direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C8×D4, C42.69C22, C8○(C4⋊C8), (C4×C8)⋊4C2, C4⋊1(C2×C8), C8○2(C4⋊C4), C4⋊C8⋊18C2, C2.3(C4×D4), C4⋊C4.11C4, C8○(C22⋊C8), C22⋊1(C2×C8), (C22×C8)⋊5C2, C4.76(C2×D4), C8○2(C22⋊C4), C22⋊C8⋊15C2, (C2×D4).11C4, (C4×D4).14C2, C2.2(C8○D4), C22⋊C4.7C4, C2.4(C22×C8), C4.51(C4○D4), (C2×C8).62C22, C23.18(C2×C4), (C2×C4).153C23, (C22×C4).95C22, C22.23(C22×C4), C4⋊C4○(C2×C8), (C2×C8)○(C2×D4), (C2×C8)○(C4×D4), (C2×C8)○(C4⋊C8), C22⋊C4○(C2×C8), (C2×C8)○(C22⋊C8), (C2×C4).35(C2×C4), SmallGroup(64,115)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8×D4
G = < a,b,c | a8=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >
Subgroups: 89 in 67 conjugacy classes, 45 normal (19 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, C23, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, C2×C8, C22×C4, C2×D4, C4×C8, C22⋊C8, C4⋊C8, C4×D4, C22×C8, C8×D4
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C23, C2×C8, C22×C4, C2×D4, C4○D4, C4×D4, C22×C8, C8○D4, C8×D4
►On 32 points
(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 15 19 30)(2 16 20 31)(3 9 21 32)(4 10 22 25)(5 11 23 26)(6 12 24 27)(7 13 17 28)(8 14 18 29)
(1 5)(2 6)(3 7)(4 8)(9 28)(10 29)(11 30)(12 31)(13 32)(14 25)(15 26)(16 27)(17 21)(18 22)(19 23)(20 24)
G:=sub<Sym(32)| (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), (1,15,19,30)(2,16,20,31)(3,9,21,32)(4,10,22,25)(5,11,23,26)(6,12,24,27)(7,13,17,28)(8,14,18,29), (1,5)(2,6)(3,7)(4,8)(9,28)(10,29)(11,30)(12,31)(13,32)(14,25)(15,26)(16,27)(17,21)(18,22)(19,23)(20,24)>;
G:=Group( (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), (1,15,19,30)(2,16,20,31)(3,9,21,32)(4,10,22,25)(5,11,23,26)(6,12,24,27)(7,13,17,28)(8,14,18,29), (1,5)(2,6)(3,7)(4,8)(9,28)(10,29)(11,30)(12,31)(13,32)(14,25)(15,26)(16,27)(17,21)(18,22)(19,23)(20,24) );
G=PermutationGroup([[(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)], [(1,15,19,30),(2,16,20,31),(3,9,21,32),(4,10,22,25),(5,11,23,26),(6,12,24,27),(7,13,17,28),(8,14,18,29)], [(1,5),(2,6),(3,7),(4,8),(9,28),(10,29),(11,30),(12,31),(13,32),(14,25),(15,26),(16,27),(17,21),(18,22),(19,23),(20,24)]])
C8×D4 is a maximal subgroup of
D4⋊C16 SD16⋊C8 C8⋊9D8 C8⋊12SD16 D4.M4(2) D4⋊2M4(2) C8⋊8D8 C8⋊7D8 C8.28D8 C8⋊11SD16 C8⋊10SD16 D4.1Q16 D4.2SD16 D4.3SD16 D4.2D8 D4.Q16 C16⋊9D4 C16⋊6D4 C42.264C23 C42.681C23 M4(2)⋊22D4 M4(2)⋊23D4 C42.291C23 C42.293C23 C42.294C23 D4⋊6M4(2) D4⋊7M4(2) C42.297C23 C42.298C23 C42.694C23 C42.301C23 D4⋊8M4(2) C42.307C23 C42.308C23 C42.309C23 D8⋊12D4 SD16⋊10D4 D8⋊13D4 SD16⋊11D4 Q16⋊12D4 Q16⋊13D4 D4⋊4D8 D4⋊7SD16 C42.461C23 C42.462C23 D4⋊8SD16 D4⋊5Q16 C42.465C23 C42.466C23 C42.467C23 C42.468C23 C42.469C23 C42.470C23 D4⋊5D8 D4⋊9SD16 C42.485C23 C42.486C23 D4⋊6Q16 C42.488C23 C42.489C23 C42.490C23 C42.491C23
D4p⋊C8: D8⋊5C8 D12⋊C8 D20⋊5C8 D20⋊2C8 D28⋊C8 ...
D2p⋊(C2×C8): C42.691C23 C42.697C23 C3⋊D4⋊C8 C5⋊5(C8×D4) C5⋊C8⋊8D4 C7⋊D4⋊C8 ...
C8×D4 is a maximal quotient of
SD16⋊C8 Q16⋊5C8 C23.21M4(2) C23.22M4(2) C4⋊C4⋊3C8 C22⋊C4⋊4C8 C42.61Q8 C42.325D4
D4p⋊C8: D8⋊5C8 D12⋊C8 D20⋊5C8 D20⋊2C8 D28⋊C8 ...
C2p.(C4×D4): C16⋊9D4 C16⋊6D4 C16○D8 D8.C8 C3⋊D4⋊C8 C5⋊5(C8×D4) C5⋊C8⋊8D4 C7⋊D4⋊C8 ...
40 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 8A | ··· | 8H | 8I | ··· | 8T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C8 | D4 | C4○D4 | C8○D4 |
| kernel | C8×D4 | C4×C8 | C22⋊C8 | C4⋊C8 | C4×D4 | C22×C8 | C22⋊C4 | C4⋊C4 | C2×D4 | D4 | C8 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 2 | 4 | 2 | 2 | 16 | 2 | 2 | 4 |
Matrix representation of C8×D4 ►in GL3(𝔽17) generated by
| 15 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 16 | 0 | 0 |
| 0 | 0 | 16 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 16 |
G:=sub<GL(3,GF(17))| [15,0,0,0,1,0,0,0,1],[16,0,0,0,0,1,0,16,0],[1,0,0,0,1,0,0,0,16] >;
C8×D4 in GAP, Magma, Sage, TeX
C_8\times D_4
% in TeX
G:=Group("C8xD4"); // GroupNames label
G:=SmallGroup(64,115);
// by ID
G=gap.SmallGroup(64,115);
# by ID
G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,86,88]);
// Polycyclic
G:=Group<a,b,c|a^8=b^4=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations