metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic7.1D4, C23.6D14, D14⋊C4⋊6C2, C22⋊C4⋊5D7, (C2×C4).8D14, C2.10(D4×D7), C14.21(C2×D4), C7⋊2(C4.4D4), C23.D7⋊5C2, (C4×Dic7)⋊12C2, (C2×Dic14)⋊3C2, C2.12(C4○D28), C14.10(C4○D4), C2.9(D4⋊2D7), (C2×C14).26C23, (C2×C28).54C22, (C22×D7).4C22, C22.44(C22×D7), (C22×C14).15C22, (C2×Dic7).27C22, (C7×C22⋊C4)⋊7C2, (C2×C7⋊D4).4C2, SmallGroup(224,80)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic7.D4
G = < a,b,c,d | a14=c4=1, b2=d2=a7, bab-1=dad-1=a-1, ac=ca, bc=cb, dbd-1=a7b, dcd-1=a7c-1 >
Subgroups: 326 in 76 conjugacy classes, 31 normal (29 characteristic)
C1, C2, C2, C4, C22, C22, C7, C2×C4, C2×C4, D4, Q8, C23, C23, D7, C14, C14, C42, C22⋊C4, C22⋊C4, C2×D4, C2×Q8, Dic7, Dic7, C28, D14, C2×C14, C2×C14, C4.4D4, Dic14, C2×Dic7, C7⋊D4, C2×C28, C22×D7, C22×C14, C4×Dic7, D14⋊C4, C23.D7, C7×C22⋊C4, C2×Dic14, C2×C7⋊D4, Dic7.D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C4.4D4, C22×D7, C4○D28, D4×D7, D4⋊2D7, Dic7.D4
►On 112 points
(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 96 8 89)(2 95 9 88)(3 94 10 87)(4 93 11 86)(5 92 12 85)(6 91 13 98)(7 90 14 97)(15 75 22 82)(16 74 23 81)(17 73 24 80)(18 72 25 79)(19 71 26 78)(20 84 27 77)(21 83 28 76)(29 64 36 57)(30 63 37 70)(31 62 38 69)(32 61 39 68)(33 60 40 67)(34 59 41 66)(35 58 42 65)(43 99 50 106)(44 112 51 105)(45 111 52 104)(46 110 53 103)(47 109 54 102)(48 108 55 101)(49 107 56 100)
(1 73 44 38)(2 74 45 39)(3 75 46 40)(4 76 47 41)(5 77 48 42)(6 78 49 29)(7 79 50 30)(8 80 51 31)(9 81 52 32)(10 82 53 33)(11 83 54 34)(12 84 55 35)(13 71 56 36)(14 72 43 37)(15 103 60 87)(16 104 61 88)(17 105 62 89)(18 106 63 90)(19 107 64 91)(20 108 65 92)(21 109 66 93)(22 110 67 94)(23 111 68 95)(24 112 69 96)(25 99 70 97)(26 100 57 98)(27 101 58 85)(28 102 59 86)
(1 80 8 73)(2 79 9 72)(3 78 10 71)(4 77 11 84)(5 76 12 83)(6 75 13 82)(7 74 14 81)(15 98 22 91)(16 97 23 90)(17 96 24 89)(18 95 25 88)(19 94 26 87)(20 93 27 86)(21 92 28 85)(29 53 36 46)(30 52 37 45)(31 51 38 44)(32 50 39 43)(33 49 40 56)(34 48 41 55)(35 47 42 54)(57 103 64 110)(58 102 65 109)(59 101 66 108)(60 100 67 107)(61 99 68 106)(62 112 69 105)(63 111 70 104)
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), (1,96,8,89)(2,95,9,88)(3,94,10,87)(4,93,11,86)(5,92,12,85)(6,91,13,98)(7,90,14,97)(15,75,22,82)(16,74,23,81)(17,73,24,80)(18,72,25,79)(19,71,26,78)(20,84,27,77)(21,83,28,76)(29,64,36,57)(30,63,37,70)(31,62,38,69)(32,61,39,68)(33,60,40,67)(34,59,41,66)(35,58,42,65)(43,99,50,106)(44,112,51,105)(45,111,52,104)(46,110,53,103)(47,109,54,102)(48,108,55,101)(49,107,56,100), (1,73,44,38)(2,74,45,39)(3,75,46,40)(4,76,47,41)(5,77,48,42)(6,78,49,29)(7,79,50