p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.45D4, C42.601C23, D4⋊C8⋊3C2, C4⋊Q8.6C4, C4.79(C2×D8), C4⋊D4.1C4, (C2×C4).126D8, C4⋊1D4.6C4, (C4×C8).3C22, C4.24(C8○D4), C42.54(C2×C4), C4.91(C2×SD16), (C4×D4).1C22, C4.7(D4⋊C4), C4⋊C8.247C22, (C2×C4).113SD16, (C22×C4).728D4, C42.12C4⋊8C2, C22.1(D4⋊C4), C23.94(C22⋊C4), (C2×C42).157C22, C2.6(C42⋊C22), C22.26C24.1C2, (C2×C4⋊C8)⋊3C2, C4⋊C4.47(C2×C4), (C2×D4).48(C2×C4), C2.4(C2×D4⋊C4), (C2×C4).1444(C2×D4), (C2×C4).74(C22⋊C4), (C2×C4).306(C22×C4), (C22×C4).179(C2×C4), C22.156(C2×C22⋊C4), C2.12((C22×C8)⋊C2), SmallGroup(128,212)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.45D4
G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=b, ab=ba, cac-1=ab2, dad-1=a-1, bc=cb, bd=db, dcd-1=a2b-1c3 >
Subgroups: 284 in 130 conjugacy classes, 54 normal (28 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, 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, C22⋊C8, C4⋊C8, C4⋊C8, C2×C42, C4×D4, C4×D4, C4⋊D4, C4⋊D4, C4.4D4, C4⋊1D4, C4⋊Q8, C22×C8, C2×C4○D4, D4⋊C8, C2×C4⋊C8, C42.12C4, C22.26C24, C42.45D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, D8, SD16, C22×C4, C2×D4, D4⋊C4, C2×C22⋊C4, C8○D4, C2×D8, C2×SD16, (C22×C8)⋊C2, C2×D4⋊C4, C42⋊C22, C42.45D4
►On 64 points
(1 9 51 32)(2 29 52 14)(3 11 53 26)(4 31 54 16)(5 13 55 28)(6 25 56 10)(7 15 49 30)(8 27 50 12)(17 36 60 44)(18 41 61 33)(19 38 62 46)(20 43 63 35)(21 40 64 48)(22 45 57 37)(23 34 58 42)(24 47 59 39)
(1 22 55 61)(2 23 56 62)(3 24 49 63)(4 17 50 64)(5 18 51 57)(6 19 52 58)(7 20 53 59)(8 21 54 60)(9 45 28 33)(10 46 29 34)(11 47 30 35)(12 48 31 36)(13 41 32 37)(14 42 25 38)(15 43 26 39)(16 44 27 40)
(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 21 22 54 55 60 61 8)(2 53 23 59 56 7 62 20)(3 58 24 6 49 19 63 52)(4 5 17 18 50 51 64 57)(9 48 45 31 28 36 33 12)(10 30 46 35 29 11 34 47)(13 44 41 27 32 40 37 16)(14 26 42 39 25 15 38 43)
G:=sub<Sym(64)| (1,9,51,32)(2,29,52,14)(3,11,53,26)(4,31,54,16)(5,13,55,28)(6,25,56,10)(7,15,49,30)(8,27,50,12)(17,36,60,44)(18,41,61,33)(19,38,62,46)(20,43,63,35)(21,40,64,48)(22,45,57,37)(23,34,58,42)(24,47,59,39), (1,22,55,61)(2,23,56,62)(3,24,49,63)(4,17,50,64)(5,18,51,57)(6,19,52,58)(7,20,53,59)(8,21,54,60)(9,45,28,33)(10,46,29,34)(11,47,30,35)(12,48,31,36)(13,41,32,37)(14,42,25,38)(15,43,26,39)(16,44,27,40), (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), (1,21,22,54,55,60,61,8)(2,53,23,59,56,7,62,20)(3,58,24,6,49,19,63,52)(4,5,17,18,50,51,64,57)(9,48,45,31,28,36,33,12)(10,30,46,35,29,11,34,47)(13,44,41,27,32,40,37,16)(14,26,42,39,25,15,38,43)>;
