direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: D4×D8, C42.446C23, C4.1332+ 1+4, D42⋊7C2, C4⋊2(C2×D8), C2.63D42, (C4×D8)⋊9C2, (C8×D4)⋊7C2, D4⋊5(C2×D4), C8⋊13(C2×D4), C8⋊7D4⋊9C2, C22⋊2(C2×D8), C22⋊D8⋊7C2, C4⋊D8⋊11C2, C8⋊4D4⋊11C2, C4⋊C8⋊61C22, C4⋊C4.252D4, (C4×C8)⋊11C22, (C2×D8)⋊8C22, (C2×D4).347D4, C2.41(D4○D8), (C4×D4)⋊21C22, (C22×D8)⋊12C2, C22⋊C4.92D4, C4.93(C22×D4), C2.18(C22×D8), C2.D8⋊59C22, D4⋊C4⋊5C22, C4⋊1D4⋊12C22, C4⋊D4⋊12C22, C4⋊C4.218C23, C22⋊C8⋊54C22, (C2×C8).178C23, (C2×C4).477C24, (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
Generators and relations for D4×D8
G = < a,b,c,d | a4=b2=c8=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 824 in 310 conjugacy classes, 104 normal (24 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, D4, C23, C23, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×C8, D8, D8, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C24, C4×C8, C22⋊C8, D4⋊C4, C4⋊C8, C2.D8, C4×D4, C4×D4, C22≀C2, C4⋊D4, C4⋊D4, C4⋊1D4, C22×C8, C2×D8, C2×D8, C2×D8, C22×D4, C22×D4, C8×D4, C4×D8, C22⋊D8, C4⋊D8, C8⋊7D4, C8⋊4D4, D42, C22×D8, D4×D8
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, C24, C2×D8, C22×D4, 2+ 1+4, D42, C22×D8, D4○D8, D4×D8
►On 32 points
(1 11 24 32)(2 12 17 25)(3 13 18 26)(4 14 19 27)(5 15 20 28)(6 16 21 29)(7 9 22 30)(8 10 23 31)
(1 28)(2 29)(3 30)(4 31)(5 32)(6 25)(7 26)(8 27)(9 18)(10 19)(11 20)(12 21)(13 22)(14 23)(15 24)(16 17)
(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 23)(2 22)(3 21)(4 20)(5 19)(6 18)(7 17)(8 24)(9 25)(10 32)(11 31)(12 30)(13 29)(14 28)(15 27)(16 26)
G:=sub<Sym(32)| (1,11,24,32)(2,12,17,25)(3,13,18,26)(4,14,19,27)(5,15,20,28)(6,16,21,29)(7,9,22,30)(8,10,23,31), (1,28)(2,29)(3,30)(4,31)(5,32)(6,25)(7,26)(8,27)(9,18)(10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,17), (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), (1,23)(2,22)(3,21)(4,20)(5,19)(6,18)(7,17)(8,24)(9,25)(10,32)(11,31)(12,30)(13,29)(14,28)(15,27)(16,26)>;
G:=Group( (1,11,24,32)(2,12,17,25)(3,13,18,26)(4,14,19,27)(5,15,20,28)(6,16,21,29)(7,9,22,30)(8,10,23,31), (1,28)(2,29)(3,30)(4,31)(5,32)(6,25)(7,26)(8,27)(9,18)(10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,17), (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), (1,23)(2,22)(3,21)(4,20)(5,19)(6,18)(7,17)(8,24)(9,25)(10,32)(11,31)(12,30)(13,29)(14,28)(15,27)(16,26) );
G=PermutationGroup([[(1,11,24,32),(2,12,17,25),(3,13,18,26),(4,14,19,27),(5,15,20,28),(6,16,21,29),(7,9,22,30),(8,10,23,31)], [(1,28),(2,29),(3,30),(4,31),(5,32),(6,25),(7,26),(8,27),(9,18),(10,19),(11,20),(12,21),(13,22),(14,23),(15,24),(16,17)], [(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)], [(1,23),(2,22),(3,21),(4,20),(5,19),(6,18),(7,17),(8,24),(9,25),(10,32),(11,31),(12,30),(13,29),(14,28),(15,27),(16,26)]])
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 |
Matrix representation of D4×D8 ►in 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] >;
D4×D8 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