p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.255D4, C42.383C23, C4○2(C8⋊Q8), C8⋊Q8⋊37C2, (C4×D8)⋊29C2, C8⋊1(C4○D4), C4○2(C8⋊2D4), C4○2(C8⋊3D4), C4○2(C8⋊D4), C8⋊3D4⋊25C2, C8⋊D4⋊58C2, C8⋊2D4⋊36C2, (C4×SD16)⋊15C2, (C4×M4(2))⋊7C2, C4⋊C4.110C23, (C2×C8).276C23, (C2×C4).369C24, (C4×C8).181C22, (C4×D4).90C22, C23.265(C2×D4), (C22×C4).475D4, C4⋊Q8.292C22, (C4×Q8).87C22, C4.150(C8⋊C22), (C2×D8).163C22, (C2×D4).124C23, (C2×Q8).112C23, C8⋊C4.126C22, C2.D8.218C22, C4.Q8.134C22, C4⋊1D4.156C22, C4⋊D4.173C22, (C2×C42).862C22, C22.629(C22×D4), C22⋊Q8.178C22, D4⋊C4.202C22, C2.42(D8⋊C22), C23.36C23⋊6C2, C22.26C24⋊13C2, (C22×C4).1049C23, Q8⋊C4.204C22, (C2×SD16).116C22, C4.4D4.144C22, C42.C2.121C22, C4○2(C42.28C22), C4○2(C42.29C22), C42.28C22⋊38C2, C42.29C22⋊24C2, (C2×M4(2)).279C22, C2.66(C22.26C24), C4.54(C2×C4○D4), (C2×C4).522(C2×D4), C2.45(C2×C8⋊C22), SmallGroup(128,1903)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 412 in 204 conjugacy classes, 90 normal (42 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×9], C22, C22 [×12], C8 [×4], C8 [×2], C2×C4 [×6], C2×C4 [×16], D4 [×16], Q8 [×4], C23, C23 [×3], C42 [×4], C42, C22⋊C4 [×9], C4⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×5], C2×C8 [×4], M4(2) [×4], D8 [×4], SD16 [×4], C22×C4 [×3], C22×C4 [×3], C2×D4, C2×D4 [×2], C2×D4 [×5], C2×Q8, C2×Q8, C4○D4 [×4], C4×C8 [×2], C8⋊C4 [×2], D4⋊C4 [×6], Q8⋊C4 [×2], C4.Q8 [×2], C2.D8 [×2], C2×C42, C42⋊C2, C4×D4, C4×D4 [×2], C4×D4 [×2], C4×Q8, C4⋊D4, C4⋊D4 [×2], C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C4.4D4, C42.C2, C42⋊2C2, C4⋊1D4, C4⋊Q8, C2×M4(2) [×2], C2×D8 [×2], C2×SD16 [×2], C2×C4○D4, C4×M4(2), C4×D8 [×2], C4×SD16 [×2], C8⋊D4 [×2], C8⋊2D4 [×2], C42.28C22, C42.29C22, C8⋊3D4, C8⋊Q8, C23.36C23, C22.26C24, C42.255D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C8⋊C22 [×2], C22×D4, C2×C4○D4 [×2], C22.26C24, C2×C8⋊C22, D8⋊C22, C42.255D4
Generators and relations
G = < a,b,c,d | a4=b4=1, c4=d2=a2, ab=ba, cac-1=a-1, dad-1=a-1b2, bc=cb, dbd-1=a2b, dcd-1=c3 >
►On 64 points
(1 46 5 42)(2 43 6 47)(3 48 7 44)(4 45 8 41)(9 53 13 49)(10 50 14 54)(11 55 15 51)(12 52 16 56)(17 34 21 38)(18 39 22 35)(19 36 23 40)(20 33 24 37)(25 63 29 59)(26 60 30 64)(27 57 31 61)(28 62 32 58)
(1 40 27 16)(2 33 28 9)(3 34 29 10)(4 35 30 11)(5 36 31 12)(6 37 32 13)(7 38 25 14)(8 39 26 15)(17 63 54 44)(18 64 55 45)(19 57 56 46)(20 58 49 47)(21 59 50 48)(22 60 51 41)(23 61 52 42)(24 62 53 43)
(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 51 5 55)(2 54 6 50)(3 49 7 53)(4 52 8 56)(9 59 13 63)(10 62 14 58)(11 57 15 61)(12 60 16 64)(17 32 21 28)(18 27 22 31)(19 30 23 26)(20 25 24 29)(33 48 37 44)(34 43 38 47)(35 46 39 42)(36 41 40 45)
G:=sub<Sym(64)| (1,46,5,42)(2,43,6,47)(3,48,7,44)(4,45,8,41)(9,53,13,49)(10,50,14,54)(11,55,15,51)(12,52,16,56)(17,34,21,38)(18,39,22,35)(19,36,23,40)(20,33,24,37)(25,63,29,59)(26,60,30,64)(27,57,31,61)(28,62,32,58), (1,40,27,16)(2,33,28,9)(3,34,29,10)(4,35,30,11)(5,36,31,12)(6,37,32,13)(7,38,25,14)(8,39,26,15)(17,63,54,44)(18,64,55,45)(19,57,56,46)(20,58,49,47)(21,59,50,48)(22,60,51,41)(23,61,52,42)(24,62,53,43), (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), (1,51,5,55)(2,54,6,50)(3,49,7,53)(4,52,8,56)(9,59,13,63)(10,62,14,58)(11,57,15,61)(12,60,16,64)(17,32,21,28)(18,27,22,31)(19,30,23,26)(20,25,24,29)(33,48,37,44)(34,43,38,47)(35,46,39,42)(36,41,40,45)>;
