direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: D4×C16, C16○(C4⋊C8), (C4×C16)⋊5C2, C4⋊1(C2×C16), C16○2(C4⋊C4), C16○(C4⋊C16), C4⋊C16⋊20C2, C2.3(C8×D4), C4⋊C8.27C4, C4⋊C4.13C8, C16○(C22⋊C8), C22⋊1(C2×C16), (C22×C16)⋊7C2, (C8×D4).19C2, (C4×D4).40C4, (C2×D4).12C8, C8.138(C2×D4), C4.175(C4×D4), C22⋊C4.8C8, C16○(C22⋊C16), C16○2(C22⋊C4), C22⋊C16⋊17C2, C2.2(D4○C16), C4.57(C8○D4), C22⋊C8.25C4, C23.22(C2×C8), C2.4(C22×C16), C8.101(C4○D4), (C4×C8).375C22, C42.272(C2×C4), (C2×C8).632C23, (C2×C16).111C22, C22.29(C22×C8), (C22×C8).501C22, C4⋊C8○(C2×C16), C4⋊C4○(C2×C16), (C2×C16)○(C4×D4), (C2×C16)○(C8×D4), (C2×D4)○(C2×C16), C22⋊C8○(C2×C16), C22⋊C4○(C2×C16), (C2×C16)○(C4⋊C16), (C2×C4).41(C2×C8), (C2×C8).162(C2×C4), (C2×C16)○(C22⋊C16), (C2×C4).617(C22×C4), (C22×C4).378(C2×C4), SmallGroup(128,899)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for D4×C16
G = < a,b,c | a16=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >
Subgroups: 116 in 87 conjugacy classes, 58 normal (26 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, C23, C16, C16, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, C2×C8, C22×C4, C2×D4, C4×C8, C22⋊C8, C4⋊C8, C2×C16, C2×C16, C2×C16, C4×D4, C22×C8, C4×C16, C22⋊C16, C4⋊C16, C8×D4, C22×C16, D4×C16
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C23, C16, C2×C8, C22×C4, C2×D4, C4○D4, C2×C16, C4×D4, C22×C8, C8○D4, C8×D4, C22×C16, D4○C16, D4×C16
►On 64 points
(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 39 19 56)(2 40 20 57)(3 41 21 58)(4 42 22 59)(5 43 23 60)(6 44 24 61)(7 45 25 62)(8 46 26 63)(9 47 27 64)(10 48 28 49)(11 33 29 50)(12 34 30 51)(13 35 31 52)(14 36 32 53)(15 37 17 54)(16 38 18 55)
(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)(33 58)(34 59)(35 60)(36 61)(37 62)(38 63)(39 64)(40 49)(41 50)(42 51)(43 52)(44 53)(45 54)(46 55)(47 56)(48 57)
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), (1,39,19,56)(2,40,20,57)(3,41,21,58)(4,42,22,59)(5,43,23,60)(6,44,24,61)(7,45,25,62)(8,46,26,63)(9,47,27,64)(10,48,28,49)(11,33,29,50)(12,34,30,51)(13,35,31,52)(14,36,32,53)(15,37,17,54)(16,38,18,55), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32)(33,58)(34,59)(35,60)(36,61)(37,62)(38,63)(39,64)(40,49)(41,50)(42,51)(43,52)(44,53)(45,54)(46,55)(47,56)(48,57)>;
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), (1,39,19,56)(2,40,20,57)(3,41,21,58)(4,42,22,59)(5,43,23,60)(6,44,24,61)(7,45,25,62)(8,46,26,63)(9,47,27,64)(10,48,28,49)(11,33,29,50)(12,34,30,51)(13,35,31,52)(14,36,32,53)(15,37,17,54)(16,38,18,55), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32)(33,58)(34,59)(35,60)(36,61)(37,62)(38,63)(39,64)(40,49)(41,50)(42,51)(43,52)(44,53)(45,54)(46,55)(47,56)(48,57) );
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)], [(1,39,19,56),(2,40,20,57),(3,41,21,58),(4,42,22,59),(5,43,23,60),(6,44,24,61),(7,45,25,62),(8,46,26,63),(9,47,27,64),(10,48,28,49),(11,33,29,50),(12,34,30,51),(13,35,31,52),(14,36,32,53),(15,37,17,54),(16,38,18,55)], [(1,9),(2,10),(3,11),(4,12),(5,13),(6,14),(7,15),(8,16),(17,25),(18,26),(19,27),(20,28),(21,29),(22,30),(23,31),(24,32),(33,58),(34,59),(35,60),(36,61),(37,62),(38,63),(39,64),(40,49),(41,50),(42,51),(43,52),(44,53),(45,54),(46,55),(47,56),(48,57)]])
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 8A | ··· | 8H | 8I | ··· | 8T | 16A | ··· | 16P | 16Q | ··· | 16AN |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 | 16 | ··· | 16 | 16 | ··· | 16 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C8 | C8 | C8 | C16 | D4 | C4○D4 | C8○D4 | D4○C16 |
| kernel | D4×C16 | C4×C16 | C22⋊C16 | C4⋊C16 | C8×D4 | C22×C16 | C22⋊C8 | C4⋊C8 | C4×D4 | C22⋊C4 | C4⋊C4 | C2×D4 | D4 | C16 | C8 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 2 | 4 | 2 | 2 | 8 | 4 | 4 | 32 | 2 | 2 | 4 | 8 |
Matrix representation of D4×C16 ►in GL3(𝔽17) generated by
| 6 | 0 | 0 |
| 0 | 6 | 0 |
| 0 | 0 | 6 |
| 16 | 0 | 0 |
| 0 | 1 | 15 |
| 0 | 1 | 16 |
| 1 | 0 | 0 |
| 0 | 16 | 2 |
| 0 | 0 | 1 |
G:=sub<GL(3,GF(17))| [6,0,0,0,6,0,0,0,6],[16,0,0,0,1,1,0,15,16],[1,0,0,0,16,0,0,2,1] >;
D4×C16 in GAP, Magma, Sage, TeX
D_4\times C_{16} % in TeX
G:=Group("D4xC16"); // GroupNames label
G:=SmallGroup(128,899);
// by ID
G=gap.SmallGroup(128,899);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,100,102,124]);
// Polycyclic
G:=Group<a,b,c|a^16=b^4=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations