p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.257D4, C42.392C23, C8⋊Q8⋊10C2, C8⋊8(C4○D4), C8⋊3D4⋊8C2, C8⋊8D4⋊16C2, C8⋊7D4⋊28C2, D8⋊C4⋊16C2, C4⋊C4.119C23, (C2×C4).378C24, (C2×C8).280C23, (C4×D4).98C22, (C2×D8).65C22, 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⋊D4.175C22, C4⋊1D4.157C22, (C2×C42).864C22, (C22×C8).280C22, (C2×SD16).26C22, C22.638(C22×D4), C22⋊Q8.180C22, D4⋊C4.136C22, C2.44(D8⋊C22), C23.36C23⋊7C2, C22.26C24⋊14C2, (C22×C4).1574C23, 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
Subgroups: 412 in 205 conjugacy classes, 90 normal (42 characteristic)
C1, C2 [×3], C2 [×5], C4 [×2], C4 [×10], C22, C22 [×2], C22 [×11], 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], C2×C8 [×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], C8⋊C4 [×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, C22×C8 [×2], C2×D8 [×2], C2×SD16 [×2], C2×C4○D4, C2×C8⋊C4, SD16⋊C4 [×2], D8⋊C4 [×2], C8⋊8D4 [×2], C8⋊7D4 [×2], C42.28C22, C42.29C22, C8⋊3D4, C8⋊Q8, C23.36C23, C22.26C24, C42.257D4
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, C42.257D4
Generators and relations
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 >
►On 64 points
(1 53 5 49)(2 54 6 50)(3 55 7 51)(4 56 8 52)(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 12)(2 38 28 9)(3 35 29 14)(4 40 30 11)(5 37 31 16)(6 34 32 13)(7 39 25 10)(8 36 26 15)(17 63 48 55)(18 60 41 52)(19 57 42 49)(20 62 43 54)(21 59 44 51)(22 64 45 56)(23 61 46 53)(24 58 47 50)
(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 41 5 45)(2 44 6 48)(3 47 7 43)(4 42 8 46)(9 59 13 63)(10 62 14 58)(11 57 15 61)(12 60 16 64)(17 28 21 32)(18 31 22 27)(19 26 23 30)(20 29 24 25)(33 52 37 56)(34 55 38 51)(35 50 39 54)(36 53 40 49)
G:=sub<Sym(64)| (1,53,5,49)(2,54,6,50)(3,55,7,51)(4,56,8,52)(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), (1,33,27,12)(2,38,28,9)(3,35,29,14)(4,40,30,11)(5,37,31,16)(6,34,32,13)(7,39,25,10)(8,36,26,15)(17,63,48,55)(18,60,41,52)(19,57,42,49)(20,62,43,54)(21,59,44,51)(22,64,45,56)(23,61,46,53)(24,58,47,50), (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), (1,41,5,45)(2,44,6,48)(3,47,7,43)(4,42,8,46)(9,59,13,63)(10,62,14,58)(11,57,15,61)(12,60,16,64)(17,28,21,32)(18,31,22,27)(19,26,23,30)(20,29,24,25)(33,52,37,56)(34,55,38,51)(35,50,39,54)(36,53,40,49)>;
G:=Group( (1,53,5,49)(2,54,6,50)(3,55,7,51)(4,56,8,52)(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), (1,33,27,12)(2,38,28,9)(3,35,29,14)(4,40,30,11)(5,37,31,16)(6,34,32,13)(7,39,25,10)(8,36,26,15)(17,63,48,55)(18,60,41,52)(19,57,42,49)(20,62,43,54)(21,59,44,51)(22,64,45,56)(23,61,46,53)(24,58,47,50), (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), (1,41,5,45)(2,44,6,48)(3,47,7,43)(4,42,8,46)(9,59,13,63)(10,62,14,58)(11,57,15,61)(12,60,16,64)(17,28,21,32)(18,31,22,27)(19,26,23,30)(20,29,24,25)(33,52,37,56)(34,55,38,51)(35,50,39,54)(36,53,40,49) );
G=PermutationGroup([(1,53,5,49),(2,54,6,50),(3,55,7,51),(4,56,8,52),(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)], [(1,33,27,12),(2,38,28,9),(3,35,29,14),(4,40,30,11),(5,37,31,16),(6,34,32,13),(7,39,25,10),(8,36,26,15),(17,63,48,55),(18,60,41,52),(19,57,42,49),(20,62,43,54),(21,59,44,51),(22,64,45,56),(23,61,46,53),(24,58,47,50)], [(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)], [(1,41,5,45),(2,44,6,48),(3,47,7,43),(4,42,8,46),(9,59,13,63),(10,62,14,58),(11,57,15,61),(12,60,16,64),(17,28,21,32),(18,31,22,27),(19,26,23,30),(20,29,24,25),(33,52,37,56),(34,55,38,51),(35,50,39,54),(36,53,40,49)])
Matrix representation ►G ⊆ 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] >;
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 |
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