metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D28⋊4C4, C28.54D4, Dic14⋊4C4, M4(2)⋊4D7, C22.3D28, C7⋊2C4≀C2, C4.3(C4×D7), C28.6(C2×C4), (C2×C14).1D4, C4○D28.2C2, (C4×Dic7)⋊1C2, (C2×C4).38D14, C4.29(C7⋊D4), (C7×M4(2))⋊8C2, C2.11(D14⋊C4), (C2×C28).15C22, C14.10(C22⋊C4), SmallGroup(224,31)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D28⋊4C4
G = < a,b,c | a28=b2=c4=1, bab=a-1, cac-1=a13, cbc-1=a19b >
Smallest permutation representation of D28⋊4C4
►On 56 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)
(1 54)(2 53)(3 52)(4 51)(5 50)(6 49)(7 48)(8 47)(9 46)(10 45)(11 44)(12 43)(13 42)(14 41)(15 40)(16 39)(17 38)(18 37)(19 36)(20 35)(21 34)(22 33)(23 32)(24 31)(25 30)(26 29)(27 56)(28 55)
(1 15)(2 28)(3 13)(4 26)(5 11)(6 24)(7 9)(8 22)(10 20)(12 18)(14 16)(17 27)(19 25)(21 23)(29 46 43 32)(30 31 44 45)(33 42 47 56)(34 55 48 41)(35 40 49 54)(36 53 50 39)(37 38 51 52)
G:=sub<Sym(56)| (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), (1,54)(2,53)(3,52)(4,51)(5,50)(6,49)(7,48)(8,47)(9,46)(10,45)(11,44)(12,43)(13,42)(14,41)(15,40)(16,39)(17,38)(18,37)(19,36)(20,35)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,56)(28,55), (1,15)(2,28)(3,13)(4,26)(5,11)(6,24)(7,9)(8,22)(10,20)(12,18)(14,16)(17,27)(19,25)(21,23)(29,46,43,32)(30,31,44,45)(33,42,47,56)(34,55,48,41)(35,40,49,54)(36,53,50,39)(37,38,51,52)>;
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), (1,54)(2,53)(3,52)(4,51)(5,50)(6,49)(7,48)(8,47)(9,46)(10,45)(11,44)(12,43)(13,42)(14,41)(15,40)(16,39)(17,38)(18,37)(19,36)(20,35)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,56)(28,55), (1,15)(2,28)(3,13)(4,26)(5,11)(6,24)(7,9)(8,22)(10,20)(12,18)(14,16)(17,27)(19,25)(21,23)(29,46,43,32)(30,31,44,45)(33,42,47,56)(34,55,48,41)(35,40,49,54)(36,53,50,39)(37,38,51,52) );
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)], [(1,54),(2,53),(3,52),(4,51),(5,50),(6,49),(7,48),(8,47),(9,46),(10,45),(11,44),(12,43),(13,42),(14,41),(15,40),(16,39),(17,38),(18,37),(19,36),(20,35),(21,34),(22,33),(23,32),(24,31),(25,30),(26,29),(27,56),(28,55)], [(1,15),(2,28),(3,13),(4,26),(5,11),(6,24),(7,9),(8,22),(10,20),(12,18),(14,16),(17,27),(19,25),(21,23),(29,46,43,32),(30,31,44,45),(33,42,47,56),(34,55,48,41),(35,40,49,54),(36,53,50,39),(37,38,51,52)]])
D28⋊4C4 is a maximal subgroup of
D28.1D4 D28⋊1D4 D28.4D4 D28.5D4 D7×C4≀C2 C42⋊D14 D56⋊10C4 D56⋊7C4 C23.20D28 C56.93D4 C56.50D4 D28⋊18D4 D28.38D4 D28.39D4 D28.40D4
D28⋊4C4 is a maximal quotient of
C42.D14 C42.2D14 C23.30D28 C22.2D56 D28⋊2C8 Dic14⋊2C8 C28.3C42
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 7A | 7B | 7C | 8A | 8B | 14A | 14B | 14C | 14D | 14E | 14F | 28A | ··· | 28F | 28G | 28H | 28I | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 14 | 14 | 14 | 14 | 14 | 14 | 28 | ··· | 28 | 28 | 28 | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 28 | 1 | 1 | 2 | 14 | 14 | 14 | 14 | 28 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | D4 | D4 | D7 | D14 | C4≀C2 | C4×D7 | C7⋊D4 | D28 | D28⋊4C4 |
| kernel | D28⋊4C4 | C4×Dic7 | C7×M4(2) | C4○D28 | Dic14 | D28 | C28 | C2×C14 | M4(2) | C2×C4 | C7 | C4 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 3 | 3 | 4 | 6 | 6 | 6 | 6 |
Matrix representation of D28⋊4C4 ►in GL4(𝔽113) generated by
| 79 | 104 | 0 | 0 |
| 88 | 0 | 0 | 0 |
| 0 | 0 | 98 | 0 |
| 0 | 0 | 0 | 15 |
| 0 | 9 | 0 | 0 |
| 88 | 0 | 0 | 0 |
| 0 | 0 | 0 | 98 |
| 0 | 0 | 15 | 0 |
| 98 | 43 | 0 | 0 |
| 0 | 15 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 15 |
G:=sub<GL(4,GF(113))| [79,88,0,0,104,0,0,0,0,0,98,0,0,0,0,15],[0,88,0,0,9,0,0,0,0,0,0,15,0,0,98,0],[98,0,0,0,43,15,0,0,0,0,112,0,0,0,0,15] >;
D28⋊4C4 in GAP, Magma, Sage, TeX
D_{28}\rtimes_4C_4 % in TeX
G:=Group("D28:4C4"); // GroupNames label
G:=SmallGroup(224,31);
// by ID
G=gap.SmallGroup(224,31);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,121,31,86,579,297,69,6917]);
// Polycyclic
G:=Group<a,b,c|a^28=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^13,c*b*c^-1=a^19*b>;
// generators/relations
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