3 | 3 | 8 | 42 | 42 | 42 | 42 | 8 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | ··· | 6 | 8 | ··· | 8 | 8 | ··· | 8 |
34 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 6 | 6 |
| type | + | + | + | - | + | - | + | - | + | + | - | ||
| image | C1 | C2 | C4 | S3 | Dic3 | D7 | Dic7 | D21 | Dic21 | S4 | A4:C4 | C7:S4 | A4:Dic7 |
| kernel | A4:Dic7 | A4xC14 | C7xA4 | C22xC14 | C2xC14 | C2xA4 | A4 | C23 | C22 | C14 | C7 | C2 | C1 |
| # reps | 1 | 1 | 2 | 1 | 1 | 3 | 3 | 6 | 6 | 2 | 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 | 336 | 336 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 336 | 0 |
| 0 | 0 | 336 | 0 | 336 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 336 | 336 | 335 |
| 0 | 0 | 0 | 0 | 1 |
| 336 | 335 | 0 | 0 | 0 |
| 98 | 195 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 148 | 0 | 0 | 0 | 0 |
| 324 | 189 | 0 | 0 | 0 |
| 0 | 0 | 336 | 0 | 0 |
| 0 | 0 | 1 | 1 | 2 |
| 0 | 0 | 0 | 0 | 336 |
G:=sub<GL(5,GF(337))| [1,0,0,0,0,0,1,0,0,0,0,0,336,0,0,0,0,0,1,336,0,0,0,0,336],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,336,0,0,0,336,0,0,0,0,0,336],[1,0,0,0,0,0,1,0,0,0,0,0,0,336,0,0,0,1,336,0,0,0,0,335,1],[336,98,0,0,0,335,195,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[148,324,0,0,0,0,189,0,0,0,0,0,336,1,0,0,0,0,1,0,0,0,0,2,336] >;
A4:Dic7 in GAP, Magma, Sage, TeX
A_4\rtimes {\rm Dic}_7 % in TeX
G:=Group("A4:Dic7"); // GroupNames label
G:=SmallGroup(336,120);
// by ID
G=gap.SmallGroup(336,120);
# by ID
G:=PCGroup([6,-2,-2,-3,-7,-2,2,12,146,1731,5044,1276,3029,2285]);
// 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=e*a*e^-1=a*b=b*a,a*d=d*a,c*b*c^-1=a,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=d^-1>;
// generators/relations
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