p-group, metabelian, nilpotent (class 3), monomial
Aliases: D4⋊5D8, C42.483C23, C4.692- (1+4), (C8×D4)⋊9C2, (C4×D8)⋊14C2, C4.45(C2×D8), D4○2(C2.D8), C8⋊14(C4○D4), C8⋊7D4⋊15C2, C4⋊C4.266D4, C8⋊2Q8⋊17C2, D4⋊6D4⋊14C2, C22.5(C2×D8), D4⋊Q8⋊14C2, (C2×D4).354D4, (C4×C8).87C22, C2.54(Q8○D8), C2.20(C22×D8), C4⋊C8.299C22, C4⋊C4.239C23, (C2×C8).195C23, (C2×C4).526C24, C22⋊C4.110D4, C22.D8⋊9C2, C23.477(C2×D4), C4⋊Q8.161C22, C2.79(D4⋊6D4), C2.D8.61C22, (C2×D8).142C22, (C4×D4).339C22, (C2×D4).248C23, C4⋊D4.97C22, C22⋊C8.185C22, (C22×C8).164C22, C22.786(C22×D4), D4⋊C4.169C22, (C22×C4).1158C23, (C2×D4)○(C2.D8), (C2×C2.D8)⋊28C2, C4.108(C2×C4○D4), (C2×C4).171(C2×D4), (C2×C4⋊C4).678C22, SmallGroup(128,2066)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 424 in 210 conjugacy classes, 96 normal (24 characteristic)
C1, C2 [×3], C2 [×6], C4 [×2], C4 [×2], C4 [×9], C22, C22 [×4], C22 [×10], C8 [×2], C8 [×3], C2×C4 [×3], C2×C4 [×2], C2×C4 [×22], D4 [×4], D4 [×12], Q8 [×4], C23 [×2], C23 [×2], C42, C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4, C4⋊C4 [×6], C4⋊C4 [×6], C2×C8 [×2], C2×C8 [×2], C2×C8 [×4], D8 [×2], C22×C4 [×2], C22×C4 [×6], C2×D4, C2×D4 [×2], C2×D4 [×4], C2×Q8 [×2], C4○D4 [×8], C4×C8, C22⋊C8 [×2], D4⋊C4 [×6], C4⋊C8, C2.D8, C2.D8 [×8], C2×C4⋊C4 [×4], C4×D4, C4×D4 [×2], C4⋊D4 [×4], C22⋊Q8 [×2], C22.D4 [×4], C4⋊Q8 [×2], C22×C8 [×2], C2×D8, C2×C4○D4 [×2], C2×C2.D8 [×2], C8×D4, C4×D8, C8⋊7D4 [×2], D4⋊Q8 [×2], C22.D8 [×4], C8⋊2Q8, D4⋊6D4 [×2], D4⋊5D8
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D8 [×4], C2×D4 [×6], C4○D4 [×2], C24, C2×D8 [×6], C22×D4, C2×C4○D4, 2- (1+4), D4⋊6D4, C22×D8, Q8○D8, D4⋊5D8
Generators and relations
G = < a,b,c,d | a4=b2=c8=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a2b, dcd=c-1 >
►On 64 points
(1 10 59 38)(2 11 60 39)(3 12 61 40)(4 13 62 33)(5 14 63 34)(6 15 64 35)(7 16 57 36)(8 9 58 37)(17 55 43 27)(18 56 44 28)(19 49 45 29)(20 50 46 30)(21 51 47 31)(22 52 48 32)(23 53 41 25)(24 54 42 26)
(1 47)(2 48)(3 41)(4 42)(5 43)(6 44)(7 45)(8 46)(9 50)(10 51)(11 52)(12 53)(13 54)(14 55)(15 56)(16 49)(17 63)(18 64)(19 57)(20 58)(21 59)(22 60)(23 61)(24 62)(25 40)(26 33)(27 34)(28 35)(29 36)(30 37)(31 38)(32 39)
(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)
(1 8)(2 7)(3 6)(4 5)(9 10)(11 16)(12 15)(13 14)(17 42)(18 41)(19 48)(20 47)(21 46)(22 45)(23 44)(24 43)(25 56)(26 55)(27 54)(28 53)(29 52)(30 51)(31 50)(32 49)(33 34)(35 40)(36 39)(37 38)(57 60)(58 59)(61 64)(62 63)
G:=sub<Sym(64)| (1,10,59,38)(2,11,60,39)(3,12,61,40)(4,13,62,33)(5,14,63,34)(6,15,64,35)(7,16,57,36)(8,9,58,37)(17,55,43,27)(18,56,44,28)(19,49,45,29)(20,50,46,30)(21,51,47,31)(22,52,48,32)(23,53,41,25)(24,54,42,26), (1,47)(2,48)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,50)(10,51)(11,52)(12,53)(13,54)(14,55)(15,56)(16,49)(17,63)(18,64)(19,57)(20,58)(21,59)(22,60)(23,61)(24,62)(25,40)(26,33)(27,34)(28,35)(29,36)(30,37)(31,38)(32,39), (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), (1,8)(2,7)(3,6)(4,5)(9,10)(11,16)(12,15)(13,14)(17,42)(18,41)(19,48)(20,47)(21,46)(22,45)(23,44)(24,43)(25,56)(26,55)(27,54)(28,53)(29,52)(30,51)(31,50)(32,49)(33,34)(35,40)(36,39)(37,38)(57,60)(58,59)(61,64)(62,63)>;
G:=Group( (1,10,59,38)(2,11,60,39)(3,12,61,40)(4,13,62,33)(5,14,63,34)(6,15,64,35)(7,16,57,36)(8,9,58,37)(17,55,43,27)(18,56,44,28)(19,49,45,29)(20,50,46,30)(21,51,47,31)(22,52,48,32)(23,53,41,25)(24,54,42,26), (1,47)(2,48)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,50)(10,51)(11,52)(12,53)(13,54)(14,55)(15,56)(16,49)(17,63)(18,64)(19,57)(20,58)(21,59)(22,60)(23,61)(24,62)(25,40)(26,33)(27,34)(28,35)(29,36)(30,37)(31,38)(32,39), (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), (1,8)(2,7)(3,6)(4,5)(9,10)(11,16)(12,15)(13,14)(17,42)(18,41)(19,48)(20,47)(21,46)(22,45)(23,44)(24,43)(25,56)(26,55)(27,54)(28,53)(29,52)(30,51)(31,50)(32,49)(33,34)(35,40)(36,39)(37,38)(57,60)(58,59)(61,64)(62,63) );
G=PermutationGroup([(1,10,59,38),(2,11,60,39),(3,12,61,40),(4,13,62,33),(5,14,63,34),(6,15,64,35),(7,16,57,36),(8,9,58,37),(17,55,43,27),(18,56,44,28),(19,49,45,29),(20,50,46,30),(21,51,47,31),(22,52,48,32),(23,53,41,25),(24,54,42,26)], [(1,47),(2,48),(3,41),(4,42),(5,43),(6,44),(7,45),(8,46),(9,50),(10,51),(11,52),(12,53),(13,54),(14,55),(15,56),(16,49),(17,63),(18,64),(19,57),(20,58),(21,59),(22,60),(23,61),(24,62),(25,40),(26,33),(27,34),(28,35),(29,36),(30,37),(31,38),(32,39)], [(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)], [(1,8),(2,7),(3,6),(4,5),(9,10),(11,16),(12,15),(13,14),(17,42),(18,41),(19,48),(20,47),(21,46),(22,45),(23,44),(24,43),(25,56),(26,55),(27,54),(28,53),(29,52),(30,51),(31,50),(32,49),(33,34),(35,40),(36,39),(37,38),(57,60),(58,59),(61,64),(62,63)])
Matrix representation ►G ⊆ GL4(𝔽17) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 4 | 4 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 4 | 8 |
| 0 | 0 | 13 | 13 |
| 14 | 3 | 0 | 0 |
| 14 | 14 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 14 | 3 | 0 | 0 |
| 3 | 3 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 16 | 16 |
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,13,4,0,0,0,4],[16,0,0,0,0,16,0,0,0,0,4,13,0,0,8,13],[14,14,0,0,3,14,0,0,0,0,1,0,0,0,0,1],[14,3,0,0,3,3,0,0,0,0,1,16,0,0,0,16] >;
35 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | ··· | 4K | 4L | 4M | 4N | 4O | 8A | 8B | 8C | 8D | 8E | ··· | 8J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
35 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | C4○D4 | D8 | 2- (1+4) | Q8○D8 |
| kernel | D4⋊5D8 | C2×C2.D8 | C8×D4 | C4×D8 | C8⋊7D4 | D4⋊Q8 | C22.D8 | C8⋊2Q8 | D4⋊6D4 | C22⋊C4 | C4⋊C4 | C2×D4 | C8 | D4 | C4 | C2 |
| # reps | 1 | 2 | 1 | 1 | 2 | 2 | 4 | 1 | 2 | 2 | 1 | 1 | 4 | 8 | 1 | 2 |
In GAP, Magma, Sage, TeX
D_4\rtimes_5D_8
% in TeX
G:=Group("D4:5D8"); // GroupNames label
G:=SmallGroup(128,2066);
// by ID
G=gap.SmallGroup(128,2066);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,758,100,346,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^8=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations