metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4⋊4D28, Q8⋊4D28, D28⋊15D4, C42⋊2D14, Dic14⋊15D4, M4(2)⋊3D14, C4≀C2⋊1D7, (C7×D4)⋊3D4, (C7×Q8)⋊3D4, C4.9(C2×D28), C28⋊4D4⋊6C2, C8⋊D14⋊8C2, C7⋊2(D4⋊4D4), C4○D4.1D14, C4.125(D4×D7), D4⋊D14⋊1C2, Dic14⋊C4⋊5C2, D4⋊8D14⋊1C2, (C4×C28)⋊11C22, C28.337(C2×D4), (C22×D7).2D4, C22.29(D4×D7), C14.27C22≀C2, C28.46D4⋊1C2, (C2×D28)⋊13C22, C4.Dic7⋊4C22, (C2×C28).262C23, C4○D28.11C22, C2.30(C22⋊D28), (C7×M4(2))⋊10C22, (C7×C4≀C2)⋊1C2, (C2×C14).26(C2×D4), (C7×C4○D4).3C22, (C2×C4).109(C22×D7), SmallGroup(448,356)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4⋊4D28
G = < a,b,c,d | a4=b2=c28=d2=1, bab=dad=a-1, ac=ca, cbc-1=a-1b, dbd=ab, dcd=c-1 >
Subgroups: 1276 in 168 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, D7, C14, C14, C42, M4(2), M4(2), D8, SD16, C2×D4, C4○D4, C4○D4, Dic7, C28, C28, D14, C2×C14, C2×C14, C4.D4, C4≀C2, C4≀C2, C4⋊1D4, C8⋊C22, 2+ 1+4, C7⋊C8, C56, Dic14, C4×D7, D28, D28, C7⋊D4, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C22×D7, C22×D7, D4⋊4D4, C56⋊C2, D56, C4.Dic7, D4⋊D7, Q8⋊D7, C4×C28, C7×M4(2), C2×D28, C2×D28, C4○D28, C4○D28, D4×D7, Q8⋊2D7, C7×C4○D4, Dic14⋊C4, C28.46D4, C7×C4≀C2, C28⋊4D4, C8⋊D14, D4⋊D14, D4⋊8D14, D4⋊4D28
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, D14, C22≀C2, D28, C22×D7, D4⋊4D4, C2×D28, D4×D7, C22⋊D28, D4⋊4D28
►On 56 points
(1 26 8 19)(2 27 9 20)(3 28 10 21)(4 15 11 22)(5 16 12 23)(6 17 13 24)(7 18 14 25)(29 50 43 36)(30 51 44 37)(31 52 45 38)(32 53 46 39)(33 54 47 40)(34 55 48 41)(35 56 49 42)
(1 56)(2 50)(3 44)(4 38)(5 32)(6 54)(7 48)(8 42)(9 36)(10 30)(11 52)(12 46)(13 40)(14 34)(15 45)(16 39)(17 33)(18 55)(19 49)(20 43)(21 37)(22 31)(23 53)(24 47)(25 41)(26 35)(27 29)(28 51)
(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)
(1 17)(2 16)(3 15)(4 28)(5 27)(6 26)(7 25)(8 24)(9 23)(10 22)(11 21)(12 20)(13 19)(14 18)(29 39)(30 38)(31 37)(32 36)(33 35)(40 56)(41 55)(42 54)(43 53)(44 52)(45 51)(46 50)(47 49)
G:=sub<Sym(56)| (1,26,8,19)(2,27,9,20)(3,28,10,21)(4,15,11,22)(5,16,12,23)(6,17,13,24)(7,18,14,25)(29,50,43,36)(30,51,44,37)(31,52,45,38)(32,53,46,39)(33,54,47,40)(34,55,48,41)(35,56,49,42), (1,56)(2,50)(3,44)(4,38)(5,32)(6,54)(7,48)(8,42)(9,36)(10,30)(11,52)(12,46)(13,40)(14,34)(15,45)(16,39)(17,33)(18,55)(19,49)(20,43)(21,37)(22,31)(23,53)(24,47)(25,41)(26,35)(27,29)(28,51), (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), (1,17)(2,16)(3,15)(4,28)(5,27)(6,26)(7,25)(8,24)(9,23)(10,22)(11,21)(12,20)(13,19)(14,18)(29,39)(30,38)(31,37)(32,36)(33,35)(40,56)(41,55)(42,54)(43,53)(44,52)(45,51)(46,50)(47,49)>;
G:=Group( (1,26,8,19)(2,27,9,20)(3,28,10,21)(4,15,11,22)(5,16,12,23)(6,17,13,24)(7,18,14,25)(29,50,43,36)(30,51,44,37)(31,52,45,38)(32,53,46,39)(33,54,47,40)(34,55,48,41)(35,56,49,42), (1,56)(2,50)(3,44)(4,38)(5,32)(6,54)(7,48)(8,42)(9,36)(10,30)(11,52)(12,46)(13,40)(14,34)(15,45)(16,39)(17,33)(18,55)(19,49)(20,43)(21,37)(22,31)(23,53)(24,47)(25,41)(26,35)(27,29)(28,51), (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), (1,17)(2,16)(3,15)(4,28)(5,27)(6,26)(7,25)(8,24)(9,23)(10,22)(11,21)(12,20)(13,19)(14,18)(29,39)(30,38)(31,37)(32,36)(33,35)(40,56)(41,55)(42,54)(43,53)(44,52)(45,51)(46,50)(47,49) );
G=PermutationGroup([[(1,26,8,19),(2,27,9,20),(3,28,10,21),(4,15,11,22),(5,16,12,23),(6,17,13,24),(7,18,14,25),(29,50,43,36),(30,51,44,37),(31,52,45,38),(32,53,46,39),(33,54,47,40),(34,55,48,41),(35,56,49,42)], [(1,56),(2,50),(3,44),(4,38),(5,32),(6,54),(7,48),(8,42),(9,36),(10,30),(11,52),(12,46),(13,40),(14,34),(15,45),(16,39),(17,33),(18,55),(19,49),(20,43),(21,37),(22,31),(23,53),(24,47),(25,41),(26,35),(27,29),(28,51)], [(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)], [(1,17),(2,16),(3,15),(4,28),(5,27),(6,26),(7,25),(8,24),(9,23),(10,22),(11,21),(12,20),(13,19),(14,18),(29,39),(30,38),(31,37),(32,36),(33,35),(40,56),(41,55),(42,54),(43,53),(44,52),(45,51),(46,50),(47,49)]])
58 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 7A | 7B | 7C | 8A | 8B | 14A | 14B | 14C | 14D | 14E | 14F | 14G | 14H | 14I | 28A | ··· | 28F | 28G | ··· | 28U | 28V | 28W | 28X | 56A | ··· | 56F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 14 | 14 | 14 | 14 | 14 | 14 | 14 | 14 | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 28 | 28 | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 4 | 28 | 28 | 28 | 56 | 2 | 2 | 4 | 4 | 4 | 28 | 2 | 2 | 2 | 8 | 56 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | ··· | 8 |
58 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D4 | D4 | D7 | D14 | D14 | D14 | D28 | D28 | D4⋊4D4 | D4×D7 | D4×D7 | D4⋊4D28 |
| kernel | D4⋊4D28 | Dic14⋊C4 | C28.46D4 | C7×C4≀C2 | C28⋊4D4 | C8⋊D14 | D4⋊D14 | D4⋊8D14 | Dic14 | D28 | C7×D4 | C7×Q8 | C22×D7 | C4≀C2 | C42 | M4(2) | C4○D4 | D4 | Q8 | C7 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 3 | 3 | 3 | 3 | 6 | 6 | 2 | 3 | 3 | 12 |
Matrix representation of D4⋊4D28 ►in GL4(𝔽113) generated by
| 96 | 105 | 0 | 0 |
| 8 | 17 | 0 | 0 |
| 0 | 0 | 17 | 8 |
| 0 | 0 | 105 | 96 |
| 0 | 0 | 17 | 8 |
| 0 | 0 | 105 | 96 |
| 96 | 105 | 0 | 0 |
| 8 | 17 | 0 | 0 |
| 103 | 24 | 0 | 0 |
| 89 | 1 | 0 | 0 |
| 0 | 0 | 23 | 36 |
| 0 | 0 | 77 | 96 |
| 90 | 77 | 0 | 0 |
| 90 | 23 | 0 | 0 |
| 0 | 0 | 103 | 24 |
| 0 | 0 | 10 | 10 |
G:=sub<GL(4,GF(113))| [96,8,0,0,105,17,0,0,0,0,17,105,0,0,8,96],[0,0,96,8,0,0,105,17,17,105,0,0,8,96,0,0],[103,89,0,0,24,1,0,0,0,0,23,77,0,0,36,96],[90,90,0,0,77,23,0,0,0,0,103,10,0,0,24,10] >;
D4⋊4D28 in GAP, Magma, Sage, TeX
D_4\rtimes_4D_{28} % in TeX
G:=Group("D4:4D28"); // GroupNames label
G:=SmallGroup(448,356);
// by ID
G=gap.SmallGroup(448,356);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,254,219,58,570,1684,851,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^28=d^2=1,b*a*b=d*a*d=a^-1,a*c=c*a,c*b*c^-1=a^-1*b,d*b*d=a*b,d*c*d=c^-1>;
// generators/relations