metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D8:5Dic7, Q16:5Dic7, SD16:3Dic7, (C7xD8):3C4, C7:5(C8oD8), C7:C8.23D4, (C7xQ16):3C4, C4oD8.5D7, C56.20(C2xC4), (C8xDic7):2C2, (C7xSD16):4C4, C14.99(C4xD4), C4.217(D4xD7), C56.C4:8C2, C4oD4.22D14, (C2xC8).254D14, C28.376(C2xD4), Q8.Dic7:3C2, D4.3(C2xDic7), Q8.3(C2xDic7), C8.11(C2xDic7), C2.16(D4xDic7), D4:2Dic7:4C2, C28.77(C22xC4), (C2xC56).44C22, C4.7(C22xDic7), (C2xC28).467C23, C22.3(D4:2D7), C4.Dic7.22C22, (C4xDic7).246C22, (C7xC4oD8).2C2, (C7xD4).10(C2xC4), (C7xQ8).10(C2xC4), (C2xC7:C8).280C22, (C7xC4oD4).9C22, (C2xC14).11(C4oD4), (C2xC4).554(C22xD7), SmallGroup(448,730)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D8:5Dic7
G = < a,b,c,d | a8=b2=c14=1, d2=c7, bab=a-1, ac=ca, ad=da, cbc-1=a4b, dbd-1=a2b, dcd-1=c-1 >
Subgroups: 340 in 106 conjugacy classes, 53 normal (31 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2xC4, C2xC4, D4, D4, Q8, C14, C14, C42, C2xC8, C2xC8, M4(2), D8, SD16, Q16, C4oD4, Dic7, C28, C28, C2xC14, C2xC14, C4xC8, C4wrC2, C8.C4, C8oD4, C4oD8, C7:C8, C7:C8, C56, C2xDic7, C2xC28, C2xC28, C7xD4, C7xD4, C7xQ8, C8oD8, C2xC7:C8, C2xC7:C8, C4.Dic7, C4.Dic7, C4xDic7, C2xC56, C7xD8, C7xSD16, C7xQ16, C7xC4oD4, C8xDic7, C56.C4, D4:2Dic7, Q8.Dic7, C7xC4oD8, D8:5Dic7
Quotients: C1, C2, C4, C22, C2xC4, D4, C23, D7, C22xC4, C2xD4, C4oD4, Dic7, D14, C4xD4, C2xDic7, C22xD7, C8oD8, D4xD7, D4:2D7, C22xDic7, D4xDic7, D8:5Dic7
►On 112 points
Generators in S112
(1 34 12 55 26 39 20 48)(2 35 13 56 27 40 21 49)(3 29 14 50 28 41 15 43)(4 30 8 51 22 42 16 44)(5 31 9 52 23 36 17 45)(6 32 10 53 24 37 18 46)(7 33 11 54 25 38 19 47)(57 86 72 104 64 93 79 111)(58 87 73 105 65 94 80 112)(59 88 74 106 66 95 81 99)(60 89 75 107 67 96 82 100)(61 90 76 108 68 97 83 101)(62 91 77 109 69 98 84 102)(63 92 78 110 70 85 71 103)
(1 106)(2 100)(3 108)(4 102)(5 110)(6 104)(7 112)(8 98)(9 92)(10 86)(11 94)(12 88)(13 96)(14 90)(15 97)(16 91)(17 85)(18 93)(19 87)(20 95)(21 89)(22 109)(23 103)(24 111)(25 105)(26 99)(27 107)(28 101)(29 76)(30 84)(31 78)(32 72)(33 80)(34 74)(35 82)(36 71)(37 79)(38 73)(39 81)(40 75)(41 83)(42 77)(43 68)(44 62)(45 70)(46 64)(47 58)(48 66)(49 60)(50 61)(51 69)(52 63)(53 57)(54 65)(55 59)(56 67)
(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)
(2 7)(3 6)(4 5)(8 9)(10 14)(11 13)(15 18)(16 17)(19 21)(22 23)(24 28)(25 27)(29 32)(30 31)(33 35)(36 42)(37 41)(38 40)(43 46)(44 45)(47 49)(50 53)(51 52)(54 56)(57 76 64 83)(58 75 65 82)(59 74 66 81)(60 73 67 80)(61 72 68 79)(62 71 69 78)(63 84 70 77)(85 109 92 102)(86 108 93 101)(87 107 94 100)(88 106 95 99)(89 105 96 112)(90 104 97 111)(91 103 98 110)
G:=sub<Sym(112)| (1,34,12,55,26,39,20,48)(2,35,13,56,27,40,21,49)(3,29,14,50,28,41,15,43)(4,30,8,51,22,42,16,44)(5,31,9,52,23,36,17,45)(6,32,10,53,24,37,18,46)(7,33,11,54,25,38,19,47)(57,86,72,104,64,93,79,111)(58,87,73,105,65,94,80,112)(59,88,74,106,66,95,81,99)(60,89,75,107,67,96,82,100)(61,90,76,108,68,97,83,101)(62,91,77,109,69,98,84,102)(63,92,78,110,70,85,71,103), (1,106)(2,100)(3,108)(4,102)(5,110)(6,104)(7,112)(8,98)(9,92)(10,86)(11,94)(12,88)(13,96)(14,90)(15,97)(16,91)(17,85)(18,93)(19,87)(20,95)(21,89)(22,109)(23,103)(24,111)(25,105)(26,99)(27,107)(28,101)(29,76)(30,84)(31,78)(32,72)(33,80)(34,74)(35,82)(36,71)(37,79)(38,73)(39,81)(40,75)(41,83)(42,77)(43,68)(44,62)(45,70)(46,64)(47,58)(48,66)(49,60)(50,61)(51,69)(52,63)(53,57)(54,65)(55,59)(56,67), (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), (2,7)(3,6)(4,5)(8,9)(10,14)(11,13)(15,18)(16,17)(19,21)(22,23)(24,28)(25,27)(29,32)(30,31)(33,35)(36,42)(37,41)(38,40)(43,46)(44,45)(47,49)(50,53)(51,52)(54,56)(57,76,64,83)(58,75,65,82)(59,74,66,81)(60,73,67,80)(61,72,68,79)(62,71,69,78)(63,84,70,77)(85,109,92,102)(86,108,93,101)(87,107,94,100)(88,106,95,99)(89,105,96,112)(90,104,97,111)(91,103,98,110)>;
G:=Group( (1,34,12,55,26,39,20,48)(2,35,13,56,27,40,21,49)(3,29,14,50,28,41,15,43)(4,30,8,51,22,42,16,44)(5,31,9,52,23,36,17,45)(6,32,10,53,24,37,18,46)(7,33,11,54,25,38,19,47)(57,86,72,104,64,93,79,111)(58,87,73,105,65,94,80,112)(59,88,74,106,66,95,81,99)(60,89,75,107,67,96,82,100)(61,90,76,108,68,97,83,101)(62,91,77,109,69,98,84,102)(63,92,78,110,70,85,71,103), (1,106)(2,100)(3,108)(4,102)(5,110)(6,104)(7,112)(8,98)(9,92)(10,86)(11,94)(12,88)(13,96)(14,90)(15,97)(16,91)(17,85)(18,93)(19,87)(20,95)(21,89)(22,109)(23,103)(24,111)(25,105)(26,99)(27,107)(28,101)(29,76)(30,84)(31,78)(32,72)(33,80)(34,74)(35,82)(36,71)(37,79)(38,73)(39,81)(40,75)(41,83)(42,77)(43,68)(44,62)(45,70)(46,64)(47,58)(48,66)(49,60)(50,61)(51,69)(52,63)(53,57)(54,65)(55,59)(56,67), (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), (2,7)(3,6)(4,5)(8,9)(10,14)(11,13)(15,18)(16,17)(19,21)(22,23)(24,28)(25,27)(29,32)(30,31)(33,35)(36,42)(37,41)(38,40)(43,46)(44,45)(47,49)(50,53)(51,52)(54,56)(57,76,64,83)(58,75,65,82)(59,74,66,81)(60,73,67,80)(61,72,68,79)(62,71,69,78)(63,84,70,77)(85,109,92,102)(86,108,93,101)(87,107,94,100)(88,106,95,99)(89,105,96,112)(90,104,97,111)(91,103,98,110) );
G=PermutationGroup([[(1,34,12,55,26,39,20,48),(2,35,13,56,27,40,21,49),(3,29,14,50,28,41,15,43),(4,30,8,51,22,42,16,44),(5,31,9,52,23,36,17,45),(6,32,10,53,24,37,18,46),(7,33,11,54,25,38,19,47),(57,86,72,104,64,93,79,111),(58,87,73,105,65,94,80,112),(59,88,74,106,66,95,81,99),(60,89,75,107,67,96,82,100),(61,90,76,108,68,97,83,101),(62,91,77,109,69,98,84,102),(63,92,78,110,70,85,71,103)], [(1,106),(2,100),(3,108),(4,102),(5,110),(6,104),(7,112),(8,98),(9,92),(10,86),(11,94),(12,88),(13,96),(14,90),(15,97),(16,91),(17,85),(18,93),(19,87),(20,95),(21,89),(22,109),(23,103),(24,111),(25,105),(26,99),(27,107),(28,101),(29,76),(30,84),(31,78),(32,72),(33,80),(34,74),(35,82),(36,71),(37,79),(38,73),(39,81),(40,75),(41,83),(42,77),(43,68),(44,62),(45,70),(46,64),(47,58),(48,66),(49,60),(50,61),(51,69),(52,63),(53,57),(54,65),(55,59),(56,67)], [(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)], [(2,7),(3,6),(4,5),(8,9),(10,14),(11,13),(15,18),(16,17),(19,21),(22,23),(24,28),(25,27),(29,32),(30,31),(33,35),(36,42),(37,41),(38,40),(43,46),(44,45),(47,49),(50,53),(51,52),(54,56),(57,76,64,83),(58,75,65,82),(59,74,66,81),(60,73,67,80),(61,72,68,79),(62,71,69,78),(63,84,70,77),(85,109,92,102),(86,108,93,101),(87,107,94,100),(88,106,95,99),(89,105,96,112),(90,104,97,111),(91,103,98,110)]])
70 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 8K | 8L | 8M | 8N | 14A | 14B | 14C | 14D | 14E | 14F | 14G | ··· | 14L | 28A | ··· | 28F | 28G | 28H | 28I | 28J | ··· | 28O | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 14 | 14 | 14 | 14 | 14 | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | 28 | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 4 | 4 | 1 | 1 | 2 | 4 | 4 | 14 | 14 | 14 | 14 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 7 | 7 | 7 | 7 | 14 | 14 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | ··· | 8 | 2 | ··· | 2 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
70 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | - | - | + | + | - | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | D7 | C4oD4 | D14 | Dic7 | Dic7 | Dic7 | D14 | C8oD8 | D4xD7 | D4:2D7 | D8:5Dic7 |
| kernel | D8:5Dic7 | C8xDic7 | C56.C4 | D4:2Dic7 | Q8.Dic7 | C7xC4oD8 | C7xD8 | C7xSD16 | C7xQ16 | C7:C8 | C4oD8 | C2xC14 | C2xC8 | D8 | SD16 | Q16 | C4oD4 | C7 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 4 | 2 | 2 | 3 | 2 | 3 | 3 | 6 | 3 | 6 | 8 | 3 | 3 | 12 |
Matrix representation of D8:5Dic7 ►in GL4(F113) generated by
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 44 | 0 |
| 0 | 0 | 36 | 18 |
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 18 | 100 |
| 0 | 0 | 77 | 95 |
| 34 | 1 | 0 | 0 |
| 53 | 88 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 32 | 1 |
| 0 | 104 | 0 | 0 |
| 25 | 0 | 0 | 0 |
| 0 | 0 | 15 | 0 |
| 0 | 0 | 2 | 1 |
G:=sub<GL(4,GF(113))| [112,0,0,0,0,112,0,0,0,0,44,36,0,0,0,18],[112,0,0,0,0,112,0,0,0,0,18,77,0,0,100,95],[34,53,0,0,1,88,0,0,0,0,112,32,0,0,0,1],[0,25,0,0,104,0,0,0,0,0,15,2,0,0,0,1] >;
D8:5Dic7 in GAP, Magma, Sage, TeX
D_8\rtimes_5{\rm Dic}_7 % in TeX
G:=Group("D8:5Dic7"); // GroupNames label
G:=SmallGroup(448,730);
// by ID
G=gap.SmallGroup(448,730);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,219,136,851,438,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^14=1,d^2=c^7,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^4*b,d*b*d^-1=a^2*b,d*c*d^-1=c^-1>;
// generators/relations