p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.417D4, C42.171C23, (C4×D4).8C4, C4⋊D4.15C4, C4.D8⋊23C2, C4.101(C4○D8), C4.10D8⋊37C2, C4⋊C8.261C22, C42.112(C2×C4), (C22×C4).240D4, C4⋊Q8.244C22, C4.110(C8⋊C22), C42.6C4⋊41C2, C42.12C4⋊22C2, C4⋊1D4.129C22, C23.64(C22⋊C4), (C2×C42).215C22, C2.18(C23.24D4), C2.13(C23.37D4), C22.26C24.16C2, C2.18(M4(2).8C22), C4⋊C4.41(C2×C4), (C2×D4).33(C2×C4), (C2×C4).1242(C2×D4), (C2×C4).165(C22×C4), (C22×C4).237(C2×C4), (C2×C4).186(C22⋊C4), C22.229(C2×C22⋊C4), SmallGroup(128,285)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.417D4
G = < a,b,c,d | a4=b4=1, c4=a2, d2=b, ab=ba, cac-1=a-1b2, ad=da, cbc-1=b-1, bd=db, dcd-1=a2bc3 >
Subgroups: 276 in 121 conjugacy classes, 46 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4×C8, C8⋊C4, C22⋊C8, C4⋊C8, C2×C42, C4×D4, C4×D4, C4⋊D4, C4⋊D4, C4.4D4, C4⋊1D4, C4⋊Q8, C2×C4○D4, C4.D8, C4.10D8, C42.12C4, C42.6C4, C22.26C24, C42.417D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C2×C22⋊C4, C4○D8, C8⋊C22, M4(2).8C22, C23.24D4, C23.37D4, C42.417D4
►On 64 points
(1 34 5 38)(2 20 6 24)(3 36 7 40)(4 22 8 18)(9 21 13 17)(10 37 14 33)(11 23 15 19)(12 39 16 35)(25 61 29 57)(26 51 30 55)(27 63 31 59)(28 53 32 49)(41 56 45 52)(42 60 46 64)(43 50 47 54)(44 62 48 58)
(1 50 11 57)(2 58 12 51)(3 52 13 59)(4 60 14 53)(5 54 15 61)(6 62 16 55)(7 56 9 63)(8 64 10 49)(17 27 36 41)(18 42 37 28)(19 29 38 43)(20 44 39 30)(21 31 40 45)(22 46 33 32)(23 25 34 47)(24 48 35 26)
(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 64 50 10 11 49 57 8)(2 7 58 56 12 9 51 63)(3 62 52 16 13 55 59 6)(4 5 60 54 14 15 53 61)(17 26 27 24 36 48 41 35)(18 34 42 47 37 23 28 25)(19 32 29 22 38 46 43 33)(20 40 44 45 39 21 30 31)
G:=sub<Sym(64)| (1,34,5,38)(2,20,6,24)(3,36,7,40)(4,22,8,18)(9,21,13,17)(10,37,14,33)(11,23,15,19)(12,39,16,35)(25,61,29,57)(26,51,30,55)(27,63,31,59)(28,53,32,49)(41,56,45,52)(42,60,46,64)(43,50,47,54)(44,62,48,58), (1,50,11,57)(2,58,12,51)(3,52,13,59)(4,60,14,53)(5,54,15,61)(6,62,16,55)(7,56,9,63)(8,64,10,49)(17,27,36,41)(18,42,37,28)(19,29,38,43)(20,44,39,30)(21,31,40,45)(22,46,33,32)(23,25,34,47)(24,48,35,26), (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), (1,64,50,10,11,49,57,8)(2,7,58,56,12,9,51,63)(3,62,52,16,13,55,59,6)(4,5,60,54,14,15,53,61)(17,26,27,24,36,48,41,35)(18,34,42,47,37,23,28,25)(19,32,29,22,38,46,43,33)(20,40,44,45,39,21,30,31)>;
G:=Group( (1,34,5,38)(2,20,6,24)(3,36,7,40)(4,22,8,18)(9,21,13,17)(10,37,14,33)(11,23,15,19)(12,39,16,35)(25,61,29,57)(26,51,30,55)(27,63,31,59)(28,53,32,49)(41,56,45,52)(42,60,46,64)(43,50,47,54)(44,62,48,58), (1,50,11,57)(2,58,12,51)(3,52,13,59)(4,60,14,53)(5,54,15,61)(6,62,16,55)(7,56,9,63)(8,64,10,49)(17,27,36,41)(18,42,37,28)(19,29,38,43)(20,44,39,30)(21,31,40,45)(22,46,33,32)(23,25,34,47)(24,48,35,26), (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), (1,64,50,10,11,49,57,8)(2,7,58,56,12,9,51,63)(3,62,52,16,13,55,59,6)(4,5,60,54,14,15,53,61)(17,26,27,24,36,48,41,35)(18,34,42,47,37,23,28,25)(19,32,29,22,38,46,43,33)(20,40,44,45,39,21,30,31) );
G=PermutationGroup([[(1,34,5,38),(2,20,6,24),(3,36,7,40),(4,22,8,18),(9,21,13,17),(10,37,14,33),(11,23,15,19),(12,39,16,35),(25,61,29,57),(26,51,30,55),(27,63,31,59),(28,53,32,49),(41,56,45,52),(42,60,46,64),(43,50,47,54),(44,62,48,58)], [(1,50,11,57),(2,58,12,51),(3,52,13,59),(4,60,14,53),(5,54,15,61),(6,62,16,55),(7,56,9,63),(8,64,10,49),(17,27,36,41),(18,42,37,28),(19,29,38,43),(20,44,39,30),(21,31,40,45),(22,46,33,32),(23,25,34,47),(24,48,35,26)], [(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)], [(1,64,50,10,11,49,57,8),(2,7,58,56,12,9,51,63),(3,62,52,16,13,55,59,6),(4,5,60,54,14,15,53,61),(17,26,27,24,36,48,41,35),(18,34,42,47,37,23,28,25),(19,32,29,22,38,46,43,33),(20,40,44,45,39,21,30,31)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | ··· | 4J | 4K | 4L | 4M | 8A | ··· | 8H | 8I | 8J | 8K | 8L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 1 | 1 | 4 | 8 | 8 | 2 | ··· | 2 | 4 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | D4 | C4○D8 | C8⋊C22 | M4(2).8C22 |
| kernel | C42.417D4 | C4.D8 | C4.10D8 | C42.12C4 | C42.6C4 | C22.26C24 | C4×D4 | C4⋊D4 | C42 | C22×C4 | C4 | C4 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 8 | 2 | 2 |
Matrix representation of C42.417D4 ►in GL6(𝔽17)
| 0 | 13 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 5 | 12 | 0 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 5 | 4 | 13 |
| 0 | 0 | 12 | 12 | 4 | 4 |
| 0 | 0 | 13 | 4 | 5 | 12 |
| 0 | 0 | 13 | 13 | 5 | 5 |
| 12 | 5 | 0 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 5 | 4 | 13 |
| 0 | 0 | 5 | 5 | 13 | 13 |
| 0 | 0 | 13 | 4 | 5 | 12 |
| 0 | 0 | 4 | 4 | 12 | 12 |
G:=sub<GL(6,GF(17))| [0,4,0,0,0,0,13,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,16,0,0,0,0,1,0,0,0],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[5,12,0,0,0,0,12,12,0,0,0,0,0,0,12,12,13,13,0,0,5,12,4,13,0,0,4,4,5,5,0,0,13,4,12,5],[12,12,0,0,0,0,5,12,0,0,0,0,0,0,12,5,13,4,0,0,5,5,4,4,0,0,4,13,5,12,0,0,13,13,12,12] >;
C42.417D4 in GAP, Magma, Sage, TeX
C_4^2._{417}D_4 % in TeX
G:=Group("C4^2.417D4"); // GroupNames label
G:=SmallGroup(128,285);
// by ID
G=gap.SmallGroup(128,285);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,758,520,1123,1018,248,1971,242]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=a^2,d^2=b,a*b=b*a,c*a*c^-1=a^-1*b^2,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=a^2*b*c^3>;
// generators/relations