metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D8.D7, C7⋊2SD32, C28.4D4, C14.9D8, C8.5D14, Dic28⋊3C2, C56.3C22, C7⋊C16⋊2C2, (C7×D8).1C2, C2.5(D4⋊D7), C4.2(C7⋊D4), SmallGroup(224,33)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D8.D7
G = < a,b,c,d | a8=b2=c7=1, d2=a4, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=a5b, dcd-1=c-1 >
Smallest permutation representation of D8.D7
►On 112 points
Generators in S112
(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)
(1 8)(2 7)(3 6)(4 5)(10 16)(11 15)(12 14)(17 23)(18 22)(19 21)(25 27)(28 32)(29 31)(34 40)(35 39)(36 38)(41 45)(42 44)(46 48)(49 51)(52 56)(53 55)(57 58)(59 64)(60 63)(61 62)(65 72)(66 71)(67 70)(68 69)(73 76)(74 75)(77 80)(78 79)(81 82)(83 88)(84 87)(85 86)(89 94)(90 93)(91 92)(95 96)(97 100)(98 99)(101 104)(102 103)(105 109)(106 108)(110 112)
(1 65 62 103 92 79 82)(2 66 63 104 93 80 83)(3 67 64 97 94 73 84)(4 68 57 98 95 74 85)(5 69 58 99 96 75 86)(6 70 59 100 89 76 87)(7 71 60 101 90 77 88)(8 72 61 102 91 78 81)(9 50 47 107 30 33 20)(10 51 48 108 31 34 21)(11 52 41 109 32 35 22)(12 53 42 110 25 36 23)(13 54 43 111 26 37 24)(14 55 44 112 27 38 17)(15 56 45 105 28 39 18)(16 49 46 106 29 40 19)
(1 44 5 48)(2 43 6 47)(3 42 7 46)(4 41 8 45)(9 63 13 59)(10 62 14 58)(11 61 15 57)(12 60 16 64)(17 99 21 103)(18 98 22 102)(19 97 23 101)(20 104 24 100)(25 77 29 73)(26 76 30 80)(27 75 31 79)(28 74 32 78)(33 93 37 89)(34 92 38 96)(35 91 39 95)(36 90 40 94)(49 67 53 71)(50 66 54 70)(51 65 55 69)(52 72 56 68)(81 105 85 109)(82 112 86 108)(83 111 87 107)(84 110 88 106)
G:=sub<Sym(112)| (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), (1,8)(2,7)(3,6)(4,5)(10,16)(11,15)(12,14)(17,23)(18,22)(19,21)(25,27)(28,32)(29,31)(34,40)(35,39)(36,38)(41,45)(42,44)(46,48)(49,51)(52,56)(53,55)(57,58)(59,64)(60,63)(61,62)(65,72)(66,71)(67,70)(68,69)(73,76)(74,75)(77,80)(78,79)(81,82)(83,88)(84,87)(85,86)(89,94)(90,93)(91,92)(95,96)(97,100)(98,99)(101,104)(102,103)(105,109)(106,108)(110,112), (1,65,62,103,92,79,82)(2,66,63,104,93,80,83)(3,67,64,97,94,73,84)(4,68,57,98,95,74,85)(5,69,58,99,96,75,86)(6,70,59,100,89,76,87)(7,71,60,101,90,77,88)(8,72,61,102,91,78,81)(9,50,47,107,30,33,20)(10,51,48,108,31,34,21)(11,52,41,109,32,35,22)(12,53,42,110,25,36,23)(13,54,43,111,26,37,24)(14,55,44,112,27,38,17)(15,56,45,105,28,39,18)(16,49,46,106,29,40,19), (1,44,5,48)(2,43,6,47)(3,42,7,46)(4,41,8,45)(9,63,13,59)(10,62,14,58)(11,61,15,57)(12,60,16,64)(17,99,21,103)(18,98,22,102)(19,97,23,101)(20,104,24,100)(25,77,29,73)(26,76,30,80)(27,75,31,79)(28,74,32,78)(33,93,37,89)(34,92,38,96)(35,91,39,95)(36,90,40,94)(49,67,53,71)(50,66,54,70)(51,65,55,69)(52,72,56,68)(81,105,85,109)(82,112,86,108)(83,111,87,107)(84,110,88,106)>;
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)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112), (1,8)(2,7)(3,6)(4,5)(10,16)(11,15)(12,14)(17,23)(18,22)(19,21)(25,27)(28,32)(29,31)(34,40)(35,39)(36,38)(41,45)(42,44)(46,48)(49,51)(52,56)(53,55)(57,58)(59,64)(60,63)(61,62)(65,72)(66,71)(67,70)(68,69)(73,76)(74,75)(77,80)(78,79)(81,82)(83,88)(84,87)(85,86)(89,94)(90,93)(91,92)(95,96)(97,100)(98,99)(101,104)(102,103)(105,109)(106,108)(110,112), (1,65,62,103,92,79,82)(2,66,63,104,93,80,83)(3,67,64,97,94,73,84)(4,68,57,98,95,74,85)(5,69,58,99,96,75,86)(6,70,59,100,89,76,87)(7,71,60,101,90,77,88)(8,72,61,102,91,78,81)(9,50,47,107,30,33,20)(10,51,48,108,31,34,21)(11,52,41,109,32,35,22)(12,53,42,110,25,36,23)(13,54,43,111,26,37,24)(14,55,44,112,27,38,17)(15,56,45,105,28,39,18)(16,49,46,106,29,40,19), (1,44,5,48)(2,43,6,47)(3,42,7,46)(4,41,8,45)(9,63,13,59)(10,62,14,58)(11,61,15,57)(12,60,16,64)(17,99,21,103)(18,98,22,102)(19,97,23,101)(20,104,24,100)(25,77,29,73)(26,76,30,80)(27,75,31,79)(28,74,32,78)(33,93,37,89)(34,92,38,96)(35,91,39,95)(36,90,40,94)(49,67,53,71)(50,66,54,70)(51,65,55,69)(52,72,56,68)(81,105,85,109)(82,112,86,108)(83,111,87,107)(84,110,88,106) );
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),(81,82,83,84,85,86,87,88),(89,90,91,92,93,94,95,96),(97,98,99,100,101,102,103,104),(105,106,107,108,109,110,111,112)], [(1,8),(2,7),(3,6),(4,5),(10,16),(11,15),(12,14),(17,23),(18,22),(19,21),(25,27),(28,32),(29,31),(34,40),(35,39),(36,38),(41,45),(42,44),(46,48),(49,51),(52,56),(53,55),(57,58),(59,64),(60,63),(61,62),(65,72),(66,71),(67,70),(68,69),(73,76),(74,75),(77,80),(78,79),(81,82),(83,88),(84,87),(85,86),(89,94),(90,93),(91,92),(95,96),(97,100),(98,99),(101,104),(102,103),(105,109),(106,108),(110,112)], [(1,65,62,103,92,79,82),(2,66,63,104,93,80,83),(3,67,64,97,94,73,84),(4,68,57,98,95,74,85),(5,69,58,99,96,75,86),(6,70,59,100,89,76,87),(7,71,60,101,90,77,88),(8,72,61,102,91,78,81),(9,50,47,107,30,33,20),(10,51,48,108,31,34,21),(11,52,41,109,32,35,22),(12,53,42,110,25,36,23),(13,54,43,111,26,37,24),(14,55,44,112,27,38,17),(15,56,45,105,28,39,18),(16,49,46,106,29,40,19)], [(1,44,5,48),(2,43,6,47),(3,42,7,46),(4,41,8,45),(9,63,13,59),(10,62,14,58),(11,61,15,57),(12,60,16,64),(17,99,21,103),(18,98,22,102),(19,97,23,101),(20,104,24,100),(25,77,29,73),(26,76,30,80),(27,75,31,79),(28,74,32,78),(33,93,37,89),(34,92,38,96),(35,91,39,95),(36,90,40,94),(49,67,53,71),(50,66,54,70),(51,65,55,69),(52,72,56,68),(81,105,85,109),(82,112,86,108),(83,111,87,107),(84,110,88,106)]])
D8.D7 is a maximal subgroup of
D8⋊D14 D16⋊3D7 D7×SD32 SD32⋊D7 D8.D14 C56.30C23 C56.31C23
D8.D7 is a maximal quotient of C8.5Dic14 C56.6D4 C14.SD32
32 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 7A | 7B | 7C | 8A | 8B | 14A | 14B | 14C | 14D | ··· | 14I | 16A | 16B | 16C | 16D | 28A | 28B | 28C | 56A | ··· | 56F |
| order | 1 | 2 | 2 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 14 | 14 | 14 | 14 | ··· | 14 | 16 | 16 | 16 | 16 | 28 | 28 | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 8 | 2 | 56 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 8 | ··· | 8 | 14 | 14 | 14 | 14 | 4 | 4 | 4 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | D4 | D7 | D8 | D14 | SD32 | C7⋊D4 | D4⋊D7 | D8.D7 |
| kernel | D8.D7 | C7⋊C16 | Dic28 | C7×D8 | C28 | D8 | C14 | C8 | C7 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 3 | 2 | 3 | 4 | 6 | 3 | 6 |
Matrix representation of D8.D7 ►in GL4(𝔽113) generated by
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 60 |
| 0 | 0 | 32 | 51 |
| 112 | 0 | 0 | 0 |
| 31 | 1 | 0 | 0 |
| 0 | 0 | 0 | 60 |
| 0 | 0 | 81 | 0 |
| 109 | 0 | 0 | 0 |
| 44 | 28 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 76 | 45 | 0 | 0 |
| 60 | 37 | 0 | 0 |
| 0 | 0 | 16 | 25 |
| 0 | 0 | 53 | 97 |
G:=sub<GL(4,GF(113))| [112,0,0,0,0,112,0,0,0,0,0,32,0,0,60,51],[112,31,0,0,0,1,0,0,0,0,0,81,0,0,60,0],[109,44,0,0,0,28,0,0,0,0,1,0,0,0,0,1],[76,60,0,0,45,37,0,0,0,0,16,53,0,0,25,97] >;
D8.D7 in GAP, Magma, Sage, TeX
D_8.D_7
% in TeX
G:=Group("D8.D7"); // GroupNames label
G:=SmallGroup(224,33);
// by ID
G=gap.SmallGroup(224,33);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,96,73,218,116,122,579,297,69,6917]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^7=1,d^2=a^4,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a^5*b,d*c*d^-1=c^-1>;
// generators/relations
Export