p-group, metabelian, nilpotent (class 3), monomial
Aliases: D4.C16, Q8.C16, C16.28D4, M6(2)⋊4C2, M4(2).4C8, M5(2).3C4, C8.10M4(2), C22.1M5(2), (C2×C32)⋊2C2, C4.3(C2×C16), C8○D4.2C4, C4○D4.1C8, D4○C16.2C2, C4.35(C22⋊C8), C8.59(C22⋊C4), C2.8(C22⋊C16), (C2×C16).101C22, (C2×C4).48(C2×C8), (C2×C8).189(C2×C4), SmallGroup(128,133)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for D4.C16
G = < a,b,c | a4=b2=1, c16=a2, bab=a-1, ac=ca, cbc-1=ab >
Smallest permutation representation of D4.C16
►On 64 points
Generators in S64
(1 41 17 57)(2 42 18 58)(3 43 19 59)(4 44 20 60)(5 45 21 61)(6 46 22 62)(7 47 23 63)(8 48 24 64)(9 49 25 33)(10 50 26 34)(11 51 27 35)(12 52 28 36)(13 53 29 37)(14 54 30 38)(15 55 31 39)(16 56 32 40)
(1 57)(2 18)(3 43)(5 61)(6 22)(7 47)(9 33)(10 26)(11 51)(13 37)(14 30)(15 55)(17 41)(19 59)(21 45)(23 63)(25 49)(27 35)(29 53)(31 39)(36 52)(40 56)(44 60)(48 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)
G:=sub<Sym(64)| (1,41,17,57)(2,42,18,58)(3,43,19,59)(4,44,20,60)(5,45,21,61)(6,46,22,62)(7,47,23,63)(8,48,24,64)(9,49,25,33)(10,50,26,34)(11,51,27,35)(12,52,28,36)(13,53,29,37)(14,54,30,38)(15,55,31,39)(16,56,32,40), (1,57)(2,18)(3,43)(5,61)(6,22)(7,47)(9,33)(10,26)(11,51)(13,37)(14,30)(15,55)(17,41)(19,59)(21,45)(23,63)(25,49)(27,35)(29,53)(31,39)(36,52)(40,56)(44,60)(48,64), (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)>;
G:=Group( (1,41,17,57)(2,42,18,58)(3,43,19,59)(4,44,20,60)(5,45,21,61)(6,46,22,62)(7,47,23,63)(8,48,24,64)(9,49,25,33)(10,50,26,34)(11,51,27,35)(12,52,28,36)(13,53,29,37)(14,54,30,38)(15,55,31,39)(16,56,32,40), (1,57)(2,18)(3,43)(5,61)(6,22)(7,47)(9,33)(10,26)(11,51)(13,37)(14,30)(15,55)(17,41)(19,59)(21,45)(23,63)(25,49)(27,35)(29,53)(31,39)(36,52)(40,56)(44,60)(48,64), (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) );
G=PermutationGroup([[(1,41,17,57),(2,42,18,58),(3,43,19,59),(4,44,20,60),(5,45,21,61),(6,46,22,62),(7,47,23,63),(8,48,24,64),(9,49,25,33),(10,50,26,34),(11,51,27,35),(12,52,28,36),(13,53,29,37),(14,54,30,38),(15,55,31,39),(16,56,32,40)], [(1,57),(2,18),(3,43),(5,61),(6,22),(7,47),(9,33),(10,26),(11,51),(13,37),(14,30),(15,55),(17,41),(19,59),(21,45),(23,63),(25,49),(27,35),(29,53),(31,39),(36,52),(40,56),(44,60),(48,64)], [(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)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 16A | ··· | 16H | 16I | 16J | 16K | 16L | 16M | 16N | 16O | 16P | 32A | ··· | 32P | 32Q | ··· | 32X |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 16 | ··· | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 32 | ··· | 32 | 32 | ··· | 32 |
| size | 1 | 1 | 2 | 4 | 1 | 1 | 2 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | C8 | C16 | C16 | D4 | M4(2) | M5(2) | D4.C16 |
| kernel | D4.C16 | C2×C32 | M6(2) | D4○C16 | M5(2) | C8○D4 | M4(2) | C4○D4 | D4 | Q8 | C16 | C8 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 2 | 2 | 4 | 16 |
Matrix representation of D4.C16 ►in GL2(𝔽97) generated by
| 79 | 43 |
| 15 | 18 |
| 79 | 43 |
| 94 | 18 |
| 46 | 32 |
| 45 | 57 |
G:=sub<GL(2,GF(97))| [79,15,43,18],[79,94,43,18],[46,45,32,57] >;
D4.C16 in GAP, Magma, Sage, TeX
D_4.C_{16} % in TeX
G:=Group("D4.C16"); // GroupNames label
G:=SmallGroup(128,133);
// by ID
G=gap.SmallGroup(128,133);
# by ID
G:=PCGroup([7,-2,2,-2,2,-2,-2,-2,56,85,723,352,1018,80,102,124]);
// Polycyclic
G:=Group<a,b,c|a^4=b^2=1,c^16=a^2,b*a*b=a^-1,a*c=c*a,c*b*c^-1=a*b>;
// generators/relations
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