p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.411D4, C42.161C23, C4○2(C4.D8), C4.97(C4○D8), C4.D8⋊28C2, C4⋊C8.257C22, C42.102(C2×C4), C4○2(C4.6Q16), C4.6Q16⋊28C2, C4.4D4.10C4, (C22×C4).234D4, C4⋊Q8.235C22, C4.24(C4.D4), C42.12C4⋊20C2, C4⋊1D4.125C22, C23.62(C22⋊C4), (C2×C42).205C22, C2.14(C23.24D4), C22.26C24.13C2, (C2×C4○D4).6C4, (C2×D4).29(C2×C4), (C2×C4)○(C4.D8), (C2×Q8).29(C2×C4), (C2×C4).1232(C2×D4), (C2×C4)○(C4.6Q16), C2.17(C2×C4.D4), (C22×C4).227(C2×C4), (C2×C4).155(C22×C4), (C2×C4).180(C22⋊C4), C22.219(C2×C22⋊C4), SmallGroup(128,275)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.411D4
G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=b, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=a2b-1c3 >
Subgroups: 276 in 123 conjugacy classes, 48 normal (14 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4×C8, C22⋊C8, C4⋊C8, C2×C42, C4×D4, C4⋊D4, C4.4D4, C4⋊1D4, C4⋊Q8, C2×C4○D4, C4.D8, C4.6Q16, C42.12C4, C22.26C24, C42.411D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4.D4, C2×C22⋊C4, C4○D8, C2×C4.D4, C23.24D4, C42.411D4
►On 64 points
(1 13 61 25)(2 14 62 26)(3 15 63 27)(4 16 64 28)(5 9 57 29)(6 10 58 30)(7 11 59 31)(8 12 60 32)(17 41 37 56)(18 42 38 49)(19 43 39 50)(20 44 40 51)(21 45 33 52)(22 46 34 53)(23 47 35 54)(24 48 36 55)
(1 18 57 34)(2 35 58 19)(3 20 59 36)(4 37 60 21)(5 22 61 38)(6 39 62 23)(7 24 63 40)(8 33 64 17)(9 46 25 49)(10 50 26 47)(11 48 27 51)(12 52 28 41)(13 42 29 53)(14 54 30 43)(15 44 31 55)(16 56 32 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 64 18 17 57 8 34 33)(2 40 35 7 58 24 19 63)(3 62 20 23 59 6 36 39)(4 38 37 5 60 22 21 61)(9 32 46 45 25 16 49 56)(10 55 50 15 26 44 47 31)(11 30 48 43 27 14 51 54)(12 53 52 13 28 42 41 29)
G:=sub<Sym(64)| (1,13,61,25)(2,14,62,26)(3,15,63,27)(4,16,64,28)(5,9,57,29)(6,10,58,30)(7,11,59,31)(8,12,60,32)(17,41,37,56)(18,42,38,49)(19,43,39,50)(20,44,40,51)(21,45,33,52)(22,46,34,53)(23,47,35,54)(24,48,36,55), (1,18,57,34)(2,35,58,19)(3,20,59,36)(4,37,60,21)(5,22,61,38)(6,39,62,23)(7,24,63,40)(8,33,64,17)(9,46,25,49)(10,50,26,47)(11,48,27,51)(12,52,28,41)(13,42,29,53)(14,54,30,43)(15,44,31,55)(16,56,32,45), (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), (1,64,18,17,57,8,34,33)(2,40,35,7,58,24,19,63)(3,62,20,23,59,6,36,39)(4,38,37,5,60,22,21,61)(9,32,46,45,25,16,49,56)(10,55,50,15,26,44,47,31)(11,30,48,43,27,14,51,54)(12,53,52,13,28,42,41,29)>;
G:=Group( (1,13,61,25)(2,14,62,26)(3,15,63,27)(4,16,64,28)(5,9,57,29)(6,10,58,30)(7,11,59,31)(8,12,60,32)(17,41,37,56)(18,42,38,49)(19,43,39,50)(20,44,40,51)(21,45,33,52)(22,46,34,53)(23,47,35,54)(24,48,36,55), (1,18,57,34)(2,35,58,19)(3,20,59,36)(4,37,60,21)(5,22,61,38)(6,39,62,23)(7,24,63,40)(8,33,64,17)(9,46,25,49)(10,50,26,47)(11,48,27,51)(12,52,28,41)(13,42,29,53)(14,54,30,43)(15,44,31,55)(16,56,32,45), (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), (1,64,18,17,57,8,34,33)(2,40,35,7,58,24,19,63)(3,62,20,23,59,6,36,39)(4,38,37,5,60,22,21,61)(9,32,46,45,25,16,49,56)(10,55,50,15,26,44,47,31)(11,30,48,43,27,14,51,54)(12,53,52,13,28,42,41,29) );
G=PermutationGroup([[(1,13,61,25),(2,14,62,26),(3,15,63,27),(4,16,64,28),(5,9,57,29),(6,10,58,30),(7,11,59,31),(8,12,60,32),(17,41,37,56),(18,42,38,49),(19,43,39,50),(20,44,40,51),(21,45,33,52),(22,46,34,53),(23,47,35,54),(24,48,36,55)], [(1,18,57,34),(2,35,58,19),(3,20,59,36),(4,37,60,21),(5,22,61,38),(6,39,62,23),(7,24,63,40),(8,33,64,17),(9,46,25,49),(10,50,26,47),(11,48,27,51),(12,52,28,41),(13,42,29,53),(14,54,30,43),(15,44,31,55),(16,56,32,45)], [(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)], [(1,64,18,17,57,8,34,33),(2,40,35,7,58,24,19,63),(3,62,20,23,59,6,36,39),(4,38,37,5,60,22,21,61),(9,32,46,45,25,16,49,56),(10,55,50,15,26,44,47,31),(11,30,48,43,27,14,51,54),(12,53,52,13,28,42,41,29)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 4M | 4N | 4O | 8A | ··· | 8P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 4 | 8 | 8 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 8 | 8 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | D4 | C4○D8 | C4.D4 |
| kernel | C42.411D4 | C4.D8 | C4.6Q16 | C42.12C4 | C22.26C24 | C4.4D4 | C2×C4○D4 | C42 | C22×C4 | C4 | C4 |
| # reps | 1 | 2 | 2 | 2 | 1 | 4 | 4 | 2 | 2 | 16 | 2 |
Matrix representation of C42.411D4 ►in GL4(𝔽17) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 1 | 0 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 14 | 3 | 0 | 0 |
| 3 | 3 | 0 | 0 |
| 0 | 0 | 10 | 10 |
| 0 | 0 | 12 | 0 |
| 3 | 3 | 0 | 0 |
| 14 | 3 | 0 | 0 |
| 0 | 0 | 7 | 7 |
| 0 | 0 | 5 | 10 |
G:=sub<GL(4,GF(17))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[0,16,0,0,1,0,0,0,0,0,16,0,0,0,0,16],[14,3,0,0,3,3,0,0,0,0,10,12,0,0,10,0],[3,14,0,0,3,3,0,0,0,0,7,5,0,0,7,10] >;
C42.411D4 in GAP, Magma, Sage, TeX
C_4^2._{411}D_4 % in TeX
G:=Group("C4^2.411D4"); // GroupNames label
G:=SmallGroup(128,275);
// by ID
G=gap.SmallGroup(128,275);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,352,1123,1018,248,1971,242]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=a^2*b^2,d^2=b,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=a^2*b^-1*c^3>;
// generators/relations