p-group, metabelian, nilpotent (class 3), monomial
Aliases: D4⋊4D8, C42.459C23, C4.392+ (1+4), (D42)⋊8C2, (C8×D4)⋊8C2, (C4×D8)⋊11C2, C4.44(C2×D8), D4⋊5(C4○D4), C22⋊D8⋊9C2, D4⋊6D4⋊7C2, C8⋊7D4⋊11C2, C4⋊D8⋊13C2, C4⋊C8⋊62C22, (C4×C8)⋊12C22, C4⋊C4.258D4, (C2×D8)⋊9C22, C4⋊Q8⋊20C22, D4○2(D4⋊C4), C22.4(C2×D8), D4⋊Q8⋊12C2, (C2×D4).350D4, C4.4D8⋊15C2, (C4×D4)⋊23C22, C22⋊C4.98D4, C2.19(C22×D8), C2.D8⋊10C22, D4⋊C4⋊6C22, C4⋊D4⋊14C22, C4⋊C4.398C23, C22⋊C8⋊55C22, (C2×C8).181C23, (C2×C4).486C24, (C22×C8)⋊12C22, C22.D8⋊7C2, C23.469(C2×D4), C2.66(D4○SD16), (C2×D4).219C23, C4⋊1D4.82C22, C2.122(D4⋊5D4), C22.746(C22×D4), (C22×C4).1130C23, (C22×D4).405C22, (C2×D4)○(D4⋊C4), (C2×C4⋊C4)⋊55C22, C4.211(C2×C4○D4), (C2×C4).163(C2×D4), (C2×D4⋊C4)⋊25C2, SmallGroup(128,2026)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 608 in 247 conjugacy classes, 96 normal (38 characteristic)
C1, C2 [×3], C2 [×9], C4 [×2], C4 [×2], C4 [×7], C22, C22 [×4], C22 [×25], C8 [×4], C2×C4 [×3], C2×C4 [×2], C2×C4 [×16], D4 [×6], D4 [×22], Q8 [×2], C23 [×2], C23 [×14], C42, C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4 [×3], C4⋊C4 [×2], C4⋊C4 [×3], C2×C8 [×2], C2×C8 [×2], C2×C8 [×2], D8 [×4], C22×C4 [×2], C22×C4 [×4], C2×D4 [×3], C2×D4 [×2], C2×D4 [×18], C2×Q8, C4○D4 [×4], C24 [×2], C4×C8, C22⋊C8 [×2], D4⋊C4 [×2], D4⋊C4 [×8], C4⋊C8, C2.D8, C2.D8 [×2], C2×C4⋊C4 [×2], C4×D4 [×3], C22≀C2 [×2], C4⋊D4 [×4], C4⋊D4, C22⋊Q8, C22.D4 [×2], C4⋊1D4, C4⋊Q8, C22×C8 [×2], C2×D8, C2×D8 [×2], C22×D4 [×2], C22×D4, C2×C4○D4, C2×D4⋊C4 [×2], C8×D4, C4×D8, C22⋊D8 [×2], C4⋊D8, C8⋊7D4 [×2], D4⋊Q8, C22.D8 [×2], C4.4D8, D42, D4⋊6D4, D4⋊4D8
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⋊5D4, C22×D8, D4○SD16, D4⋊4D8
Generators and relations
G = < a,b,c,d | a4=b2=c8=d2=1, bab=a-1, ac=ca, ad=da, cbc-1=dbd=a2b, dcd=c-1 >
►On 32 points
(1 14 32 19)(2 15 25 20)(3 16 26 21)(4 9 27 22)(5 10 28 23)(6 11 29 24)(7 12 30 17)(8 13 31 18)
(1 23)(2 11)(3 17)(4 13)(5 19)(6 15)(7 21)(8 9)(10 32)(12 26)(14 28)(16 30)(18 27)(20 29)(22 31)(24 25)
(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)
(1 31)(2 30)(3 29)(4 28)(5 27)(6 26)(7 25)(8 32)(9 23)(10 22)(11 21)(12 20)(13 19)(14 18)(15 17)(16 24)
G:=sub<Sym(32)| (1,14,32,19)(2,15,25,20)(3,16,26,21)(4,9,27,22)(5,10,28,23)(6,11,29,24)(7,12,30,17)(8,13,31,18), (1,23)(2,11)(3,17)(4,13)(5,19)(6,15)(7,21)(8,9)(10,32)(12,26)(14,28)(16,30)(18,27)(20,29)(22,31)(24,25), (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), (1,31)(2,30)(3,29)(4,28)(5,27)(6,26)(7,25)(8,32)(9,23)(10,22)(11,21)(12,20)(13,19)(14,18)(15,17)(16,24)>;
