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×C4).381C24, (C2×C8).282C23, (C4×C8).186C22, (C2×D8).66C22, (C22×C4).170D4, C23.268(C2×D4), SD16⋊C4⋊26C2, (C4×Q8).98C22, C4.Q8.32C22, (C2×D4).135C23, (C4×D4).101C22, (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
Subgroups: 340 in 185 conjugacy classes, 88 normal (34 characteristic)
C1, C2, C2 [×2], C2 [×3], C4 [×2], C4 [×11], C22, C22 [×9], C8 [×4], C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×14], D4 [×8], Q8 [×4], C23, C23 [×2], C42 [×2], C42 [×2], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×4], C4⋊C4 [×8], C2×C8 [×4], M4(2) [×4], D8 [×2], SD16 [×4], Q16 [×2], C22×C4, C22×C4 [×2], C22×C4 [×2], C2×D4 [×2], C2×D4 [×2], C2×Q8 [×2], C4×C8 [×2], C8⋊C4 [×2], D4⋊C4 [×4], Q8⋊C4 [×4], C4.Q8 [×2], C2.D8 [×2], C2×C42, C42⋊C2 [×2], C4×D4 [×2], C4×D4 [×2], C4×Q8 [×2], C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4 [×2], C4.4D4 [×2], C42.C2 [×2], C42⋊2C2 [×2], C2×M4(2) [×2], C2×D8, C2×SD16 [×2], C2×Q16, C4×M4(2), SD16⋊C4 [×2], Q16⋊C4, D8⋊C4, C8⋊D4 [×2], C8⋊2D4, C8.D4, C42.78C22 [×2], C8.12D4, C8.5Q8, C23.36C23 [×2], C42.260D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C22×D4, C2×C4○D4 [×2], C22.26C24, D8⋊C22 [×2], C42.260D4
Generators and relations
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 >
►On 64 points
(1 42 5 46)(2 47 6 43)(3 44 7 48)(4 41 8 45)(9 53 13 49)(10 50 14 54)(11 55 15 51)(12 52 16 56)(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 52 48)(18 58 53 45)(19 63 54 42)(20 60 55 47)(21 57 56 44)(22 62 49 41)(23 59 50 46)(24 64 51 43)
(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 56 5 52)(2 51 6 55)(3 54 7 50)(4 49 8 53)(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 41 37 45)(34 44 38 48)(35 47 39 43)(36 42 40 46)
G:=sub<Sym(64)| (1,42,5,46)(2,47,6,43)(3,44,7,48)(4,41,8,45)(9,53,13,49)(10,50,14,54)(11,55,15,51)(12,52,16,56)(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)(17,61,52,48)(18,58,53,45)(19,63,54,42)(20,60,55,47)(21,57,56,44)(22,62,49,41)(23,59,50,46)(24,64,51,43), (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), (1,56,5,52)(2,51,6,55)(3,54,7,50)(4,49,8,53)(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)(33,41,37,45)(34,44,38,48)(35,47,39,43)(36,42,40,46)>;
G:=Group( (1,42,5,46)(2,47,6,43)(3,44,7,48)(4,41,8,45)(9,53,13,49)(10,50,14,54)(11,55,15,51)(12,52,16,56)(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)(17,61,52,48)(18,58,53,45)(19,63,54,42)(20,60,55,47)(21,57,56,44)(22,62,49,41)(23,59,50,46)(24,64,51,43), (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), (1,56,5,52)(2,51,6,55)(3,54,7,50)(4,49,8,53)(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)(33,41,37,45)(34,44,38,48)(35,47,39,43)(36,42,40,46) );
G=PermutationGroup([(1,42,5,46),(2,47,6,43),(3,44,7,48),(4,41,8,45),(9,53,13,49),(10,50,14,54),(11,55,15,51),(12,52,16,56),(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),(17,61,52,48),(18,58,53,45),(19,63,54,42),(20,60,55,47),(21,57,56,44),(22,62,49,41),(23,59,50,46),(24,64,51,43)], [(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)], [(1,56,5,52),(2,51,6,55),(3,54,7,50),(4,49,8,53),(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),(33,41,37,45),(34,44,38,48),(35,47,39,43),(36,42,40,46)])
Matrix representation ►G ⊆ 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] >;
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 |
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