G:=Group( (1,9,51,32)(2,29,52,14)(3,11,53,26)(4,31,54,16)(5,13,55,28)(6,25,56,10)(7,15,49,30)(8,27,50,12)(17,36,60,44)(18,41,61,33)(19,38,62,46)(20,43,63,35)(21,40,64,48)(22,45,57,37)(23,34,58,42)(24,47,59,39), (1,22,55,61)(2,23,56,62)(3,24,49,63)(4,17,50,64)(5,18,51,57)(6,19,52,58)(7,20,53,59)(8,21,54,60)(9,45,28,33)(10,46,29,34)(11,47,30,35)(12,48,31,36)(13,41,32,37)(14,42,25,38)(15,43,26,39)(16,44,27,40), (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), (1,21,22,54,55,60,61,8)(2,53,23,59,56,7,62,20)(3,58,24,6,49,19,63,52)(4,5,17,18,50,51,64,57)(9,48,45,31,28,36,33,12)(10,30,46,35,29,11,34,47)(13,44,41,27,32,40,37,16)(14,26,42,39,25,15,38,43) );
G=PermutationGroup([[(1,9,51,32),(2,29,52,14),(3,11,53,26),(4,31,54,16),(5,13,55,28),(6,25,56,10),(7,15,49,30),(8,27,50,12),(17,36,60,44),(18,41,61,33),(19,38,62,46),(20,43,63,35),(21,40,64,48),(22,45,57,37),(23,34,58,42),(24,47,59,39)], [(1,22,55,61),(2,23,56,62),(3,24,49,63),(4,17,50,64),(5,18,51,57),(6,19,52,58),(7,20,53,59),(8,21,54,60),(9,45,28,33),(10,46,29,34),(11,47,30,35),(12,48,31,36),(13,41,32,37),(14,42,25,38),(15,43,26,39),(16,44,27,40)], [(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)], [(1,21,22,54,55,60,61,8),(2,53,23,59,56,7,62,20),(3,58,24,6,49,19,63,52),(4,5,17,18,50,51,64,57),(9,48,45,31,28,36,33,12),(10,30,46,35,29,11,34,47),(13,44,41,27,32,40,37,16),(14,26,42,39,25,15,38,43)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 4K | 4L | 4M | 4N | 8A | ··· | 8P |
| order | 1 | 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 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 8 | 8 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | D4 | D8 | SD16 | C8○D4 | C42⋊C22 |
| kernel | C42.45D4 | D4⋊C8 | C2×C4⋊C8 | C42.12C4 | C22.26C24 | C4⋊D4 | C4⋊1D4 | C4⋊Q8 | C42 | C22×C4 | C2×C4 | C2×C4 | C4 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 2 |
Matrix representation of C42.45D4 ►in GL4(𝔽17) generated by
| 0 | 1 | 0 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 2 |
| 0 | 0 | 0 | 1 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 5 | 12 | 0 | 0 |
| 5 | 5 | 0 | 0 |
| 0 | 0 | 2 | 13 |
| 0 | 0 | 2 | 15 |
| 5 | 12 | 0 | 0 |
| 12 | 12 | 0 | 0 |
| 0 | 0 | 2 | 13 |
| 0 | 0 | 0 | 15 |
G:=sub<GL(4,GF(17))| [0,16,0,0,1,0,0,0,0,0,16,0,0,0,2,1],[16,0,0,0,0,16,0,0,0,0,4,0,0,0,0,4],[5,5,0,0,12,5,0,0,0,0,2,2,0,0,13,15],[5,12,0,0,12,12,0,0,0,0,2,0,0,0,13,15] >;
C42.45D4 in GAP, Magma, Sage, TeX
C_4^2._{45}D_4 % in TeX
G:=Group("C4^2.45D4"); // GroupNames label
G:=SmallGroup(128,212);
// by ID
G=gap.SmallGroup(128,212);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,723,1123,570,136,172]);
// 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,c*a*c^-1=a*b^2,d*a*d^-1=a^-1,b*c=c*b,b*d=d*b,d*c*d^-1=a^2*b^-1*c^3>;
// generators/relations