p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.257D4, C42.392C23, C8⋊Q8⋊10C2, C8⋊8(C4○D4), C8⋊3D4⋊8C2, C8⋊7D4⋊28C2, C8⋊8D4⋊16C2, D8⋊C4⋊16C2, C4⋊C4.119C23, (C2×C8).280C23, (C2×C4).378C24, (C2×D8).65C22, (C4×D4).98C22, C23.397(C2×D4), (C22×C4).477D4, C4⋊Q8.294C22, SD16⋊C4⋊23C2, (C4×Q8).95C22, C2.D8.97C22, C4.Q8.30C22, (C2×D4).132C23, C22.1(C8⋊C22), (C2×Q8).120C23, C8⋊C4.135C22, C4⋊1D4.157C22, C4⋊D4.175C22, (C2×C42).864C22, (C22×C8).280C22, (C2×SD16).26C22, C22.638(C22×D4), C22⋊Q8.180C22, D4⋊C4.136C22, C2.44(D8⋊C22), (C22×C4).1574C23, C23.36C23⋊7C2, C22.26C24⋊14C2, Q8⋊C4.129C22, C4.4D4.146C22, C42.C2.123C22, C42.28C22⋊35C2, C42.29C22⋊22C2, C2.75(C22.26C24), (C2×C8⋊C4)⋊11C2, C4.63(C2×C4○D4), (C2×C4).702(C2×D4), C2.46(C2×C8⋊C22), SmallGroup(128,1912)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.257D4
G = < a,b,c,d | a4=b4=1, c4=d2=a2, ab=ba, ac=ca, dad-1=ab2, cbc-1=a2b, bd=db, dcd-1=c3 >
Subgroups: 412 in 205 conjugacy classes, 90 normal (42 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, D8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C8⋊C4, C8⋊C4, D4⋊C4, Q8⋊C4, C4.Q8, C2.D8, C2×C42, C42⋊C2, C4×D4, C4×D4, C4×D4, C4×Q8, C4⋊D4, C4⋊D4, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C4.4D4, C42.C2, C42⋊2C2, C4⋊1D4, C4⋊Q8, C22×C8, C2×D8, C2×SD16, C2×C4○D4, C2×C8⋊C4, SD16⋊C4, D8⋊C4, C8⋊8D4, C8⋊7D4, C42.28C22, C42.29C22, C8⋊3D4, C8⋊Q8, C23.36C23, C22.26C24, C42.257D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C8⋊C22, C22×D4, C2×C4○D4, C22.26C24, C2×C8⋊C22, D8⋊C22, C42.257D4
►On 64 points
(1 55 5 51)(2 56 6 52)(3 49 7 53)(4 50 8 54)(9 43 13 47)(10 44 14 48)(11 45 15 41)(12 46 16 42)(17 39 21 35)(18 40 22 36)(19 33 23 37)(20 34 24 38)(25 59 29 63)(26 60 30 64)(27 61 31 57)(28 62 32 58)
(1 33 27 14)(2 38 28 11)(3 35 29 16)(4 40 30 13)(5 37 31 10)(6 34 32 15)(7 39 25 12)(8 36 26 9)(17 63 42 49)(18 60 43 54)(19 57 44 51)(20 62 45 56)(21 59 46 53)(22 64 47 50)(23 61 48 55)(24 58 41 52)
(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 43 5 47)(2 46 6 42)(3 41 7 45)(4 44 8 48)(9 61 13 57)(10 64 14 60)(11 59 15 63)(12 62 16 58)(17 28 21 32)(18 31 22 27)(19 26 23 30)(20 29 24 25)(33 54 37 50)(34 49 38 53)(35 52 39 56)(36 55 40 51)
G:=sub<Sym(64)| (1,55,5,51)(2,56,6,52)(3,49,7,53)(4,50,8,54)(9,43,13,47)(10,44,14,48)(11,45,15,41)(12,46,16,42)(17,39,21,35)(18,40,22,36)(19,33,23,37)(20,34,24,38)(25,59,29,63)(26,60,30,64)(27,61,31,57)(28,62,32,58), (1,33,27,14)(2,38,28,11)(3,35,29,16)(4,40,30,13)(5,37,31,10)(6,34,32,15)(7,39,25,12)(8,36,26,9)(17,63,42,49)(18,60,43,54)(19,57,44,51)(20,62,45,56)(21,59,46,53)(22,64,47,50)(23,61,48,55)(24,58,41,52), (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), (1,43,5,47)(2,46,6,42)(3,41,7,45)(4,44,8,48)(9,61,13,57)(10,64,14,60)(11,59,15,63)(12,62,16,58)(17,28,21,32)(18,31,22,27)(19,26,23,30)(20,29,24,25)(33,54,37,50)(34,49,38,53)(35,52,39,56)(36,55,40,51)>;
G:=Group( (1,55,5,51)(2,56,6,52)(3,49,7,53)(4,50,8,54)(9,43,13,47)(10,44,14,48)(11,45,15,41)(12,46,16,42)(17,39,21,35)(18,40,22,36)(19,33,23,37)(20,34,24,38)(25,59,29,63)(26,60,30,64)(27,61,31,57)(28,62,32,58), (1,33,27,14)(2,38,28,11)(3,35,29,16)(4,40,30,13)(5,37,31,10)(6,34,32,15)(7,39,25,12)(8,36,26,9)(17,63,42,49)(18,60,43,54)(19,57,44,51)(20,62,45,56)(21,59,46,53)(22,64,47,50)(23,61,48,55)(24,58,41,52), (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), (1,43,5,47)(2,46,6,42)(3,41,7,45)(4,44,8,48)(9,61,13,57)(10,64,14,60)(11,59,15,63)(12,62,16,58)(17,28,21,32)(18,31,22,27)(19,26,23,30)(20,29,24,25)(33,54,37,50)(34,49,38,53)(35,52,39,56)(36,55,40,51) );
G=PermutationGroup([[(1,55,5,51),(2,56,6,52),(3,49,7,53),(4,50,8,54),(9,43,13,47),(10,44,14,48),(11,45,15,41),(12,46,16,42),(17,39,21,35),(18,40,22,36),(19,33,23,37),(20,34,24,38),(25,59,29,63),(26,60,30,64),(27,61,31,57),(28,62,32,58)], [(1,33,27,14),(2,38,28,11),(3,35,29,16),(4,40,30,13),(5,37,31,10),(6,34,32,15),(7,39,25,12),(8,36,26,9),(17,63,42,49),(18,60,43,54),(19,57,44,51),(20,62,45,56),(21,59,46,53),(22,64,47,50),(23,61,48,55),(24,58,41,52)], [(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)], [(1,43,5,47),(2,46,6,42),(3,41,7,45),(4,44,8,48),(9,61,13,57),(10,64,14,60),(11,59,15,63),(12,62,16,58),(17,28,21,32),(18,31,22,27),(19,26,23,30),(20,29,24,25),(33,54,37,50),(34,49,38,53),(35,52,39,56),(36,55,40,51)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | ··· | 4H | 4I | 4J | 4K | ··· | 4O | 8A | ··· | 8H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 8 | 8 | 2 | ··· | 2 | 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 | C42.257D4 | C2×C8⋊C4 | SD16⋊C4 | D8⋊C4 | C8⋊8D4 | C8⋊7D4 | C42.28C22 | C42.29C22 | C8⋊3D4 | C8⋊Q8 | C23.36C23 | C22.26C24 | C42 | C22×C4 | C8 | C22 | C2 |
| # reps | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 2 | 2 |
Matrix representation of C42.257D4 ►in GL6(𝔽17)
| 4 | 8 | 0 | 0 | 0 | 0 |
| 13 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 15 |
| 0 | 0 | 0 | 0 | 1 | 16 |
| 0 | 0 | 16 | 2 | 0 | 0 |
| 0 | 0 | 16 | 1 | 0 | 0 |
| 16 | 15 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 15 | 2 | 7 | 10 |
| 0 | 0 | 16 | 0 | 12 | 0 |
| 0 | 0 | 7 | 10 | 2 | 15 |
| 0 | 0 | 12 | 0 | 1 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 11 | 0 | 8 |
| 0 | 0 | 14 | 0 | 4 | 0 |
| 0 | 0 | 0 | 8 | 0 | 6 |
| 0 | 0 | 4 | 0 | 3 | 0 |
G:=sub<GL(6,GF(17))| [4,13,0,0,0,0,8,13,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,16,16,0,0,0,0,2,1,0,0,1,1,0,0,0,0,15,16,0,0],[16,1,0,0,0,0,15,1,0,0,0,0,0,0,15,16,7,12,0,0,2,0,10,0,0,0,7,12,2,1,0,0,10,0,15,0],[16,1,0,0,0,0,0,1,0,0,0,0,0,0,0,14,0,4,0,0,11,0,8,0,0,0,0,4,0,3,0,0,8,0,6,0] >;
C42.257D4 in GAP, Magma, Sage, TeX
C_4^2._{257}D_4 % in TeX
G:=Group("C4^2.257D4"); // GroupNames label
G:=SmallGroup(128,1912);
// by ID
G=gap.SmallGroup(128,1912);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,184,521,80,4037,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=d^2=a^2,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^2,c*b*c^-1=a^2*b,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations