p-group, metabelian, nilpotent (class 4), monomial
Aliases: C4.4D16, C4.3SD32, C8.18SD16, C42.333D4, (C4×C16)⋊6C2, C2.9(C2×D16), (C2×C4).81D8, C2.D16⋊3C2, C8⋊2Q8⋊11C2, (C2×C8).250D4, C8⋊4D4.9C2, C8.50(C4○D4), C2.14(C2×SD32), C4.14(C2×SD16), (C2×C16).72C22, (C2×C8).538C23, (C4×C8).399C22, C4.7(C4.4D4), C2.9(C4.4D8), (C2×D8).12C22, C22.124(C2×D8), C2.D8.23C22, (C2×C4).806(C2×D4), SmallGroup(128,972)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C4.4D16
G = < a,b,c | a4=b16=1, c2=a2, ab=ba, cac-1=a-1, cbc-1=a2b-1 >
Subgroups: 264 in 80 conjugacy classes, 36 normal (18 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C16, C42, C4⋊C4, C2×C8, D8, C2×D4, C2×Q8, C4×C8, C2.D8, C2.D8, C2×C16, C4⋊1D4, C4⋊Q8, C2×D8, C2×D8, C4×C16, C2.D16, C8⋊4D4, C8⋊2Q8, C4.4D16
Quotients: C1, C2, C22, D4, C23, D8, SD16, C2×D4, C4○D4, D16, SD32, C4.4D4, C2×D8, C2×SD16, C4.4D8, C2×D16, C2×SD32, C4.4D16
►On 64 points
(1 32 33 56)(2 17 34 57)(3 18 35 58)(4 19 36 59)(5 20 37 60)(6 21 38 61)(7 22 39 62)(8 23 40 63)(9 24 41 64)(10 25 42 49)(11 26 43 50)(12 27 44 51)(13 28 45 52)(14 29 46 53)(15 30 47 54)(16 31 48 55)
(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 48 33 16)(2 15 34 47)(3 46 35 14)(4 13 36 45)(5 44 37 12)(6 11 38 43)(7 42 39 10)(8 9 40 41)(17 54 57 30)(18 29 58 53)(19 52 59 28)(20 27 60 51)(21 50 61 26)(22 25 62 49)(23 64 63 24)(31 56 55 32)
G:=sub<Sym(64)| (1,32,33,56)(2,17,34,57)(3,18,35,58)(4,19,36,59)(5,20,37,60)(6,21,38,61)(7,22,39,62)(8,23,40,63)(9,24,41,64)(10,25,42,49)(11,26,43,50)(12,27,44,51)(13,28,45,52)(14,29,46,53)(15,30,47,54)(16,31,48,55), (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), (1,48,33,16)(2,15,34,47)(3,46,35,14)(4,13,36,45)(5,44,37,12)(6,11,38,43)(7,42,39,10)(8,9,40,41)(17,54,57,30)(18,29,58,53)(19,52,59,28)(20,27,60,51)(21,50,61,26)(22,25,62,49)(23,64,63,24)(31,56,55,32)>;
G:=Group( (1,32,33,56)(2,17,34,57)(3,18,35,58)(4,19,36,59)(5,20,37,60)(6,21,38,61)(7,22,39,62)(8,23,40,63)(9,24,41,64)(10,25,42,49)(11,26,43,50)(12,27,44,51)(13,28,45,52)(14,29,46,53)(15,30,47,54)(16,31,48,55), (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), (1,48,33,16)(2,15,34,47)(3,46,35,14)(4,13,36,45)(5,44,37,12)(6,11,38,43)(7,42,39,10)(8,9,40,41)(17,54,57,30)(18,29,58,53)(19,52,59,28)(20,27,60,51)(21,50,61,26)(22,25,62,49)(23,64,63,24)(31,56,55,32) );
G=PermutationGroup([[(1,32,33,56),(2,17,34,57),(3,18,35,58),(4,19,36,59),(5,20,37,60),(6,21,38,61),(7,22,39,62),(8,23,40,63),(9,24,41,64),(10,25,42,49),(11,26,43,50),(12,27,44,51),(13,28,45,52),(14,29,46,53),(15,30,47,54),(16,31,48,55)], [(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)], [(1,48,33,16),(2,15,34,47),(3,46,35,14),(4,13,36,45),(5,44,37,12),(6,11,38,43),(7,42,39,10),(8,9,40,41),(17,54,57,30),(18,29,58,53),(19,52,59,28),(20,27,60,51),(21,50,61,26),(22,25,62,49),(23,64,63,24),(31,56,55,32)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4F | 4G | 4H | 8A | ··· | 8H | 16A | ··· | 16P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 8 | ··· | 8 | 16 | ··· | 16 |
| size | 1 | 1 | 1 | 1 | 16 | 16 | 2 | ··· | 2 | 16 | 16 | 2 | ··· | 2 | 2 | ··· | 2 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | D4 | D4 | SD16 | C4○D4 | D8 | D16 | SD32 |
| kernel | C4.4D16 | C4×C16 | C2.D16 | C8⋊4D4 | C8⋊2Q8 | C42 | C2×C8 | C8 | C8 | C2×C4 | C4 | C4 |
| # reps | 1 | 1 | 4 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 8 | 8 |
Matrix representation of C4.4D16 ►in GL4(𝔽17) generated by
| 0 | 16 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 16 | 10 | 0 | 0 |
| 7 | 16 | 0 | 0 |
| 0 | 0 | 13 | 6 |
| 0 | 0 | 11 | 13 |
| 16 | 10 | 0 | 0 |
| 10 | 1 | 0 | 0 |
| 0 | 0 | 6 | 4 |
| 0 | 0 | 4 | 11 |
G:=sub<GL(4,GF(17))| [0,1,0,0,16,0,0,0,0,0,16,0,0,0,0,16],[16,7,0,0,10,16,0,0,0,0,13,11,0,0,6,13],[16,10,0,0,10,1,0,0,0,0,6,4,0,0,4,11] >;
C4.4D16 in GAP, Magma, Sage, TeX
C_4._4D_{16} % in TeX
G:=Group("C4.4D16"); // GroupNames label
G:=SmallGroup(128,972);
// by ID
G=gap.SmallGroup(128,972);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,141,120,422,58,1123,360,3924,102,4037,124]);
// Polycyclic
G:=Group<a,b,c|a^4=b^16=1,c^2=a^2,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=a^2*b^-1>;
// generators/relations