p-group, metabelian, nilpotent (class 3), monomial
Aliases: (C2×C8).51D4, (C2×C4).23D8, C4⋊C4.108D4, (C2×D4).13Q8, C22.87(C2×D8), C2.15(C4⋊D8), C2.15(C8⋊7D4), C2.20(C8⋊D4), C2.8(D4⋊Q8), C23.924(C2×D4), (C22×C4).156D4, C2.11(D4.Q8), C4.37(C22⋊Q8), C4.155(C4⋊D4), C22.4Q16⋊26C2, (C22×C8).82C22, C4.35(C42⋊2C2), C22.118(C4○D8), (C2×C42).378C22, C2.21(Q8.D4), (C22×D4).90C22, C2.8(C23.Q8), C22.245(C4⋊D4), C22.146(C8⋊C22), (C22×C4).1458C23, C23.65C23⋊8C2, C22.111(C22⋊Q8), C22.135(C8.C22), C24.3C22.18C2, (C2×C4⋊C8)⋊21C2, (C2×C2.D8)⋊9C2, (C2×C4).285(C2×Q8), (C2×C4).1057(C2×D4), (C2×D4⋊C4).16C2, (C2×C4).622(C4○D4), (C2×C4⋊C4).143C22, SmallGroup(128,799)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for (C2×C8).51D4
G = < a,b,c,d | a2=b8=c4=1, d2=b6, dbd-1=ab=ba, ac=ca, ad=da, cbc-1=b-1, dcd-1=b6c-1 >
Subgroups: 344 in 142 conjugacy classes, 52 normal (44 characteristic)
C1, C2 [×7], C2 [×2], C4 [×4], C4 [×7], C22 [×7], C22 [×10], C8 [×3], C2×C4 [×6], C2×C4 [×2], C2×C4 [×15], D4 [×6], C23, C23 [×8], C42 [×2], C22⋊C4 [×4], C4⋊C4 [×2], C4⋊C4 [×7], C2×C8 [×2], C2×C8 [×5], C22×C4 [×3], C22×C4 [×3], C2×D4 [×2], C2×D4 [×5], C24, C2.C42, D4⋊C4 [×4], C4⋊C8 [×2], C2.D8 [×2], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4 [×3], C2×C4⋊C4, C22×C8 [×2], C22×D4, C22.4Q16, C23.65C23, C24.3C22, C2×D4⋊C4 [×2], C2×C4⋊C8, C2×C2.D8, (C2×C8).51D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], Q8 [×2], C23, D8 [×2], C2×D4 [×3], C2×Q8, C4○D4 [×3], C4⋊D4 [×3], C22⋊Q8 [×3], C42⋊2C2, C2×D8, C4○D8, C8⋊C22, C8.C22, C23.Q8, C4⋊D8, Q8.D4, C8⋊7D4, C8⋊D4, D4⋊Q8, D4.Q8, (C2×C8).51D4
►On 64 points
(1 44)(2 45)(3 46)(4 47)(5 48)(6 41)(7 42)(8 43)(9 25)(10 26)(11 27)(12 28)(13 29)(14 30)(15 31)(16 32)(17 58)(18 59)(19 60)(20 61)(21 62)(22 63)(23 64)(24 57)(33 55)(34 56)(35 49)(36 50)(37 51)(38 52)(39 53)(40 54)
(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 10 53 17)(2 9 54 24)(3 16 55 23)(4 15 56 22)(5 14 49 21)(6 13 50 20)(7 12 51 19)(8 11 52 18)(25 40 57 45)(26 39 58 44)(27 38 59 43)(28 37 60 42)(29 36 61 41)(30 35 62 48)(31 34 63 47)(32 33 64 46)
(1 23 7 21 5 19 3 17)(2 57 8 63 6 61 4 59)(9 38 15 36 13 34 11 40)(10 53 16 51 14 49 12 55)(18 45 24 43 22 41 20 47)(25 52 31 50 29 56 27 54)(26 39 32 37 30 35 28 33)(42 62 48 60 46 58 44 64)
G:=sub<Sym(64)| (1,44)(2,45)(3,46)(4,47)(5,48)(6,41)(7,42)(8,43)(9,25)(10,26)(11,27)(12,28)(13,29)(14,30)(15,31)(16,32)(17,58)(18,59)(19,60)(20,61)(21,62)(22,63)(23,64)(24,57)(33,55)(34,56)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54), (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), (1,10,53,17)(2,9,54,24)(3,16,55,23)(4,15,56,22)(5,14,49,21)(6,13,50,20)(7,12,51,19)(8,11,52,18)(25,40,57,45)(26,39,58,44)(27,38,59,43)(28,37,60,42)(29,36,61,41)(30,35,62,48)(31,34,63,47)(32,33,64,46), (1,23,7,21,5,19,3,17)(2,57,8,63,6,61,4,59)(9,38,15,36,13,34,11,40)(10,53,16,51,14,49,12,55)(18,45,24,43,22,41,20,47)(25,52,31,50,29,56,27,54)(26,39,32,37,30,35,28,33)(42,62,48,60,46,58,44,64)>;
G:=Group( (1,44)(2,45)(3,46)(4,47)(5,48)(6,41)(7,42)(8,43)(9,25)(10,26)(11,27)(12,28)(13,29)(14,30)(15,31)(16,32)(17,58)(18,59)(19,60)(20,61)(21,62)(22,63)(23,64)(24,57)(33,55)(34,56)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54), (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), (1,10,53,17)(2,9,54,24)(3,16,55,23)(4,15,56,22)(5,14,49,21)(6,13,50,20)(7,12,51,19)(8,11,52,18)(25,40,57,45)(26,39,58,44)(27,38,59,43)(28,37,60,42)(29,36,61,41)(30,35,62,48)(31,34,63,47)(32,33,64,46), (1,23,7,21,5,19,3,17)(2,57,8,63,6,61,4,59)(9,38,15,36,13,34,11,40)(10,53,16,51,14,49,12,55)(18,45,24,43,22,41,20,47)(25,52,31,50,29,56,27,54)(26,39,32,37,30,35,28,33)(42,62,48,60,46,58,44,64) );
G=PermutationGroup([(1,44),(2,45),(3,46),(4,47),(5,48),(6,41),(7,42),(8,43),(9,25),(10,26),(11,27),(12,28),(13,29),(14,30),(15,31),(16,32),(17,58),(18,59),(19,60),(20,61),(21,62),(22,63),(23,64),(24,57),(33,55),(34,56),(35,49),(36,50),(37,51),(38,52),(39,53),(40,54)], [(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)], [(1,10,53,17),(2,9,54,24),(3,16,55,23),(4,15,56,22),(5,14,49,21),(6,13,50,20),(7,12,51,19),(8,11,52,18),(25,40,57,45),(26,39,58,44),(27,38,59,43),(28,37,60,42),(29,36,61,41),(30,35,62,48),(31,34,63,47),(32,33,64,46)], [(1,23,7,21,5,19,3,17),(2,57,8,63,6,61,4,59),(9,38,15,36,13,34,11,40),(10,53,16,51,14,49,12,55),(18,45,24,43,22,41,20,47),(25,52,31,50,29,56,27,54),(26,39,32,37,30,35,28,33),(42,62,48,60,46,58,44,64)])
32 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 8A | ··· | 8H |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | Q8 | D8 | C4○D4 | C4○D8 | C8⋊C22 | C8.C22 |
| kernel | (C2×C8).51D4 | C22.4Q16 | C23.65C23 | C24.3C22 | C2×D4⋊C4 | C2×C4⋊C8 | C2×C2.D8 | C4⋊C4 | C2×C8 | C22×C4 | C2×D4 | C2×C4 | C2×C4 | C22 | C22 | C22 |
| # reps | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 6 | 4 | 1 | 1 |
Matrix representation of (C2×C8).51D4 ►in GL6(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 16 | 15 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 14 | 0 | 0 |
| 0 | 0 | 3 | 14 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 8 |
| 0 | 0 | 0 | 0 | 4 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 5 | 0 | 0 |
| 0 | 0 | 5 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 11 |
| 0 | 0 | 0 | 0 | 3 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 16 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 5 | 0 | 0 |
| 0 | 0 | 12 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 11 |
| 0 | 0 | 0 | 0 | 3 | 0 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,0,0,0,0,0,15,1,0,0,0,0,0,0,14,3,0,0,0,0,14,14,0,0,0,0,0,0,16,4,0,0,0,0,8,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,5,5,0,0,0,0,5,12,0,0,0,0,0,0,0,3,0,0,0,0,11,0],[1,16,0,0,0,0,0,16,0,0,0,0,0,0,5,12,0,0,0,0,5,5,0,0,0,0,0,0,0,3,0,0,0,0,11,0] >;
(C2×C8).51D4 in GAP, Magma, Sage, TeX
(C_2\times C_8)._{51}D_4 % in TeX
G:=Group("(C2xC8).51D4"); // GroupNames label
G:=SmallGroup(128,799);
// by ID
G=gap.SmallGroup(128,799);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,168,141,512,422,387,2019,521,248,2804,718,172,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^8=c^4=1,d^2=b^6,d*b*d^-1=a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*c*d^-1=b^6*c^-1>;
// generators/relations