p-group, metabelian, nilpotent (class 3), monomial
Aliases: D4⋊5Q16, C42.464C23, C4.442+ 1+4, (C8×D4).9C2, (D4×Q8).5C2, C4⋊C4.263D4, (C4×Q16)⋊11C2, C4.28(C2×Q16), D4○3(Q8⋊C4), C4.Q16⋊12C2, C4⋊2Q16⋊14C2, (C2×D4).353D4, (C4×C8).83C22, D4⋊3Q8.4C2, C22⋊Q16⋊9C2, C8.18D4⋊12C2, C22.5(C2×Q16), C4⋊C4.401C23, C4⋊C8.296C22, (C2×C8).183C23, (C2×C4).491C24, Q8.16(C4○D4), C22⋊C4.103D4, C4.SD16⋊14C2, C23.472(C2×D4), C4⋊Q8.142C22, C2.19(C22×Q16), C2.D8.53C22, C2.68(D4○SD16), (C4×D4).332C22, C23.48D4⋊8C2, (C4×Q8).148C22, (C2×Q8).208C23, C2.127(D4⋊5D4), C22⋊Q8.71C22, C22⋊C8.184C22, (C22×C8).160C22, Q8⋊C4.11C22, (C2×Q16).132C22, C22.751(C22×D4), (C22×C4).1135C23, (C22×Q8).338C22, C4.216(C2×C4○D4), (C2×C4).168(C2×D4), (C2×Q8⋊C4)⋊25C2, (C2×C4⋊C4).661C22, SmallGroup(128,2031)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for D4⋊5Q16
G = < a,b,c,d | a4=b2=c8=1, d2=c4, bab=a-1, ac=ca, ad=da, cbc-1=a2b, bd=db, dcd-1=c-1 >
Subgroups: 352 in 194 conjugacy classes, 96 normal (38 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, Q8, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×C8, Q16, C22×C4, C22×C4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4×C8, C22⋊C8, Q8⋊C4, Q8⋊C4, C4⋊C8, C2.D8, C2.D8, C2×C4⋊C4, C4×D4, C4×D4, C4×Q8, C22⋊Q8, C22⋊Q8, C42.C2, C4⋊Q8, C4⋊Q8, C22×C8, C2×Q16, C2×Q16, C22×Q8, C2×Q8⋊C4, C8×D4, C4×Q16, C22⋊Q16, C4⋊2Q16, C8.18D4, C4.Q16, C23.48D4, C4.SD16, D4×Q8, D4⋊3Q8, D4⋊5Q16
Quotients: C1, C2, C22, D4, C23, Q16, C2×D4, C4○D4, C24, C2×Q16, C22×D4, C2×C4○D4, 2+ 1+4, D4⋊5D4, C22×Q16, D4○SD16, D4⋊5Q16
►On 64 points
(1 15 59 40)(2 16 60 33)(3 9 61 34)(4 10 62 35)(5 11 63 36)(6 12 64 37)(7 13 57 38)(8 14 58 39)(17 30 45 53)(18 31 46 54)(19 32 47 55)(20 25 48 56)(21 26 41 49)(22 27 42 50)(23 28 43 51)(24 29 44 52)
(1 36)(2 12)(3 38)(4 14)(5 40)(6 16)(7 34)(8 10)(9 57)(11 59)(13 61)(15 63)(17 26)(18 50)(19 28)(20 52)(21 30)(22 54)(23 32)(24 56)(25 44)(27 46)(29 48)(31 42)(33 64)(35 58)(37 60)(39 62)(41 53)(43 55)(45 49)(47 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)(57 58 59 60 61 62 63 64)
(1 28 5 32)(2 27 6 31)(3 26 7 30)(4 25 8 29)(9 41 13 45)(10 48 14 44)(11 47 15 43)(12 46 16 42)(17 34 21 38)(18 33 22 37)(19 40 23 36)(20 39 24 35)(49 57 53 61)(50 64 54 60)(51 63 55 59)(52 62 56 58)
G:=sub<Sym(64)| (1,15,59,40)(2,16,60,33)(3,9,61,34)(4,10,62,35)(5,11,63,36)(6,12,64,37)(7,13,57,38)(8,14,58,39)(17,30,45,53)(18,31,46,54)(19,32,47,55)(20,25,48,56)(21,26,41,49)(22,27,42,50)(23,28,43,51)(24,29,44,52), (1,36)(2,12)(3,38)(4,14)(5,40)(6,16)(7,34)(8,10)(9,57)(11,59)(13,61)(15,63)(17,26)(18,50)(19,28)(20,52)(21,30)(22,54)(23,32)(24,56)(25,44)(27,46)(29,48)(31,42)(33,64)(35,58)(37,60)(39,62)(41,53)(43,55)(45,49)(47,51), (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)(57,58,59,60,61,62,63,64), (1,28,5,32)(2,27,6,31)(3,26,7,30)(4,25,8,29)(9,41,13,45)(10,48,14,44)(11,47,15,43)(12,46,16,42)(17,34,21,38)(18,33,22,37)(19,40,23,36)(20,39,24,35)(49,57,53,61)(50,64,54,60)(51,63,55,59)(52,62,56,58)>;
G:=Group( (1,15,59,40)(2,16,60,33)(3,9,61,34)(4,10,62,35)(5,11,63,36)(6,12,64,37)(7,13,57,38)(8,14,58,39)(17,30,45,53)(18,31,46,54)(19,32,47,55)(20,25,48,56)(21,26,41,49)(22,27,42,50)(23,28,43,51)(24,29,44,52), (1,36)(2,12)(3,38)(4,14)(5,40)(6,16)(7,34)(8,10)(9,57)(11,59)(13,61)(15,63)(17,26)(18,50)(19,28)(20,52)(21,30)(22,54)(23,32)(24,56)(25,44)(27,46)(29,48)(31,42)(33,64)(35,58)(37,60)(39,62)(41,53)(43,55)(45,49)(47,51), (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)(57,58,59,60,61,62,63,64), (1,28,5,32)(2,27,6,31)(3,26,7,30)(4,25,8,29)(9,41,13,45)(10,48,14,44)(11,47,15,43)(12,46,16,42)(17,34,21,38)(18,33,22,37)(19,40,23,36)(20,39,24,35)(49,57,53,61)(50,64,54,60)(51,63,55,59)(52,62,56,58) );
G=PermutationGroup([[(1,15,59,40),(2,16,60,33),(3,9,61,34),(4,10,62,35),(5,11,63,36),(6,12,64,37),(7,13,57,38),(8,14,58,39),(17,30,45,53),(18,31,46,54),(19,32,47,55),(20,25,48,56),(21,26,41,49),(22,27,42,50),(23,28,43,51),(24,29,44,52)], [(1,36),(2,12),(3,38),(4,14),(5,40),(6,16),(7,34),(8,10),(9,57),(11,59),(13,61),(15,63),(17,26),(18,50),(19,28),(20,52),(21,30),(22,54),(23,32),(24,56),(25,44),(27,46),(29,48),(31,42),(33,64),(35,58),(37,60),(39,62),(41,53),(43,55),(45,49),(47,51)], [(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),(57,58,59,60,61,62,63,64)], [(1,28,5,32),(2,27,6,31),(3,26,7,30),(4,25,8,29),(9,41,13,45),(10,48,14,44),(11,47,15,43),(12,46,16,42),(17,34,21,38),(18,33,22,37),(19,40,23,36),(20,39,24,35),(49,57,53,61),(50,64,54,60),(51,63,55,59),(52,62,56,58)]])
35 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4K | 4L | ··· | 4Q | 8A | 8B | 8C | 8D | 8E | ··· | 8J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
35 irreducible representations
| dim | 1 | 1 | 1 | 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 | C2 | C2 | C2 | D4 | D4 | D4 | Q16 | C4○D4 | 2+ 1+4 | D4○SD16 |
| kernel | D4⋊5Q16 | C2×Q8⋊C4 | C8×D4 | C4×Q16 | C22⋊Q16 | C4⋊2Q16 | C8.18D4 | C4.Q16 | C23.48D4 | C4.SD16 | D4×Q8 | D4⋊3Q8 | C22⋊C4 | C4⋊C4 | C2×D4 | D4 | Q8 | C4 | C2 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 2 | 1 | 1 | 8 | 4 | 1 | 2 |
Matrix representation of D4⋊5Q16 ►in GL4(𝔽17) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 2 |
| 0 | 0 | 16 | 16 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 2 |
| 0 | 0 | 0 | 16 |
| 14 | 3 | 0 | 0 |
| 14 | 14 | 0 | 0 |
| 0 | 0 | 4 | 8 |
| 0 | 0 | 13 | 13 |
| 16 | 10 | 0 | 0 |
| 10 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,1,16,0,0,2,16],[16,0,0,0,0,16,0,0,0,0,1,0,0,0,2,16],[14,14,0,0,3,14,0,0,0,0,4,13,0,0,8,13],[16,10,0,0,10,1,0,0,0,0,1,0,0,0,0,1] >;
D4⋊5Q16 in GAP, Magma, Sage, TeX
D_4\rtimes_5Q_{16} % in TeX
G:=Group("D4:5Q16"); // GroupNames label
G:=SmallGroup(128,2031);
// by ID
G=gap.SmallGroup(128,2031);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,560,253,758,352,346,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^8=1,d^2=c^4,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^2*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations