direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: D4×D8, C42.446C23, C4.1332+ (1+4), C4⋊2(C2×D8), (D42)⋊7C2, D4⋊5(C2×D4), (C4×D8)⋊9C2, (C8×D4)⋊7C2, C8⋊13(C2×D4), C8⋊7D4⋊9C2, C2.63(D42), C22⋊2(C2×D8), C22⋊D8⋊7C2, C4⋊D8⋊11C2, C8⋊4D4⋊11C2, C4⋊C8⋊61C22, (C4×C8)⋊11C22, C4⋊C4.252D4, (C2×D8)⋊8C22, (C2×D4).347D4, C2.41(D4○D8), (C4×D4)⋊21C22, (C22×D8)⋊12C2, C22⋊C4.92D4, C2.18(C22×D8), C4.93(C22×D4), C2.D8⋊59C22, D4⋊C4⋊5C22, C4⋊1D4⋊12C22, C4⋊C4.218C23, C4⋊D4⋊12C22, C22⋊C8⋊54C22, (C2×C4).477C24, (C2×C8).178C23, (C22×C8)⋊11C22, C23.463(C2×D4), (C2×D4).416C23, (C22×D4)⋊28C22, C22.737(C22×D4), (C22×C4).1121C23, (C2×D4)○(C2×D8), (C2×C4).160(C2×D4), SmallGroup(128,2011)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 824 in 310 conjugacy classes, 104 normal (24 characteristic)
C1, C2 [×3], C2 [×12], C4 [×2], C4 [×2], C4 [×5], C22, C22 [×4], C22 [×40], C8 [×2], C8 [×3], C2×C4 [×3], C2×C4 [×2], C2×C4 [×10], D4 [×8], D4 [×32], C23 [×2], C23 [×26], C42, C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4, C4⋊C4 [×2], C2×C8 [×2], C2×C8 [×2], C2×C8 [×4], D8 [×4], D8 [×14], C22×C4 [×2], C22×C4 [×2], C2×D4, C2×D4 [×6], C2×D4 [×32], C24 [×4], C4×C8, C22⋊C8 [×2], D4⋊C4 [×6], C4⋊C8, C2.D8, C4×D4, C4×D4 [×2], C22≀C2 [×4], C4⋊D4 [×4], C4⋊D4 [×2], C4⋊1D4 [×2], C22×C8 [×2], C2×D8, C2×D8 [×8], C2×D8 [×8], C22×D4 [×4], C22×D4 [×2], C8×D4, C4×D8, C22⋊D8 [×4], C4⋊D8 [×2], C8⋊7D4 [×2], C8⋊4D4, D42 [×2], C22×D8 [×2], D4×D8
Quotients:
C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], D8 [×4], C2×D4 [×12], C24, C2×D8 [×6], C22×D4 [×2], 2+ (1+4), D42, C22×D8, D4○D8, D4×D8
Generators and relations
G = < a,b,c,d | a4=b2=c8=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
►On 32 points
(1 11 22 32)(2 12 23 25)(3 13 24 26)(4 14 17 27)(5 15 18 28)(6 16 19 29)(7 9 20 30)(8 10 21 31)
(1 28)(2 29)(3 30)(4 31)(5 32)(6 25)(7 26)(8 27)(9 24)(10 17)(11 18)(12 19)(13 20)(14 21)(15 22)(16 23)
(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 21)(2 20)(3 19)(4 18)(5 17)(6 24)(7 23)(8 22)(9 25)(10 32)(11 31)(12 30)(13 29)(14 28)(15 27)(16 26)
G:=sub<Sym(32)| (1,11,22,32)(2,12,23,25)(3,13,24,26)(4,14,17,27)(5,15,18,28)(6,16,19,29)(7,9,20,30)(8,10,21,31), (1,28)(2,29)(3,30)(4,31)(5,32)(6,25)(7,26)(8,27)(9,24)(10,17)(11,18)(12,19)(13,20)(14,21)(15,22)(16,23), (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), (1,21)(2,20)(3,19)(4,18)(5,17)(6,24)(7,23)(8,22)(9,25)(10,32)(11,31)(12,30)(13,29)(14,28)(15,27)(16,26)>;
G:=Group( (1,11,22,32)(2,12,23,25)(3,13,24,26)(4,14,17,27)(5,15,18,28)(6,16,19,29)(7,9,20,30)(8,10,21,31), (1,28)(2,29)(3,30)(4,31)(5,32)(6,25)(7,26)(8,27)(9,24)(10,17)(11,18)(12,19)(13,20)(14,21)(15,22)(16,23), (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), (1,21)(2,20)(3,19)(4,18)(5,17)(6,24)(7,23)(8,22)(9,25)(10,32)(11,31)(12,30)(13,29)(14,28)(15,27)(16,26) );
G=PermutationGroup([(1,11,22,32),(2,12,23,25),(3,13,24,26),(4,14,17,27),(5,15,18,28),(6,16,19,29),(7,9,20,30),(8,10,21,31)], [(1,28),(2,29),(3,30),(4,31),(5,32),(6,25),(7,26),(8,27),(9,24),(10,17),(11,18),(12,19),(13,20),(14,21),(15,22),(16,23)], [(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)], [(1,21),(2,20),(3,19),(4,18),(5,17),(6,24),(7,23),(8,22),(9,25),(10,32),(11,31),(12,30),(13,29),(14,28),(15,27),(16,26)])
Matrix representation ►G ⊆ GL4(𝔽17) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 14 | 2 |
| 0 | 0 | 12 | 3 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 14 | 2 |
| 0 | 0 | 13 | 3 |
| 14 | 3 | 0 | 0 |
| 14 | 14 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 14 | 3 | 0 | 0 |
| 3 | 3 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,14,12,0,0,2,3],[1,0,0,0,0,1,0,0,0,0,14,13,0,0,2,3],[14,14,0,0,3,14,0,0,0,0,16,0,0,0,0,16],[14,3,0,0,3,3,0,0,0,0,1,0,0,0,0,1] >;
35 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 2N | 2O | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 8A | 8B | 8C | 8D | 8E | ··· | 8J |
| order | 1 | 2 | 2 | 2 | 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 | 4 | 4 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 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 | D4 | D8 | 2+ (1+4) | D4○D8 |
| kernel | D4×D8 | C8×D4 | C4×D8 | C22⋊D8 | C4⋊D8 | C8⋊7D4 | C8⋊4D4 | D42 | C22×D8 | C22⋊C4 | C4⋊C4 | D8 | C2×D4 | D4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 4 | 2 | 2 | 1 | 2 | 2 | 2 | 1 | 4 | 1 | 8 | 1 | 2 |
In GAP, Magma, Sage, TeX
D_4\times D_8
% in TeX
G:=Group("D4xD8"); // GroupNames label
G:=SmallGroup(128,2011);
// by ID
G=gap.SmallGroup(128,2011);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,2019,346,2804,1411,375,172]);
// 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,b*d=d*b,d*c*d=c^-1>;
// generators/relations