p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.360D4, C42.714C23, (C2×C8)⋊31D4, C4○(C8⋊5D4), C4○(C8⋊4D4), C4⋊1(C4○D8), C8.53(C2×D4), C8⋊4D4⋊24C2, C8⋊5D4⋊30C2, C4○(C4⋊Q16), C4.5(C22×D4), C4⋊Q16⋊25C2, C4○(C8.12D4), C8.12D4⋊27C2, C4.55(C4⋊1D4), (C4×C8).416C22, (C2×C4).345C24, (C2×C8).594C23, C23.389(C2×D4), (C22×C4).614D4, C4⋊Q8.279C22, (C2×Q8).99C23, (C2×D8).130C22, (C2×D4).111C23, C22.2(C4⋊1D4), C4⋊1D4.151C22, C22.26C24⋊8C2, (C22×C8).558C22, (C2×Q16).126C22, C22.605(C22×D4), (C22×C4).1560C23, (C2×C42).1129C22, (C2×SD16).148C22, C4.4D4.139C22, (C2×C4×C8)⋊30C2, (C2×C4○D8)⋊8C2, (C2×C4)○(C8⋊5D4), (C2×C4)○(C8⋊4D4), C2.31(C2×C4○D8), (C2×C4)○(C4⋊Q16), (C2×C4).855(C2×D4), C2.24(C2×C4⋊1D4), (C2×C4)○(C8.12D4), (C2×C4○D4).153C22, SmallGroup(128,1879)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 564 in 286 conjugacy classes, 112 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×8], C4 [×6], C22, C22 [×2], C22 [×14], C8 [×8], C2×C4 [×2], C2×C4 [×8], C2×C4 [×16], D4 [×24], Q8 [×8], C23, C23 [×4], C42 [×2], C42 [×2], 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], C4×C8 [×2], C4×C8 [×2], 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×C4×C8, C8⋊5D4 [×2], C8⋊4D4, C4⋊Q16, C8.12D4 [×4], C22.26C24 [×2], C2×C4○D8 [×4], C42.360D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C24, C4⋊1D4 [×4], C4○D8 [×4], C22×D4 [×3], C2×C4⋊1D4, C2×C4○D8 [×2], C42.360D4
Generators and relations
G = < a,b,c,d | a4=b4=1, c4=d2=b2, ab=ba, ac=ca, dad-1=a-1, bc=cb, bd=db, dcd-1=c3 >
►On 64 points
(1 63 55 16)(2 64 56 9)(3 57 49 10)(4 58 50 11)(5 59 51 12)(6 60 52 13)(7 61 53 14)(8 62 54 15)(17 37 31 45)(18 38 32 46)(19 39 25 47)(20 40 26 48)(21 33 27 41)(22 34 28 42)(23 35 29 43)(24 36 30 44)
(1 21 5 17)(2 22 6 18)(3 23 7 19)(4 24 8 20)(9 42 13 46)(10 43 14 47)(11 44 15 48)(12 45 16 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 47 5 43)(2 42 6 46)(3 45 7 41)(4 48 8 44)(9 22 13 18)(10 17 14 21)(11 20 15 24)(12 23 16 19)(25 59 29 63)(26 62 30 58)(27 57 31 61)(28 60 32 64)(33 49 37 53)(34 52 38 56)(35 55 39 51)(36 50 40 54)
G:=sub<Sym(64)| (1,63,55,16)(2,64,56,9)(3,57,49,10)(4,58,50,11)(5,59,51,12)(6,60,52,13)(7,61,53,14)(8,62,54,15)(17,37,31,45)(18,38,32,46)(19,39,25,47)(20,40,26,48)(21,33,27,41)(22,34,28,42)(23,35,29,43)(24,36,30,44), (1,21,5,17)(2,22,6,18)(3,23,7,19)(4,24,8,20)(9,42,13,46)(10,43,14,47)(11,44,15,48)(12,45,16,41)(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), (1,47,5,43)(2,42,6,46)(3,45,7,41)(4,48,8,44)(9,22,13,18)(10,17,14,21)(11,20,15,24)(12,23,16,19)(25,59,29,63)(26,62,30,58)(27,57,31,61)(28,60,32,64)(33,49,37,53)(34,52,38,56)(35,55,39,51)(36,50,40,54)>;
G:=Group( (1,63,55,16)(2,64,56,9)(3,57,49,10)(4,58,50,11)(5,59,51,12)(6,60,52,13)(7,61,53,14)(8,62,54,15)(17,37,31,45)(18,38,32,46)(19,39,25,47)(20,40,26,48)(21,33,27,41)(22,34,28,42)(23,35,29,43)(24,36,30,44), (1,21,5,17)(2,22,6,18)(3,23,7,19)(4,24,8,20)(9,42,13,46)(10,43,14,47)(11,44,15,48)(12,45,16,41)(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), (1,47,5,43)(2,42,6,46)(3,45,7,41)(4,48,8,44)(9,22,13,18)(10,17,14,21)(11,20,15,24)(12,23,16,19)(25,59,29,63)(26,62,30,58)(27,57,31,61)(28,60,32,64)(33,49,37,53)(34,52,38,56)(35,55,39,51)(36,50,40,54) );
G=PermutationGroup([(1,63,55,16),(2,64,56,9),(3,57,49,10),(4,58,50,11),(5,59,51,12),(6,60,52,13),(7,61,53,14),(8,62,54,15),(17,37,31,45),(18,38,32,46),(19,39,25,47),(20,40,26,48),(21,33,27,41),(22,34,28,42),(23,35,29,43),(24,36,30,44)], [(1,21,5,17),(2,22,6,18),(3,23,7,19),(4,24,8,20),(9,42,13,46),(10,43,14,47),(11,44,15,48),(12,45,16,41),(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)], [(1,47,5,43),(2,42,6,46),(3,45,7,41),(4,48,8,44),(9,22,13,18),(10,17,14,21),(11,20,15,24),(12,23,16,19),(25,59,29,63),(26,62,30,58),(27,57,31,61),(28,60,32,64),(33,49,37,53),(34,52,38,56),(35,55,39,51),(36,50,40,54)])
Matrix representation ►G ⊆ GL4(𝔽17) generated by
| 0 | 16 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 16 | 15 |
| 0 | 0 | 1 | 1 |
| 13 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 5 | 12 | 0 | 0 |
| 5 | 5 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 13 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 16 | 15 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(17))| [0,1,0,0,16,0,0,0,0,0,16,1,0,0,15,1],[13,0,0,0,0,13,0,0,0,0,1,0,0,0,0,1],[5,5,0,0,12,5,0,0,0,0,16,0,0,0,0,16],[13,0,0,0,0,4,0,0,0,0,16,0,0,0,15,1] >;
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 4O | 4P | 4Q | 4R | 8A | ··· | 8P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 8 | 8 | 8 | 8 | 2 | ··· | 2 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | C4○D8 |
| kernel | C42.360D4 | C2×C4×C8 | C8⋊5D4 | C8⋊4D4 | C4⋊Q16 | C8.12D4 | C22.26C24 | C2×C4○D8 | C42 | C2×C8 | C22×C4 | C4 |
| # reps | 1 | 1 | 2 | 1 | 1 | 4 | 2 | 4 | 2 | 8 | 2 | 16 |
In GAP, Magma, Sage, TeX
C_4^2._{360}D_4 % in TeX
G:=Group("C4^2.360D4"); // GroupNames label
G:=SmallGroup(128,1879);
// by ID
G=gap.SmallGroup(128,1879);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,184,248,2804,172]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=d^2=b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,b*c=c*b,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations