direct product, p-group, metabelian, nilpotent (class 3), monomial
Aliases: C8×D8, C82⋊4C2, C42.632C23, C8⋊4(C2×C8), C8○(D4⋊C8), D4⋊1(C2×C8), (C8×D4)⋊1C2, C8○(C8⋊1C8), C8○(C2.D8), C2.4(C8×D4), C2.1(C4×D8), D4⋊C8⋊38C2, C8⋊1C8⋊32C2, C4.85(C2×D8), C8○(D4⋊C4), (C4×D8).21C2, (C2×D8).15C4, C4.7(C8○D4), C2.1(C8○D8), (C2×C8).217D4, C4.7(C22×C8), C2.D8.22C4, C22.77(C4×D4), D4⋊C4.13C4, C4.125(C4○D8), C4⋊C8.270C22, (C4×C8).366C22, (C4×D4).268C22, (C2×C8)○(C4×D8), (C2×C8)○(D4⋊C8), (C2×C8)○(C8⋊1C8), C4⋊C4.126(C2×C4), (C2×C8).154(C2×C4), (C2×D4).148(C2×C4), (C2×C4).1468(C2×D4), (C2×C4).493(C4○D4), (C2×C4).324(C22×C4), 2-Sylow(CO(3,9)), SmallGroup(128,307)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8×D8
G = < a,b,c | a8=b8=c2=1, ab=ba, ac=ca, cbc=b-1 >
Subgroups: 184 in 98 conjugacy classes, 52 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, D4, C23, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, D8, C22×C4, C2×D4, C4×C8, C22⋊C8, D4⋊C4, C4⋊C8, C2.D8, C4×D4, C22×C8, C2×D8, C82, D4⋊C8, C8⋊1C8, C8×D4, C4×D8, C8×D8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C23, C2×C8, D8, C22×C4, C2×D4, C4○D4, C4×D4, C22×C8, C8○D4, C2×D8, C4○D8, C8×D4, C4×D8, C8○D8, C8×D8
►On 64 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)(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 55 16 21 31 63 39 48)(2 56 9 22 32 64 40 41)(3 49 10 23 25 57 33 42)(4 50 11 24 26 58 34 43)(5 51 12 17 27 59 35 44)(6 52 13 18 28 60 36 45)(7 53 14 19 29 61 37 46)(8 54 15 20 30 62 38 47)
(1 5)(2 6)(3 7)(4 8)(9 36)(10 37)(11 38)(12 39)(13 40)(14 33)(15 34)(16 35)(17 63)(18 64)(19 57)(20 58)(21 59)(22 60)(23 61)(24 62)(25 29)(26 30)(27 31)(28 32)(41 52)(42 53)(43 54)(44 55)(45 56)(46 49)(47 50)(48 51)
G:=sub<Sym(64)| (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), (1,55,16,21,31,63,39,48)(2,56,9,22,32,64,40,41)(3,49,10,23,25,57,33,42)(4,50,11,24,26,58,34,43)(5,51,12,17,27,59,35,44)(6,52,13,18,28,60,36,45)(7,53,14,19,29,61,37,46)(8,54,15,20,30,62,38,47), (1,5)(2,6)(3,7)(4,8)(9,36)(10,37)(11,38)(12,39)(13,40)(14,33)(15,34)(16,35)(17,63)(18,64)(19,57)(20,58)(21,59)(22,60)(23,61)(24,62)(25,29)(26,30)(27,31)(28,32)(41,52)(42,53)(43,54)(44,55)(45,56)(46,49)(47,50)(48,51)>;
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)(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), (1,55,16,21,31,63,39,48)(2,56,9,22,32,64,40,41)(3,49,10,23,25,57,33,42)(4,50,11,24,26,58,34,43)(5,51,12,17,27,59,35,44)(6,52,13,18,28,60,36,45)(7,53,14,19,29,61,37,46)(8,54,15,20,30,62,38,47), (1,5)(2,6)(3,7)(4,8)(9,36)(10,37)(11,38)(12,39)(13,40)(14,33)(15,34)(16,35)(17,63)(18,64)(19,57)(20,58)(21,59)(22,60)(23,61)(24,62)(25,29)(26,30)(27,31)(28,32)(41,52)(42,53)(43,54)(44,55)(45,56)(46,49)(47,50)(48,51) );
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),(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)], [(1,55,16,21,31,63,39,48),(2,56,9,22,32,64,40,41),(3,49,10,23,25,57,33,42),(4,50,11,24,26,58,34,43),(5,51,12,17,27,59,35,44),(6,52,13,18,28,60,36,45),(7,53,14,19,29,61,37,46),(8,54,15,20,30,62,38,47)], [(1,5),(2,6),(3,7),(4,8),(9,36),(10,37),(11,38),(12,39),(13,40),(14,33),(15,34),(16,35),(17,63),(18,64),(19,57),(20,58),(21,59),(22,60),(23,61),(24,62),(25,29),(26,30),(27,31),(28,32),(41,52),(42,53),(43,54),(44,55),(45,56),(46,49),(47,50),(48,51)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 8A | ··· | 8H | 8I | ··· | 8AB | 8AC | ··· | 8AJ |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C8 | D4 | D8 | C4○D4 | C8○D4 | C4○D8 | C8○D8 |
| kernel | C8×D8 | C82 | D4⋊C8 | C8⋊1C8 | C8×D4 | C4×D8 | D4⋊C4 | C2.D8 | C2×D8 | D8 | C2×C8 | C8 | C2×C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 4 | 2 | 2 | 16 | 2 | 4 | 2 | 4 | 4 | 8 |
Matrix representation of C8×D8 ►in GL4(𝔽17) generated by
| 15 | 0 | 0 | 0 |
| 0 | 15 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 13 |
| 9 | 16 | 0 | 0 |
| 14 | 8 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 11 | 2 |
| 16 | 0 | 0 | 0 |
| 16 | 1 | 0 | 0 |
| 0 | 0 | 1 | 4 |
| 0 | 0 | 0 | 16 |
G:=sub<GL(4,GF(17))| [15,0,0,0,0,15,0,0,0,0,13,0,0,0,0,13],[9,14,0,0,16,8,0,0,0,0,9,11,0,0,0,2],[16,16,0,0,0,1,0,0,0,0,1,0,0,0,4,16] >;
C8×D8 in GAP, Magma, Sage, TeX
C_8\times D_8
% in TeX
G:=Group("C8xD8"); // GroupNames label
G:=SmallGroup(128,307);
// by ID
G=gap.SmallGroup(128,307);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,100,1123,570,136,172]);
// Polycyclic
G:=Group<a,b,c|a^8=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations