p-group, metabelian, nilpotent (class 4), monomial
Aliases: D8⋊1C8, C4.16D16, C4.14SD32, C8.6M4(2), C42.309D4, C4.1C4≀C2, (C4×C16)⋊2C2, C8.7(C2×C8), C8⋊1C8⋊1C2, (C2×D8).5C4, (C4×D8).1C2, C2.D8.6C4, C2.6(D4⋊C8), (C2×C4).158D8, (C2×C8).295D4, C4.1(C22⋊C8), (C2×C4).59SD16, C2.1(C2.D16), (C4×C8).384C22, C2.1(D8.C4), C22.40(D4⋊C4), (C2×C8).165(C2×C4), (C2×C4).211(C22⋊C4), SmallGroup(128,63)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C4.16D16
G = < a,b,c | a4=b16=1, c2=a, ab=ba, ac=ca, cbc-1=ab-1 >
Smallest permutation representation of C4.16D16
►On 64 points
Generators in S64
(1 62 47 30)(2 63 48 31)(3 64 33 32)(4 49 34 17)(5 50 35 18)(6 51 36 19)(7 52 37 20)(8 53 38 21)(9 54 39 22)(10 55 40 23)(11 56 41 24)(12 57 42 25)(13 58 43 26)(14 59 44 27)(15 60 45 28)(16 61 46 29)
(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 29 62 16 47 61 30 46)(2 15 63 60 48 45 31 28)(3 59 64 44 33 27 32 14)(4 43 49 26 34 13 17 58)(5 25 50 12 35 57 18 42)(6 11 51 56 36 41 19 24)(7 55 52 40 37 23 20 10)(8 39 53 22 38 9 21 54)
G:=sub<Sym(64)| (1,62,47,30)(2,63,48,31)(3,64,33,32)(4,49,34,17)(5,50,35,18)(6,51,36,19)(7,52,37,20)(8,53,38,21)(9,54,39,22)(10,55,40,23)(11,56,41,24)(12,57,42,25)(13,58,43,26)(14,59,44,27)(15,60,45,28)(16,61,46,29), (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), (1,29,62,16,47,61,30,46)(2,15,63,60,48,45,31,28)(3,59,64,44,33,27,32,14)(4,43,49,26,34,13,17,58)(5,25,50,12,35,57,18,42)(6,11,51,56,36,41,19,24)(7,55,52,40,37,23,20,10)(8,39,53,22,38,9,21,54)>;
G:=Group( (1,62,47,30)(2,63,48,31)(3,64,33,32)(4,49,34,17)(5,50,35,18)(6,51,36,19)(7,52,37,20)(8,53,38,21)(9,54,39,22)(10,55,40,23)(11,56,41,24)(12,57,42,25)(13,58,43,26)(14,59,44,27)(15,60,45,28)(16,61,46,29), (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), (1,29,62,16,47,61,30,46)(2,15,63,60,48,45,31,28)(3,59,64,44,33,27,32,14)(4,43,49,26,34,13,17,58)(5,25,50,12,35,57,18,42)(6,11,51,56,36,41,19,24)(7,55,52,40,37,23,20,10)(8,39,53,22,38,9,21,54) );
G=PermutationGroup([[(1,62,47,30),(2,63,48,31),(3,64,33,32),(4,49,34,17),(5,50,35,18),(6,51,36,19),(7,52,37,20),(8,53,38,21),(9,54,39,22),(10,55,40,23),(11,56,41,24),(12,57,42,25),(13,58,43,26),(14,59,44,27),(15,60,45,28),(16,61,46,29)], [(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)], [(1,29,62,16,47,61,30,46),(2,15,63,60,48,45,31,28),(3,59,64,44,33,27,32,14),(4,43,49,26,34,13,17,58),(5,25,50,12,35,57,18,42),(6,11,51,56,36,41,19,24),(7,55,52,40,37,23,20,10),(8,39,53,22,38,9,21,54)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 16A | ··· | 16P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 16 | ··· | 16 |
| size | 1 | 1 | 1 | 1 | 8 | 8 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 8 | 8 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 2 | ··· | 2 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | D4 | D4 | M4(2) | D8 | SD16 | C4≀C2 | D16 | SD32 | D8.C4 |
| kernel | C4.16D16 | C8⋊1C8 | C4×C16 | C4×D8 | C2.D8 | C2×D8 | D8 | C42 | C2×C8 | C8 | C2×C4 | C2×C4 | C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 8 |
Matrix representation of C4.16D16 ►in GL4(𝔽17) generated by
| 13 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 10 | 7 | 0 | 0 |
| 10 | 10 | 0 | 0 |
| 0 | 0 | 3 | 5 |
| 0 | 0 | 7 | 13 |
| 7 | 10 | 0 | 0 |
| 10 | 10 | 0 | 0 |
| 0 | 0 | 4 | 8 |
| 0 | 0 | 7 | 13 |
G:=sub<GL(4,GF(17))| [13,0,0,0,0,13,0,0,0,0,4,0,0,0,0,4],[10,10,0,0,7,10,0,0,0,0,3,7,0,0,5,13],[7,10,0,0,10,10,0,0,0,0,4,7,0,0,8,13] >;
C4.16D16 in GAP, Magma, Sage, TeX
C_4._{16}D_{16} % in TeX
G:=Group("C4.16D16"); // GroupNames label
G:=SmallGroup(128,63);
// by ID
G=gap.SmallGroup(128,63);
# by ID
G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,422,219,436,136,2804,1411,172]);
// Polycyclic
G:=Group<a,b,c|a^4=b^16=1,c^2=a,a*b=b*a,a*c=c*a,c*b*c^-1=a*b^-1>;
// generators/relations
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