metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C56.14D4, C28.11D8, D8.2Dic7, (C7×D8).2C4, (C2×D8).5D7, C56.25(C2×C4), (C14×D8).1C2, (C2×C8).49D14, C28.C8⋊1C2, C56.C4⋊2C2, C8.1(C2×Dic7), C4.14(D4⋊D7), (C2×C28).115D4, C7⋊3(M5(2)⋊C2), C8.24(C7⋊D4), (C2×C56).29C22, (C2×C14).30SD16, C4.2(C23.D7), C28.14(C22⋊C4), C2.7(D4⋊Dic7), C22.6(D4.D7), C14.27(D4⋊C4), (C2×C4).24(C7⋊D4), SmallGroup(448,120)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D8.Dic7
G = < a,b,c,d | a8=b2=1, c14=a4, d2=a4c7, bab=a-1, ac=ca, dad-1=a3, cbc-1=a4b, dbd-1=a5b, dcd-1=c13 >
Subgroups: 228 in 62 conjugacy classes, 27 normal (23 characteristic)
C1, C2, C2, C4, C22, C22, C7, C8, C8, C2×C4, D4, C23, C14, C14, C16, C2×C8, M4(2), D8, D8, C2×D4, C28, C2×C14, C2×C14, C8.C4, M5(2), C2×D8, C7⋊C8, C56, C2×C28, C7×D4, C22×C14, M5(2)⋊C2, C7⋊C16, C4.Dic7, C2×C56, C7×D8, C7×D8, D4×C14, C28.C8, C56.C4, C14×D8, D8.Dic7
Quotients: C1, C2, C4, C22, C2×C4, D4, D7, C22⋊C4, D8, SD16, Dic7, D14, D4⋊C4, C2×Dic7, C7⋊D4, M5(2)⋊C2, D4⋊D7, D4.D7, C23.D7, D4⋊Dic7, D8.Dic7
►On 112 points
Generators in S112
(1 35 8 42 15 49 22 56)(2 36 9 43 16 50 23 29)(3 37 10 44 17 51 24 30)(4 38 11 45 18 52 25 31)(5 39 12 46 19 53 26 32)(6 40 13 47 20 54 27 33)(7 41 14 48 21 55 28 34)(57 106 78 99 71 92 64 85)(58 107 79 100 72 93 65 86)(59 108 80 101 73 94 66 87)(60 109 81 102 74 95 67 88)(61 110 82 103 75 96 68 89)(62 111 83 104 76 97 69 90)(63 112 84 105 77 98 70 91)
(1 56)(2 43)(3 30)(4 45)(5 32)(6 47)(7 34)(8 49)(9 36)(10 51)(11 38)(12 53)(13 40)(14 55)(15 42)(16 29)(17 44)(18 31)(19 46)(20 33)(21 48)(22 35)(23 50)(24 37)(25 52)(26 39)(27 54)(28 41)(57 78)(58 65)(59 80)(60 67)(61 82)(62 69)(63 84)(64 71)(66 73)(68 75)(70 77)(72 79)(74 81)(76 83)(85 99)(87 101)(89 103)(91 105)(93 107)(95 109)(97 111)
(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 78 22 71 15 64 8 57)(2 63 23 84 16 77 9 70)(3 76 24 69 17 62 10 83)(4 61 25 82 18 75 11 68)(5 74 26 67 19 60 12 81)(6 59 27 80 20 73 13 66)(7 72 28 65 21 58 14 79)(29 98 50 91 43 112 36 105)(30 111 51 104 44 97 37 90)(31 96 52 89 45 110 38 103)(32 109 53 102 46 95 39 88)(33 94 54 87 47 108 40 101)(34 107 55 100 48 93 41 86)(35 92 56 85 49 106 42 99)
