p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.360D4, C42.714C23, (C2×C8)⋊31D4, C4⋊1(C4○D8), C4○(C8⋊5D4), C4○(C8⋊4D4), 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×C8).594C23, (C2×C4).345C24, (C22×C4).614D4, C23.389(C2×D4), C4⋊Q8.279C22, (C2×Q8).99C23, (C2×D4).111C23, (C2×D8).130C22, C22.2(C4⋊1D4), C4⋊1D4.151C22, (C22×C8).558C22, C22.26C24⋊8C2, (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
Generators and relations for C42.360D4
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 >
Subgroups: 564 in 286 conjugacy classes, 112 normal (16 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C2×C8, D8, SD16, Q16, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4×C8, C4×C8, C2×C42, C4×D4, C4⋊D4, C4.4D4, C4⋊1D4, C4⋊Q8, C22×C8, C2×D8, C2×SD16, C2×Q16, C4○D8, C2×C4○D4, C2×C4×C8, C8⋊5D4, C8⋊4D4, C4⋊Q16, C8.12D4, C22.26C24, C2×C4○D8, C42.360D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C4⋊1D4, C4○D8, C22×D4, C2×C4⋊1D4, C2×C4○D8, C42.360D4
►On 64 points
(1 63 55 10)(2 64 56 11)(3 57 49 12)(4 58 50 13)(5 59 51 14)(6 60 52 15)(7 61 53 16)(8 62 54 9)(17 37 31 47)(18 38 32 48)(19 39 25 41)(20 40 26 42)(21 33 27 43)(22 34 28 44)(23 35 29 45)(24 36 30 46)
(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 41 5 45)(2 44 6 48)(3 47 7 43)(4 42 8 46)(9 24 13 20)(10 19 14 23)(11 22 15 18)(12 17 16 21)(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)| (1,63,55,10)(2,64,56,11)(3,57,49,12)(4,58,50,13)(5,59,51,14)(6,60,52,15)(7,61,53,16)(8,62,54,9)(17,37,31,47)(18,38,32,48)(19,39,25,41)(20,40,26,42)(21,33,27,43)(22,34,28,44)(23,35,29,45)(24,36,30,46), (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), (1,41,5,45)(2,44,6,48)(3,47,7,43)(4,42,8,46)(9,24,13,20)(10,19,14,23)(11,22,15,18)(12,17,16,21)(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( (1,63,55,10)(2,64,56,11)(3,57,49,12)(4,58,50,13)(5,59,51,14)(6,60,52,15)(7,61,53,16)(8,62,54,9)(17,37,31,47)(18,38,32,48)(19,39,25,41)(20,40,26,42)(21,33,27,43)(22,34,28,44)(23,35,29,45)(24,36,30,46), (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), (1,41,5,45)(2,44,6,48)(3,47,7,43)(4,42,8,46)(9,24,13,20)(10,19,14,23)(11,22,15,18)(12,17,16,21)(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([[(1,63,55,10),(2,64,56,11),(3,57,49,12),(4,58,50,13),(5,59,51,14),(6,60,52,15),(7,61,53,16),(8,62,54,9),(17,37,31,47),(18,38,32,48),(19,39,25,41),(20,40,26,42),(21,33,27,43),(22,34,28,44),(23,35,29,45),(24,36,30,46)], [(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)], [(1,41,5,45),(2,44,6,48),(3,47,7,43),(4,42,8,46),(9,24,13,20),(10,19,14,23),(11,22,15,18),(12,17,16,21),(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)]])
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 |
Matrix representation of C42.360D4 ►in 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] >;
C42.360D4 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