G:=Group( (1,46,5,42)(2,43,6,47)(3,48,7,44)(4,45,8,41)(9,53,13,49)(10,50,14,54)(11,55,15,51)(12,52,16,56)(17,34,21,38)(18,39,22,35)(19,36,23,40)(20,33,24,37)(25,63,29,59)(26,60,30,64)(27,57,31,61)(28,62,32,58), (1,40,27,16)(2,33,28,9)(3,34,29,10)(4,35,30,11)(5,36,31,12)(6,37,32,13)(7,38,25,14)(8,39,26,15)(17,63,54,44)(18,64,55,45)(19,57,56,46)(20,58,49,47)(21,59,50,48)(22,60,51,41)(23,61,52,42)(24,62,53,43), (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), (1,51,5,55)(2,54,6,50)(3,49,7,53)(4,52,8,56)(9,59,13,63)(10,62,14,58)(11,57,15,61)(12,60,16,64)(17,32,21,28)(18,27,22,31)(19,30,23,26)(20,25,24,29)(33,48,37,44)(34,43,38,47)(35,46,39,42)(36,41,40,45) );
G=PermutationGroup([(1,46,5,42),(2,43,6,47),(3,48,7,44),(4,45,8,41),(9,53,13,49),(10,50,14,54),(11,55,15,51),(12,52,16,56),(17,34,21,38),(18,39,22,35),(19,36,23,40),(20,33,24,37),(25,63,29,59),(26,60,30,64),(27,57,31,61),(28,62,32,58)], [(1,40,27,16),(2,33,28,9),(3,34,29,10),(4,35,30,11),(5,36,31,12),(6,37,32,13),(7,38,25,14),(8,39,26,15),(17,63,54,44),(18,64,55,45),(19,57,56,46),(20,58,49,47),(21,59,50,48),(22,60,51,41),(23,61,52,42),(24,62,53,43)], [(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)], [(1,51,5,55),(2,54,6,50),(3,49,7,53),(4,52,8,56),(9,59,13,63),(10,62,14,58),(11,57,15,61),(12,60,16,64),(17,32,21,28),(18,27,22,31),(19,30,23,26),(20,25,24,29),(33,48,37,44),(34,43,38,47),(35,46,39,42),(36,41,40,45)])
Matrix representation ►G ⊆ GL6(𝔽17)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 16 | 16 |
| 0 | 0 | 0 | 5 | 9 | 0 |
| 0 | 0 | 0 | 16 | 12 | 0 |
| 0 | 0 | 9 | 1 | 0 | 12 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 8 | 0 | 0 |
| 0 | 0 | 13 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 8 |
| 0 | 0 | 0 | 0 | 13 | 13 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 16 | 15 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 4 | 14 | 11 |
| 0 | 0 | 15 | 13 | 0 | 3 |
| 0 | 0 | 3 | 6 | 0 | 13 |
| 0 | 0 | 0 | 14 | 15 | 0 |
G:=sub<GL(6,GF(17))| [0,1,0,0,0,0,1,0,0,0,0,0,0,0,5,0,0,9,0,0,0,5,16,1,0,0,16,9,12,0,0,0,16,0,0,12],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,13,0,0,0,0,8,13,0,0,0,0,0,0,4,13,0,0,0,0,8,13],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,1,0,0,0,0,15,1,0,0,1,0,0,0,0,0,0,1,0,0],[0,4,0,0,0,0,13,0,0,0,0,0,0,0,4,15,3,0,0,0,4,13,6,14,0,0,14,0,0,15,0,0,11,3,13,0] >;
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | ··· | 4P | 8A | ··· | 8H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 4 | 8 | 8 | 8 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 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 | C2 | C2 | D4 | D4 | C4○D4 | C8⋊C22 | D8⋊C22 |
| kernel | C42.255D4 | C4×M4(2) | C4×D8 | C4×SD16 | C8⋊D4 | C8⋊2D4 | C42.28C22 | C42.29C22 | C8⋊3D4 | C8⋊Q8 | C23.36C23 | C22.26C24 | C42 | C22×C4 | C8 | C4 | C2 |
| # reps | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{255}D_4 % in TeX
G:=Group("C4^2.255D4"); // GroupNames label
G:=SmallGroup(128,1903);
// by ID
G=gap.SmallGroup(128,1903);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,723,184,80,4037,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=d^2=a^2,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a^-1*b^2,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=c^3>;
// generators/relations