G:=sub<Sym(112)| (1,35,8,42,15,49,22,56)(2,36,9,43,16,50,23,29)(3,37,10,44,17,51,24,30)(4,38,11,45,18,52,25,31)(5,39,12,46,19,53,26,32)(6,40,13,47,20,54,27,33)(7,41,14,48,21,55,28,34)(57,106,78,99,71,92,64,85)(58,107,79,100,72,93,65,86)(59,108,80,101,73,94,66,87)(60,109,81,102,74,95,67,88)(61,110,82,103,75,96,68,89)(62,111,83,104,76,97,69,90)(63,112,84,105,77,98,70,91), (1,56)(2,43)(3,30)(4,45)(5,32)(6,47)(7,34)(8,49)(9,36)(10,51)(11,38)(12,53)(13,40)(14,55)(15,42)(16,29)(17,44)(18,31)(19,46)(20,33)(21,48)(22,35)(23,50)(24,37)(25,52)(26,39)(27,54)(28,41)(57,78)(58,65)(59,80)(60,67)(61,82)(62,69)(63,84)(64,71)(66,73)(68,75)(70,77)(72,79)(74,81)(76,83)(85,99)(87,101)(89,103)(91,105)(93,107)(95,109)(97,111), (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), (1,78,22,71,15,64,8,57)(2,63,23,84,16,77,9,70)(3,76,24,69,17,62,10,83)(4,61,25,82,18,75,11,68)(5,74,26,67,19,60,12,81)(6,59,27,80,20,73,13,66)(7,72,28,65,21,58,14,79)(29,98,50,91,43,112,36,105)(30,111,51,104,44,97,37,90)(31,96,52,89,45,110,38,103)(32,109,53,102,46,95,39,88)(33,94,54,87,47,108,40,101)(34,107,55,100,48,93,41,86)(35,92,56,85,49,106,42,99)>;
G:=Group( (1,35,8,42,15,49,22,56)(2,36,9,43,16,50,23,29)(3,37,10,44,17,51,24,30)(4,38,11,45,18,52,25,31)(5,39,12,46,19,53,26,32)(6,40,13,47,20,54,27,33)(7,41,14,48,21,55,28,34)(57,106,78,99,71,92,64,85)(58,107,79,100,72,93,65,86)(59,108,80,101,73,94,66,87)(60,109,81,102,74,95,67,88)(61,110,82,103,75,96,68,89)(62,111,83,104,76,97,69,90)(63,112,84,105,77,98,70,91), (1,56)(2,43)(3,30)(4,45)(5,32)(6,47)(7,34)(8,49)(9,36)(10,51)(11,38)(12,53)(13,40)(14,55)(15,42)(16,29)(17,44)(18,31)(19,46)(20,33)(21,48)(22,35)(23,50)(24,37)(25,52)(26,39)(27,54)(28,41)(57,78)(58,65)(59,80)(60,67)(61,82)(62,69)(63,84)(64,71)(66,73)(68,75)(70,77)(72,79)(74,81)(76,83)(85,99)(87,101)(89,103)(91,105)(93,107)(95,109)(97,111), (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), (1,78,22,71,15,64,8,57)(2,63,23,84,16,77,9,70)(3,76,24,69,17,62,10,83)(4,61,25,82,18,75,11,68)(5,74,26,67,19,60,12,81)(6,59,27,80,20,73,13,66)(7,72,28,65,21,58,14,79)(29,98,50,91,43,112,36,105)(30,111,51,104,44,97,37,90)(31,96,52,89,45,110,38,103)(32,109,53,102,46,95,39,88)(33,94,54,87,47,108,40,101)(34,107,55,100,48,93,41,86)(35,92,56,85,49,106,42,99) );
G=PermutationGroup([[(1,35,8,42,15,49,22,56),(2,36,9,43,16,50,23,29),(3,37,10,44,17,51,24,30),(4,38,11,45,18,52,25,31),(5,39,12,46,19,53,26,32),(6,40,13,47,20,54,27,33),(7,41,14,48,21,55,28,34),(57,106,78,99,71,92,64,85),(58,107,79,100,72,93,65,86),(59,108,80,101,73,94,66,87),(60,109,81,102,74,95,67,88),(61,110,82,103,75,96,68,89),(62,111,83,104,76,97,69,90),(63,112,84,105,77,98,70,91)], [(1,56),(2,43),(3,30),(4,45),(5,32),(6,47),(7,34),(8,49),(9,36),(10,51),(11,38),(12,53),(13,40),(14,55),(15,42),(16,29),(17,44),(18,31),(19,46),(20,33),(21,48),(22,35),(23,50),(24,37),(25,52),(26,39),(27,54),(28,41),(57,78),(58,65),(59,80),(60,67),(61,82),(62,69),(63,84),(64,71),(66,73),(68,75),(70,77),(72,79),(74,81),(76,83),(85,99),(87,101),(89,103),(91,105),(93,107),(95,109),(97,111)], [(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)], [(1,78,22,71,15,64,8,57),(2,63,23,84,16,77,9,70),(3,76,24,69,17,62,10,83),(4,61,25,82,18,75,11,68),(5,74,26,67,19,60,12,81),(6,59,27,80,20,73,13,66),(7,72,28,65,21,58,14,79),(29,98,50,91,43,112,36,105),(30,111,51,104,44,97,37,90),(31,96,52,89,45,110,38,103),(32,109,53,102,46,95,39,88),(33,94,54,87,47,108,40,101),(34,107,55,100,48,93,41,86),(35,92,56,85,49,106,42,99)]])
58 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 14A | ··· | 14I | 14J | ··· | 14U | 16A | 16B | 16C | 16D | 28A | ··· | 28F | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 16 | 16 | 16 | 16 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 8 | 8 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 56 | 56 | 2 | ··· | 2 | 8 | ··· | 8 | 28 | 28 | 28 | 28 | 4 | ··· | 4 | 4 | ··· | 4 |
58 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | - | |||||
| image | C1 | C2 | C2 | C2 | C4 | D4 | D4 | D7 | D8 | SD16 | D14 | Dic7 | C7⋊D4 | C7⋊D4 | M5(2)⋊C2 | D4⋊D7 | D4.D7 | D8.Dic7 |
| kernel | D8.Dic7 | C28.C8 | C56.C4 | C14×D8 | C7×D8 | C56 | C2×C28 | C2×D8 | C28 | C2×C14 | C2×C8 | D8 | C8 | C2×C4 | C7 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 3 | 2 | 2 | 3 | 6 | 6 | 6 | 2 | 3 | 3 | 12 |
Matrix representation of D8.Dic7 ►in GL4(𝔽113) generated by
| 62 | 51 | 0 | 0 |
| 31 | 0 | 0 | 0 |
| 16 | 97 | 0 | 107 |
| 68 | 0 | 19 | 51 |
| 62 | 51 | 0 | 0 |
| 31 | 51 | 0 | 0 |
| 85 | 97 | 1 | 80 |
| 0 | 0 | 0 | 112 |
| 97 | 32 | 0 | 0 |
| 97 | 16 | 0 | 0 |
| 5 | 52 | 7 | 108 |
| 19 | 0 | 110 | 106 |
| 35 | 0 | 48 | 0 |
| 0 | 0 | 0 | 1 |
| 111 | 33 | 78 | 0 |
| 112 | 1 | 0 | 0 |
G:=sub<GL(4,GF(113))| [62,31,16,68,51,0,97,0,0,0,0,19,0,0,107,51],[62,31,85,0,51,51,97,0,0,0,1,0,0,0,80,112],[97,97,5,19,32,16,52,0,0,0,7,110,0,0,108,106],[35,0,111,112,0,0,33,1,48,0,78,0,0,1,0,0] >;
D8.Dic7 in GAP, Magma, Sage, TeX
D_8.{\rm Dic}_7 % in TeX
G:=Group("D8.Dic7"); // GroupNames label
G:=SmallGroup(448,120);
// by ID
G=gap.SmallGroup(448,120);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,141,387,184,675,794,80,1684,851,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=1,c^14=a^4,d^2=a^4*c^7,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^3,c*b*c^-1=a^4*b,d*b*d^-1=a^5*b,d*c*d^-1=c^13>;
// generators/relations