p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.268D4, C42.729C23, C4.1022+ (1+4), C8⋊5D4⋊23C2, C4⋊Q16⋊9C2, D4.7D4⋊3C2, C4.29(C4○D8), C4.4D8⋊12C2, Q8.D4⋊1C2, C4⋊C8.312C22, C4⋊C4.149C23, (C2×C4).408C24, (C4×C8).265C22, (C2×C8).326C23, C4.SD16⋊26C2, C23.285(C2×D4), (C22×C4).498D4, C4⋊Q8.302C22, D4⋊C4.1C22, (C2×D4).157C23, C4.27(C8.C22), (C4×Q8).101C22, (C2×Q8).145C23, (C2×Q16).26C22, C42.12C4⋊34C2, C4⋊1D4.163C22, C22⋊C8.193C22, (C2×C42).875C22, (C2×SD16).85C22, C22.668(C22×D4), C22⋊Q8.193C22, (C22×C4).1079C23, Q8⋊C4.100C22, C4.4D4.150C22, C23.37C23⋊17C2, C2.79(C22.29C24), C22.26C24.41C2, C2.42(C2×C4○D8), (C2×C4).538(C2×D4), C2.55(C2×C8.C22), (C2×C4○D4).172C22, SmallGroup(128,1942)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 396 in 197 conjugacy classes, 88 normal (34 characteristic)
C1, C2 [×3], C2 [×3], C4 [×6], C4 [×9], C22, C22 [×9], C8 [×4], C2×C4 [×6], C2×C4 [×16], D4 [×12], Q8 [×10], C23, C23 [×2], C42 [×4], C42 [×2], C22⋊C4 [×6], C4⋊C4 [×2], C4⋊C4 [×9], C2×C8 [×4], SD16 [×4], Q16 [×4], C22×C4 [×3], C22×C4 [×2], C2×D4 [×2], C2×D4 [×4], C2×Q8 [×4], C2×Q8, C4○D4 [×4], C4×C8 [×2], C22⋊C8 [×2], D4⋊C4 [×4], Q8⋊C4 [×4], C4⋊C8 [×2], C2×C42, C42⋊C2, C4×D4 [×2], C4×Q8 [×2], C4×Q8, C4⋊D4 [×2], C22⋊Q8 [×2], C22⋊Q8, C4.4D4 [×2], C42.C2, C4⋊1D4, C4⋊Q8 [×3], C2×SD16 [×4], C2×Q16 [×4], C2×C4○D4 [×2], C42.12C4, D4.7D4 [×4], Q8.D4 [×4], C4.4D8, C4.SD16, C8⋊5D4, C4⋊Q16, C22.26C24, C23.37C23, C42.268D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C4○D8 [×2], C8.C22 [×2], C22×D4, 2+ (1+4) [×2], C22.29C24, C2×C4○D8, C2×C8.C22, C42.268D4
Generators and relations
G = < a,b,c,d | a4=b4=1, c4=d2=b2, ab=ba, ac=ca, dad-1=a-1, cbc-1=dbd-1=a2b, dcd-1=c3 >
►On 64 points
(1 46 51 59)(2 47 52 60)(3 48 53 61)(4 41 54 62)(5 42 55 63)(6 43 56 64)(7 44 49 57)(8 45 50 58)(9 24 29 37)(10 17 30 38)(11 18 31 39)(12 19 32 40)(13 20 25 33)(14 21 26 34)(15 22 27 35)(16 23 28 36)
(1 14 5 10)(2 27 6 31)(3 16 7 12)(4 29 8 25)(9 50 13 54)(11 52 15 56)(17 46 21 42)(18 60 22 64)(19 48 23 44)(20 62 24 58)(26 55 30 51)(28 49 32 53)(33 41 37 45)(34 63 38 59)(35 43 39 47)(36 57 40 61)
(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 4 5 8)(2 7 6 3)(9 10 13 14)(11 16 15 12)(17 33 21 37)(18 36 22 40)(19 39 23 35)(20 34 24 38)(25 26 29 30)(27 32 31 28)(41 63 45 59)(42 58 46 62)(43 61 47 57)(44 64 48 60)(49 56 53 52)(50 51 54 55)
G:=sub<Sym(64)| (1,46,51,59)(2,47,52,60)(3,48,53,61)(4,41,54,62)(5,42,55,63)(6,43,56,64)(7,44,49,57)(8,45,50,58)(9,24,29,37)(10,17,30,38)(11,18,31,39)(12,19,32,40)(13,20,25,33)(14,21,26,34)(15,22,27,35)(16,23,28,36), (1,14,5,10)(2,27,6,31)(3,16,7,12)(4,29,8,25)(9,50,13,54)(11,52,15,56)(17,46,21,42)(18,60,22,64)(19,48,23,44)(20,62,24,58)(26,55,30,51)(28,49,32,53)(33,41,37,45)(34,63,38,59)(35,43,39,47)(36,57,40,61), (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), (1,4,5,8)(2,7,6,3)(9,10,13,14)(11,16,15,12)(17,33,21,37)(18,36,22,40)(19,39,23,35)(20,34,24,38)(25,26,29,30)(27,32,31,28)(41,63,45,59)(42,58,46,62)(43,61,47,57)(44,64,48,60)(49,56,53,52)(50,51,54,55)>;
