p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.247D4, C42.372C23, (C2×C8)⋊16D4, C4○(C8⋊3D4), C8.28(C2×D4), C8⋊3D4⋊24C2, C4○(C8.2D4), C4.8(C22×D4), C8.2D4⋊24C2, C4.56(C4⋊1D4), (C2×C4).348C24, (C2×C8).266C23, (C22×C4).467D4, C23.390(C2×D4), C4⋊Q8.280C22, (C2×D4).114C23, (C2×D8).162C22, C22.3(C4⋊1D4), (C2×Q8).102C23, C8⋊C4.117C22, C4⋊1D4.152C22, (C22×C8).270C22, C22.26C24⋊9C2, (C2×C42).854C22, (C2×Q16).157C22, C22.608(C22×D4), C2.38(D8⋊C22), (C22×C4).1563C23, (C2×SD16).115C22, C4.4D4.141C22, (C2×C4○D8)⋊20C2, (C2×C4)○(C8⋊3D4), (C2×C8⋊C4)⋊10C2, (C2×C4)○(C8.2D4), (C2×C4).519(C2×D4), C2.27(C2×C4⋊1D4), (C2×C4○D4).154C22, SmallGroup(128,1882)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 564 in 282 conjugacy classes, 108 normal (10 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×4], C4 [×8], C22, C22 [×2], C22 [×14], C8 [×8], C2×C4 [×2], C2×C4 [×8], C2×C4 [×14], D4 [×24], Q8 [×8], C23, C23 [×4], C42 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C2×C8 [×12], D8 [×8], SD16 [×16], Q16 [×8], C22×C4, C22×C4 [×2], C22×C4 [×4], C2×D4 [×4], C2×D4 [×8], C2×Q8 [×4], C4○D4 [×16], C8⋊C4 [×4], C2×C42, C4×D4 [×4], C4⋊D4 [×4], C4.4D4 [×4], C4⋊1D4 [×2], C4⋊Q8 [×2], C22×C8 [×2], C2×D8 [×4], C2×SD16 [×8], C2×Q16 [×4], C4○D8 [×16], C2×C4○D4 [×4], C2×C8⋊C4, C8⋊3D4 [×4], C8.2D4 [×4], C22.26C24 [×2], C2×C4○D8 [×4], C42.247D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C24, C4⋊1D4 [×4], C22×D4 [×3], C2×C4⋊1D4, D8⋊C22 [×2], C42.247D4
Generators and relations
G = < a,b,c,d | a4=b4=1, c4=d2=b2, ab=ba, cac-1=ab2, dad-1=a-1b2, bc=cb, bd=db, dcd-1=c3 >
►On 64 points
(1 63 55 46)(2 60 56 43)(3 57 49 48)(4 62 50 45)(5 59 51 42)(6 64 52 47)(7 61 53 44)(8 58 54 41)(9 34 32 18)(10 39 25 23)(11 36 26 20)(12 33 27 17)(13 38 28 22)(14 35 29 19)(15 40 30 24)(16 37 31 21)
(1 12 5 16)(2 13 6 9)(3 14 7 10)(4 15 8 11)(17 42 21 46)(18 43 22 47)(19 44 23 48)(20 45 24 41)(25 49 29 53)(26 50 30 54)(27 51 31 55)(28 52 32 56)(33 59 37 63)(34 60 38 64)(35 61 39 57)(36 62 40 58)
(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 19 13 23)(10 22 14 18)(11 17 15 21)(12 20 16 24)(25 38 29 34)(26 33 30 37)(27 36 31 40)(28 39 32 35)(49 64 53 60)(50 59 54 63)(51 62 55 58)(52 57 56 61)
G:=sub<Sym(64)| (1,63,55,46)(2,60,56,43)(3,57,49,48)(4,62,50,45)(5,59,51,42)(6,64,52,47)(7,61,53,44)(8,58,54,41)(9,34,32,18)(10,39,25,23)(11,36,26,20)(12,33,27,17)(13,38,28,22)(14,35,29,19)(15,40,30,24)(16,37,31,21), (1,12,5,16)(2,13,6,9)(3,14,7,10)(4,15,8,11)(17,42,21,46)(18,43,22,47)(19,44,23,48)(20,45,24,41)(25,49,29,53)(26,50,30,54)(27,51,31,55)(28,52,32,56)(33,59,37,63)(34,60,38,64)(35,61,39,57)(36,62,40,58), (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)(9,19,13,23)(10,22,14,18)(11,17,15,21)(12,20,16,24)(25,38,29,34)(26,33,30,37)(27,36,31,40)(28,39,32,35)(49,64,53,60)(50,59,54,63)(51,62,55,58)(52,57,56,61)>;
G:=Group( (1,63,55,46)(2,60,56,43)(3,57,49,48)(4,62,50,45)(5,59,51,42)(6,64,52,47)(7,61,53,44)(8,58,54,41)(9,34,32,18)(10,39,25,23)(11,36,26,20)(12,33,27,17)(13,38,28,22)(14,35,29,19)(15,40,30,24)(16,37,31,21), (1,12,5,16)(2,13,6,9)(3,14,7,10)(4,15,8,11)(17,42,21,46)(18,43,22,47)(19,44,23,48)(20,45,24,41)(25,49,29,53)(26,50,30,54)(27,51,31,55)(28,52,32,56)(33,59,37,63)(34,60,38,64)(35,61,39,57)(36,62,40,58), (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)(9,19,13,23)(10,22,14,18)(11,17,15,21)(12,20,16,24)(25,38,29,34)(26,33,30,37)(27,36,31,40)(28,39,32,35)(49,64,53,60)(50,59,54,63)(51,62,55,58)(52,57,56,61) );
G=PermutationGroup([(1,63,55,46),(2,60,56,43),(3,57,49,48),(4,62,50,45),(5,59,51,42),(6,64,52,47),(7,61,53,44),(8,58,54,41),(9,34,32,18),(10,39,25,23),(11,36,26,20),(12,33,27,17),(13,38,28,22),(14,35,29,19),(15,40,30,24),(16,37,31,21)], [(1,12,5,16),(2,13,6,9),(3,14,7,10),(4,15,8,11),(17,42,21,46),(18,43,22,47),(19,44,23,48),(20,45,24,41),(25,49,29,53),(26,50,30,54),(27,51,31,55),(28,52,32,56),(33,59,37,63),(34,60,38,64),(35,61,39,57),(36,62,40,58)], [(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),(9,19,13,23),(10,22,14,18),(11,17,15,21),(12,20,16,24),(25,38,29,34),(26,33,30,37),(27,36,31,40),(28,39,32,35),(49,64,53,60),(50,59,54,63),(51,62,55,58),(52,57,56,61)])
Matrix representation ►G ⊆ GL6(𝔽17)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 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 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 16 | 14 | 14 |
| 0 | 0 | 1 | 16 | 3 | 14 |
| 0 | 0 | 3 | 3 | 1 | 1 |
| 0 | 0 | 14 | 3 | 16 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 3 | 1 | 1 |
| 0 | 0 | 3 | 14 | 1 | 16 |
| 0 | 0 | 16 | 16 | 14 | 14 |
| 0 | 0 | 16 | 1 | 14 | 3 |
G:=sub<GL(6,GF(17))| [0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,16,1,3,14,0,0,16,16,3,3,0,0,14,3,1,16,0,0,14,14,1,1],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,3,3,16,16,0,0,3,14,16,1,0,0,1,1,14,14,0,0,1,16,14,3] >;
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 8A | ··· | 8H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 8 | 8 | 8 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D8⋊C22 |
| kernel | C42.247D4 | C2×C8⋊C4 | C8⋊3D4 | C8.2D4 | C22.26C24 | C2×C4○D8 | C42 | C2×C8 | C22×C4 | C2 |
| # reps | 1 | 1 | 4 | 4 | 2 | 4 | 2 | 8 | 2 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{247}D_4 % in TeX
G:=Group("C4^2.247D4"); // GroupNames label
G:=SmallGroup(128,1882);
// by ID
G=gap.SmallGroup(128,1882);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,723,184,248,2804,172]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=d^2=b^2,a*b=b*a,c*a*c^-1=a*b^2,d*a*d^-1=a^-1*b^2,b*c=c*b,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations