p-group, metabelian, nilpotent (class 5), monomial
Aliases: D16.1C4, C16.22D4, C4.18D16, Q32.1C4, C8.20SD16, C22.1SD32, (C2×C32)⋊4C2, (C2×C4).67D8, C16.12(C2×C4), C4○D16.1C2, (C2×C8).265D4, C8.4Q8⋊1C2, C2.8(C2.D16), C8.15(C22⋊C4), (C2×C16).92C22, C4.15(D4⋊C4), SmallGroup(128,149)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for D16.C4
G = < a,b,c | a16=b2=1, c4=a8, bab=cac-1=a-1, cbc-1=a13b >
Smallest permutation representation of D16.C4
►On 64 points
Generators in S64
(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 16)(2 15)(3 14)(4 13)(5 12)(6 11)(7 10)(8 9)(17 32)(18 31)(19 30)(20 29)(21 28)(22 27)(23 26)(24 25)(33 39)(34 38)(35 37)(40 48)(41 47)(42 46)(43 45)(49 57)(50 56)(51 55)(52 54)(58 64)(59 63)(60 62)
(1 37 25 62 9 45 17 54)(2 36 26 61 10 44 18 53)(3 35 27 60 11 43 19 52)(4 34 28 59 12 42 20 51)(5 33 29 58 13 41 21 50)(6 48 30 57 14 40 22 49)(7 47 31 56 15 39 23 64)(8 46 32 55 16 38 24 63)
G:=sub<Sym(64)| (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), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25)(33,39)(34,38)(35,37)(40,48)(41,47)(42,46)(43,45)(49,57)(50,56)(51,55)(52,54)(58,64)(59,63)(60,62), (1,37,25,62,9,45,17,54)(2,36,26,61,10,44,18,53)(3,35,27,60,11,43,19,52)(4,34,28,59,12,42,20,51)(5,33,29,58,13,41,21,50)(6,48,30,57,14,40,22,49)(7,47,31,56,15,39,23,64)(8,46,32,55,16,38,24,63)>;
G:=Group( (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), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25)(33,39)(34,38)(35,37)(40,48)(41,47)(42,46)(43,45)(49,57)(50,56)(51,55)(52,54)(58,64)(59,63)(60,62), (1,37,25,62,9,45,17,54)(2,36,26,61,10,44,18,53)(3,35,27,60,11,43,19,52)(4,34,28,59,12,42,20,51)(5,33,29,58,13,41,21,50)(6,48,30,57,14,40,22,49)(7,47,31,56,15,39,23,64)(8,46,32,55,16,38,24,63) );
G=PermutationGroup([[(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)], [(1,16),(2,15),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9),(17,32),(18,31),(19,30),(20,29),(21,28),(22,27),(23,26),(24,25),(33,39),(34,38),(35,37),(40,48),(41,47),(42,46),(43,45),(49,57),(50,56),(51,55),(52,54),(58,64),(59,63),(60,62)], [(1,37,25,62,9,45,17,54),(2,36,26,61,10,44,18,53),(3,35,27,60,11,43,19,52),(4,34,28,59,12,42,20,51),(5,33,29,58,13,41,21,50),(6,48,30,57,14,40,22,49),(7,47,31,56,15,39,23,64),(8,46,32,55,16,38,24,63)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 8A | 8B | 8C | 8D | 8E | 8F | 16A | ··· | 16H | 32A | ··· | 32P |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 16 | ··· | 16 | 32 | ··· | 32 |
| size | 1 | 1 | 2 | 16 | 1 | 1 | 2 | 16 | 2 | 2 | 2 | 2 | 16 | 16 | 2 | ··· | 2 | 2 | ··· | 2 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | D4 | D4 | SD16 | D8 | D16 | SD32 | D16.C4 |
| kernel | D16.C4 | C8.4Q8 | C2×C32 | C4○D16 | D16 | Q32 | C16 | C2×C8 | C8 | C2×C4 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 4 | 4 | 16 |
Matrix representation of D16.C4 ►in GL2(𝔽97) generated by
| 69 | 93 |
| 2 | 73 |
| 69 | 93 |
| 26 | 28 |
| 39 | 86 |
| 4 | 58 |
G:=sub<GL(2,GF(97))| [69,2,93,73],[69,26,93,28],[39,4,86,58] >;
D16.C4 in GAP, Magma, Sage, TeX
D_{16}.C_4 % in TeX
G:=Group("D16.C4"); // GroupNames label
G:=SmallGroup(128,149);
// by ID
G=gap.SmallGroup(128,149);
# by ID
G:=PCGroup([7,-2,2,-2,2,-2,-2,-2,56,85,422,219,268,248,1684,851,242,4037,2028,124]);
// Polycyclic
G:=Group<a,b,c|a^16=b^2=1,c^4=a^8,b*a*b=c*a*c^-1=a^-1,c*b*c^-1=a^13*b>;
// generators/relations
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