,30)(8,80,51,31)(9,81,52,32)(10,82,53,33)(11,83,54,34)(12,84,55,35)(13,71,56,36)(14,72,43,37)(15,103,60,87)(16,104,61,88)(17,105,62,89)(18,106,63,90)(19,107,64,91)(20,108,65,92)(21,109,66,93)(22,110,67,94)(23,111,68,95)(24,112,69,96)(25,99,70,97)(26,100,57,98)(27,101,58,85)(28,102,59,86), (1,80,8,73)(2,79,9,72)(3,78,10,71)(4,77,11,84)(5,76,12,83)(6,75,13,82)(7,74,14,81)(15,98,22,91)(16,97,23,90)(17,96,24,89)(18,95,25,88)(19,94,26,87)(20,93,27,86)(21,92,28,85)(29,53,36,46)(30,52,37,45)(31,51,38,44)(32,50,39,43)(33,49,40,56)(34,48,41,55)(35,47,42,54)(57,103,64,110)(58,102,65,109)(59,101,66,108)(60,100,67,107)(61,99,68,106)(62,112,69,105)(63,111,70,104)>;
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), (1,96,8,89)(2,95,9,88)(3,94,10,87)(4,93,11,86)(5,92,12,85)(6,91,13,98)(7,90,14,97)(15,75,22,82)(16,74,23,81)(17,73,24,80)(18,72,25,79)(19,71,26,78)(20,84,27,77)(21,83,28,76)(29,64,36,57)(30,63,37,70)(31,62,38,69)(32,61,39,68)(33,60,40,67)(34,59,41,66)(35,58,42,65)(43,99,50,106)(44,112,51,105)(45,111,52,104)(46,110,53,103)(47,109,54,102)(48,108,55,101)(49,107,56,100), (1,73,44,38)(2,74,45,39)(3,75,46,40)(4,76,47,41)(5,77,48,42)(6,78,49,29)(7,79,50,30)(8,80,51,31)(9,81,52,32)(10,82,53,33)(11,83,54,34)(12,84,55,35)(13,71,56,36)(14,72,43,37)(15,103,60,87)(16,104,61,88)(17,105,62,89)(18,106,63,90)(19,107,64,91)(20,108,65,92)(21,109,66,93)(22,110,67,94)(23,111,68,95)(24,112,69,96)(25,99,70,97)(26,100,57,98)(27,101,58,85)(28,102,59,86), (1,80,8,73)(2,79,9,72)(3,78,10,71)(4,77,11,84)(5,76,12,83)(6,75,13,82)(7,74,14,81)(15,98,22,91)(16,97,23,90)(17,96,24,89)(18,95,25,88)(19,94,26,87)(20,93,27,86)(21,92,28,85)(29,53,36,46)(30,52,37,45)(31,51,38,44)(32,50,39,43)(33,49,40,56)(34,48,41,55)(35,47,42,54)(57,103,64,110)(58,102,65,109)(59,101,66,108)(60,100,67,107)(61,99,68,106)(62,112,69,105)(63,111,70,104) );
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)], [(1,96,8,89),(2,95,9,88),(3,94,10,87),(4,93,11,86),(5,92,12,85),(6,91,13,98),(7,90,14,97),(15,75,22,82),(16,74,23,81),(17,73,24,80),(18,72,25,79),(19,71,26,78),(20,84,27,77),(21,83,28,76),(29,64,36,57),(30,63,37,70),(31,62,38,69),(32,61,39,68),(33,60,40,67),(34,59,41,66),(35,58,42,65),(43,99,50,106),(44,112,51,105),(45,111,52,104),(46,110,53,103),(47,109,54,102),(48,108,55,101),(49,107,56,100)], [(1,73,44,38),(2,74,45,39),(3,75,46,40),(4,76,47,41),(5,77,48,42),(6,78,49,29),(7,79,50,30),(8,80,51,31),(9,81,52,32),(10,82,53,33),(11,83,54,34),(12,84,55,35),(13,71,56,36),(14,72,43,37),(15,103,60,87),(16,104,61,88),(17,105,62,89),(18,106,63,90),(19,107,64,91),(20,108,65,92),(21,109,66,93),(22,110,67,94),(23,111,68,95),(24,112,69,96),(25,99,70,97),(26,100,57,98),(27,101,58,85),(28,102,59,86)], [(1,80,8,73),(2,79,9,72),(3,78,10,71),(4,77,11,84),(5,76,12,83),(6,75,13,82),(7,74,14,81),(15,98,22,91),(16,97,23,90),(17,96,24,89),(18,95,25,88),(19,94,26,87),(20,93,27,86),(21,92,28,85),(29,53,36,46),(30,52,37,45),(31,51,38,44),(32,50,39,43),(33,49,40,56),(34,48,41,55),(35,47,42,54),(57,103,64,110),(58,102,65,109),(59,101,66,108),(60,100,67,107),(61,99,68,106),(62,112,69,105),(63,111,70,104)]])
Dic7.D4 is a maximal subgroup of
C24.27D14 C24.30D14 C24.31D14 C42.93D14 C42.97D14 C42.98D14 C42.99D14 C42.102D14 C42.228D14 Dic14⋊23D4 C42⋊16D14 C42.114D14 C42⋊17D14 C42.115D14 C42.117D14 C24.33D14 C24.34D14 C24.35D14 C24⋊4D14 C28⋊(C4○D4) Dic14⋊19D4 C14.382+ 1+4 C14.402+ 1+4 C14.422+ 1+4 C14.452+ 1+4 C14.462+ 1+4 C14.492+ 1+4 C14.162- 1+4 Dic14⋊22D4 C14.222- 1+4 C14.232- 1+4 C14.242- 1+4 C14.582+ 1+4 C14.262- 1+4 C14.792- 1+4 C4⋊C4.197D14 C14.1212+ 1+4 C14.612+ 1+4 C14.1222+ 1+4 C14.842- 1+4 C14.672+ 1+4 C14.862- 1+4 C42.137D14 C42.138D14 D7×C4.4D4 Dic14⋊10D4 C42.143D14 C42.144D14 C42⋊22D14 C42.160D14 C42⋊23D14 C42⋊24D14 C42.189D14 C42.164D14 C42.165D14
Dic7.D4 is a maximal quotient of
(C2×C28)⋊Q8 C7⋊(C42⋊8C4) (C2×Dic7).Q8 (C2×C4).Dic14 D14⋊C4⋊5C4 C2.(C4×D28) (C2×C4).20D28 (C22×D7).9D4 Dic7.SD16 C4⋊C4.D14 C28⋊Q8⋊C2 (C8×Dic7)⋊C2 Dic7.1Q16 Q8⋊C4⋊D7 C56⋊C4.C2 Q8⋊Dic7⋊C2 C24.3D14 C23⋊Dic14 C24.8D14 C24.9D14 C24.13D14 C24.14D14 C23.16D28
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 4 | 28 | 2 | 2 | 4 | 14 | 14 | 14 | 14 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D7 | C4○D4 | D14 | D14 | C4○D28 | D4×D7 | D4⋊2D7 |
| kernel | Dic7.D4 | C4×Dic7 | D14⋊C4 | C23.D7 | C7×C22⋊C4 | C2×Dic14 | C2×C7⋊D4 | Dic7 | C22⋊C4 | C14 | C2×C4 | C23 | C2 | C2 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 3 | 4 | 6 | 3 | 12 | 3 | 3 |
Matrix representation of Dic7.D4 ►in GL4(𝔽29) generated by
| 26 | 28 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 5 | 2 | 0 | 0 |
| 16 | 24 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 17 | 27 |
| 0 | 0 | 0 | 12 |
| 17 | 22 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 2 |
| 0 | 0 | 1 | 17 |
G:=sub<GL(4,GF(29))| [26,1,0,0,28,0,0,0,0,0,1,0,0,0,0,1],[5,16,0,0,2,24,0,0,0,0,1,0,0,0,0,1],[12,0,0,0,0,12,0,0,0,0,17,0,0,0,27,12],[17,0,0,0,22,12,0,0,0,0,12,1,0,0,2,17] >;
Dic7.D4 in GAP, Magma, Sage, TeX
{\rm Dic}_7.D_4 % in TeX
G:=Group("Dic7.D4"); // GroupNames label
G:=SmallGroup(224,80);
// by ID
G=gap.SmallGroup(224,80);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,217,55,506,188,6917]);
// Polycyclic
G:=Group<a,b,c,d|a^14=c^4=1,b^2=d^2=a^7,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a^7*b,d*c*d^-1=a^7*c^-1>;
// generators/relations