G:=Group( (1,14,32,19)(2,15,25,20)(3,16,26,21)(4,9,27,22)(5,10,28,23)(6,11,29,24)(7,12,30,17)(8,13,31,18), (1,23)(2,11)(3,17)(4,13)(5,19)(6,15)(7,21)(8,9)(10,32)(12,26)(14,28)(16,30)(18,27)(20,29)(22,31)(24,25), (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), (1,31)(2,30)(3,29)(4,28)(5,27)(6,26)(7,25)(8,32)(9,23)(10,22)(11,21)(12,20)(13,19)(14,18)(15,17)(16,24) );
G=PermutationGroup([(1,14,32,19),(2,15,25,20),(3,16,26,21),(4,9,27,22),(5,10,28,23),(6,11,29,24),(7,12,30,17),(8,13,31,18)], [(1,23),(2,11),(3,17),(4,13),(5,19),(6,15),(7,21),(8,9),(10,32),(12,26),(14,28),(16,30),(18,27),(20,29),(22,31),(24,25)], [(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)], [(1,31),(2,30),(3,29),(4,28),(5,27),(6,26),(7,25),(8,32),(9,23),(10,22),(11,21),(12,20),(13,19),(14,18),(15,17),(16,24)])
Matrix representation ►G ⊆ GL4(𝔽17) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 2 |
| 0 | 0 | 16 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 2 |
| 0 | 0 | 0 | 1 |
| 0 | 11 | 0 | 0 |
| 3 | 11 | 0 | 0 |
| 0 | 0 | 13 | 8 |
| 0 | 0 | 13 | 4 |
| 0 | 11 | 0 | 0 |
| 14 | 0 | 0 | 0 |
| 0 | 0 | 4 | 9 |
| 0 | 0 | 4 | 13 |
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,16,16,0,0,2,1],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,2,1],[0,3,0,0,11,11,0,0,0,0,13,13,0,0,8,4],[0,14,0,0,11,0,0,0,0,0,4,4,0,0,9,13] >;
35 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 4A | 4B | 4C | 4D | 4E | ··· | 4I | 4J | 4K | 4L | 8A | 8B | 8C | 8D | 8E | ··· | 8J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 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 | D8 | C4○D4 | 2+ (1+4) | D4○SD16 |
| kernel | D4⋊4D8 | C2×D4⋊C4 | C8×D4 | C4×D8 | C22⋊D8 | C4⋊D8 | C8⋊7D4 | D4⋊Q8 | C22.D8 | C4.4D8 | D42 | D4⋊6D4 | C22⋊C4 | C4⋊C4 | C2×D4 | D4 | D4 | C4 | C2 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 2 | 1 | 1 | 8 | 4 | 1 | 2 |
In GAP, Magma, Sage, TeX
D_4\rtimes_4D_8
% in TeX
G:=Group("D4:4D8"); // GroupNames label
G:=SmallGroup(128,2026);
// by ID
G=gap.SmallGroup(128,2026);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,758,346,4037,1027,124]);
// 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,c*b*c^-1=d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations