p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.260D4, C42.394C23, C8⋊D4⋊11C2, C8.D4⋊8C2, C8⋊2D4.5C2, D8⋊C4⋊17C2, C8.13(C4○D4), C8.5Q8⋊20C2, Q16⋊C4⋊17C2, C8.12D4⋊20C2, C4⋊C4.122C23, (C4×M4(2))⋊10C2, (C2×C8).282C23, (C4×C8).186C22, (C2×C4).381C24, (C2×D8).66C22, (C22×C4).170D4, C23.268(C2×D4), SD16⋊C4⋊26C2, (C4×Q8).98C22, C4.Q8.32C22, (C4×D4).101C22, (C2×D4).135C23, (C2×Q16).67C22, (C2×Q8).123C23, C2.D8.100C22, C8⋊C4.138C22, C4⋊D4.178C22, (C2×C42).867C22, (C2×SD16).29C22, C22.641(C22×D4), C22⋊Q8.183C22, D4⋊C4.139C22, C2.46(D8⋊C22), C23.36C23⋊8C2, (C22×C4).1059C23, Q8⋊C4.132C22, C4.4D4.148C22, C42.C2.125C22, C42.78C22⋊31C2, (C2×M4(2)).289C22, C2.78(C22.26C24), C4.66(C2×C4○D4), (C2×C4).524(C2×D4), SmallGroup(128,1915)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.260D4
G = < a,b,c,d | a4=b4=1, c4=d2=a2, ab=ba, cac-1=a-1, dad-1=a-1b2, cbc-1=dbd-1=a2b, dcd-1=c3 >
Subgroups: 340 in 185 conjugacy classes, 88 normal (34 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, M4(2), D8, SD16, Q16, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4×C8, C8⋊C4, D4⋊C4, Q8⋊C4, C4.Q8, C2.D8, C2×C42, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C2×M4(2), C2×D8, C2×SD16, C2×Q16, C4×M4(2), SD16⋊C4, Q16⋊C4, D8⋊C4, C8⋊D4, C8⋊2D4, C8.D4, C42.78C22, C8.12D4, C8.5Q8, C23.36C23, C42.260D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22×D4, C2×C4○D4, C22.26C24, D8⋊C22, C42.260D4
►On 64 points
(1 44 5 48)(2 41 6 45)(3 46 7 42)(4 43 8 47)(9 55 13 51)(10 52 14 56)(11 49 15 53)(12 54 16 50)(17 40 21 36)(18 37 22 33)(19 34 23 38)(20 39 24 35)(25 57 29 61)(26 62 30 58)(27 59 31 63)(28 64 32 60)
(1 38 31 14)(2 35 32 11)(3 40 25 16)(4 37 26 13)(5 34 27 10)(6 39 28 15)(7 36 29 12)(8 33 30 9)(17 61 54 42)(18 58 55 47)(19 63 56 44)(20 60 49 41)(21 57 50 46)(22 62 51 43)(23 59 52 48)(24 64 53 45)
(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 50 5 54)(2 53 6 49)(3 56 7 52)(4 51 8 55)(9 62 13 58)(10 57 14 61)(11 60 15 64)(12 63 16 59)(17 31 21 27)(18 26 22 30)(19 29 23 25)(20 32 24 28)(33 43 37 47)(34 46 38 42)(35 41 39 45)(36 44 40 48)
G:=sub<Sym(64)| (1,44,5,48)(2,41,6,45)(3,46,7,42)(4,43,8,47)(9,55,13,51)(10,52,14,56)(11,49,15,53)(12,54,16,50)(17,40,21,36)(18,37,22,33)(19,34,23,38)(20,39,24,35)(25,57,29,61)(26,62,30,58)(27,59,31,63)(28,64,32,60), (1,38,31,14)(2,35,32,11)(3,40,25,16)(4,37,26,13)(5,34,27,10)(6,39,28,15)(7,36,29,12)(8,33,30,9)(17,61,54,42)(18,58,55,47)(19,63,56,44)(20,60,49,41)(21,57,50,46)(22,62,51,43)(23,59,52,48)(24,64,53,45), (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), (1,50,5,54)(2,53,6,49)(3,56,7,52)(4,51,8,55)(9,62,13,58)(10,57,14,61)(11,60,15,64)(12,63,16,59)(17,31,21,27)(18,26,22,30)(19,29,23,25)(20,32,24,28)(33,43,37,47)(34,46,38,42)(35,41,39,45)(36,44,40,48)>;
G:=Group( (1,44,5,48)(2,41,6,45)(3,46,7,42)(4,43,8,47)(9,55,13,51)(10,52,14,56)(11,49,15,53)(12,54,16,50)(17,40,21,36)(18,37,22,33)(19,34,23,38)(20,39,24,35)(25,57,29,61)(26,62,30,58)(27,59,31,63)(28,64,32,60), (1,38,31,14)(2,35,32,11)(3,40,25,16)(4,37,26,13)(5,34,27,10)(6,39,28,15)(7,36,29,12)(8,33,30,9)(17,61,54,42)(18,58,55,47)(19,63,56,44)(20,60,49,41)(21,57,50,46)(22,62,51,43)(23,59,52,48)(24,64,53,45), (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), (1,50,5,54)(2,53,6,49)(3,56,7,52)(4,51,8,55)(9,62,13,58)(10,57,14,61)(11,60,15,64)(12,63,16,59)(17,31,21,27)(18,26,22,30)(19,29,23,25)(20,32,24,28)(33,43,37,47)(34,46,38,42)(35,41,39,45)(36,44,40,48) );
G=PermutationGroup([[(1,44,5,48),(2,41,6,45),(3,46,7,42),(4,43,8,47),(9,55,13,51),(10,52,14,56),(11,49,15,53),(12,54,16,50),(17,40,21,36),(18,37,22,33),(19,34,23,38),(20,39,24,35),(25,57,29,61),(26,62,30,58),(27,59,31,63),(28,64,32,60)], [(1,38,31,14),(2,35,32,11),(3,40,25,16),(4,37,26,13),(5,34,27,10),(6,39,28,15),(7,36,29,12),(8,33,30,9),(17,61,54,42),(18,58,55,47),(19,63,56,44),(20,60,49,41),(21,57,50,46),(22,62,51,43),(23,59,52,48),(24,64,53,45)], [(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)], [(1,50,5,54),(2,53,6,49),(3,56,7,52),(4,51,8,55),(9,62,13,58),(10,57,14,61),(11,60,15,64),(12,63,16,59),(17,31,21,27),(18,26,22,30),(19,29,23,25),(20,32,24,28),(33,43,37,47),(34,46,38,42),(35,41,39,45),(36,44,40,48)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | ··· | 4J | 4K | 4L | ··· | 4Q | 8A | ··· | 8H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 4 | 8 | 8 | 2 | ··· | 2 | 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 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | C4○D4 | D8⋊C22 |
| kernel | C42.260D4 | C4×M4(2) | SD16⋊C4 | Q16⋊C4 | D8⋊C4 | C8⋊D4 | C8⋊2D4 | C8.D4 | C42.78C22 | C8.12D4 | C8.5Q8 | C23.36C23 | C42 | C22×C4 | C8 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 2 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 8 | 4 |
Matrix representation of C42.260D4 ►in GL6(𝔽17)
| 16 | 2 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 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 | 0 | 0 | 5 | 5 |
| 0 | 0 | 0 | 0 | 12 | 5 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 5 | 12 | 0 | 0 |
| 13 | 8 | 0 | 0 | 0 | 0 |
| 13 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 0 | 4 | 0 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,4,0,0,0,0,0,0,4,0,0],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,5,0,0,0,0,12,12,0,0,5,12,0,0,0,0,5,5,0,0],[13,13,0,0,0,0,8,4,0,0,0,0,0,0,0,13,0,0,0,0,13,0,0,0,0,0,0,0,0,4,0,0,0,0,4,0] >;
C42.260D4 in GAP, Magma, Sage, TeX
C_4^2._{260}D_4 % in TeX
G:=Group("C4^2.260D4"); // GroupNames label
G:=SmallGroup(128,1915);
// by ID
G=gap.SmallGroup(128,1915);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,456,758,723,184,521,80,4037,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=d^2=a^2,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a^-1*b^2,c*b*c^-1=d*b*d^-1=a^2*b,d*c*d^-1=c^3>;
// generators/relations