G:=Group( (1,46,51,59)(2,47,52,60)(3,48,53,61)(4,41,54,62)(5,42,55,63)(6,43,56,64)(7,44,49,57)(8,45,50,58)(9,24,29,37)(10,17,30,38)(11,18,31,39)(12,19,32,40)(13,20,25,33)(14,21,26,34)(15,22,27,35)(16,23,28,36), (1,14,5,10)(2,27,6,31)(3,16,7,12)(4,29,8,25)(9,50,13,54)(11,52,15,56)(17,46,21,42)(18,60,22,64)(19,48,23,44)(20,62,24,58)(26,55,30,51)(28,49,32,53)(33,41,37,45)(34,63,38,59)(35,43,39,47)(36,57,40,61), (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), (1,4,5,8)(2,7,6,3)(9,10,13,14)(11,16,15,12)(17,33,21,37)(18,36,22,40)(19,39,23,35)(20,34,24,38)(25,26,29,30)(27,32,31,28)(41,63,45,59)(42,58,46,62)(43,61,47,57)(44,64,48,60)(49,56,53,52)(50,51,54,55) );
G=PermutationGroup([(1,46,51,59),(2,47,52,60),(3,48,53,61),(4,41,54,62),(5,42,55,63),(6,43,56,64),(7,44,49,57),(8,45,50,58),(9,24,29,37),(10,17,30,38),(11,18,31,39),(12,19,32,40),(13,20,25,33),(14,21,26,34),(15,22,27,35),(16,23,28,36)], [(1,14,5,10),(2,27,6,31),(3,16,7,12),(4,29,8,25),(9,50,13,54),(11,52,15,56),(17,46,21,42),(18,60,22,64),(19,48,23,44),(20,62,24,58),(26,55,30,51),(28,49,32,53),(33,41,37,45),(34,63,38,59),(35,43,39,47),(36,57,40,61)], [(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)], [(1,4,5,8),(2,7,6,3),(9,10,13,14),(11,16,15,12),(17,33,21,37),(18,36,22,40),(19,39,23,35),(20,34,24,38),(25,26,29,30),(27,32,31,28),(41,63,45,59),(42,58,46,62),(43,61,47,57),(44,64,48,60),(49,56,53,52),(50,51,54,55)])
Matrix representation ►G ⊆ GL6(𝔽17)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 12 | 5 | 0 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 5 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 0 | 0 | 12 | 5 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 12 | 5 | 0 | 0 | 0 | 0 |
| 5 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 5 |
| 0 | 0 | 0 | 0 | 5 | 5 |
| 0 | 0 | 12 | 5 | 0 | 0 |
| 0 | 0 | 5 | 5 | 0 | 0 |
G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,1,0,0],[12,12,0,0,0,0,5,12,0,0,0,0,0,0,0,0,12,12,0,0,0,0,5,12,0,0,12,12,0,0,0,0,5,12,0,0],[12,5,0,0,0,0,5,5,0,0,0,0,0,0,0,0,12,5,0,0,0,0,5,5,0,0,12,5,0,0,0,0,5,5,0,0] >;
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | ··· | 4J | 4K | 4L | ··· | 4Q | 8A | ··· | 8H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 4 | 8 | 8 | 2 | ··· | 2 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D8 | C8.C22 | 2+ (1+4) |
| kernel | C42.268D4 | C42.12C4 | D4.7D4 | Q8.D4 | C4.4D8 | C4.SD16 | C8⋊5D4 | C4⋊Q16 | C22.26C24 | C23.37C23 | C42 | C22×C4 | C4 | C4 | C4 |
| # reps | 1 | 1 | 4 | 4 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{268}D_4 % in TeX
G:=Group("C4^2.268D4"); // GroupNames label
G:=SmallGroup(128,1942);
// by ID
G=gap.SmallGroup(128,1942);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,120,758,219,675,248,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=d^2=b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=a^2*b,d*c*d^-1=c^3>;
// generators/relations