direct product, p-group, metabelian, nilpotent (class 4), monomial
Aliases: C4×D16, C42.329D4, C16⋊7(C2×C4), (C4×C16)⋊8C2, D8⋊2(C2×C4), (C4×D8)⋊1C2, C2.13(C4×D8), C2.3(C2×D16), C4.25(C4×D4), C4○2(C16⋊3C4), C16⋊3C4⋊14C2, (C2×D16).7C2, (C2×C4).172D8, (C2×C8).231D4, C4○2(C2.D16), C2.D16⋊21C2, C4.11(C4○D8), C2.3(C4○D16), C8.38(C4○D4), C8.35(C22×C4), C22.61(C2×D8), (C4×C8).396C22, (C2×C16).69C22, (C2×C8).502C23, (C2×D8).103C22, C2.D8.150C22, (C2×C4).768(C2×D4), SmallGroup(128,904)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C4×D16
G = < a,b,c | a4=b16=c2=1, ab=ba, ac=ca, cbc=b-1 >
Subgroups: 244 in 87 conjugacy classes, 42 normal (22 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, C23, C16, C16, C42, C22⋊C4, C4⋊C4, C2×C8, D8, D8, C22×C4, C2×D4, C4×C8, D4⋊C4, C2.D8, C2×C16, D16, C4×D4, C2×D8, C4×C16, C2.D16, C16⋊3C4, C4×D8, C2×D16, C4×D16
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D8, C22×C4, C2×D4, C4○D4, D16, C4×D4, C2×D8, C4○D8, C4×D8, C2×D16, C4○D16, C4×D16
►On 64 points
(1 35 23 49)(2 36 24 50)(3 37 25 51)(4 38 26 52)(5 39 27 53)(6 40 28 54)(7 41 29 55)(8 42 30 56)(9 43 31 57)(10 44 32 58)(11 45 17 59)(12 46 18 60)(13 47 19 61)(14 48 20 62)(15 33 21 63)(16 34 22 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 23)(2 22)(3 21)(4 20)(5 19)(6 18)(7 17)(8 32)(9 31)(10 30)(11 29)(12 28)(13 27)(14 26)(15 25)(16 24)(33 51)(34 50)(35 49)(36 64)(37 63)(38 62)(39 61)(40 60)(41 59)(42 58)(43 57)(44 56)(45 55)(46 54)(47 53)(48 52)
G:=sub<Sym(64)| (1,35,23,49)(2,36,24,50)(3,37,25,51)(4,38,26,52)(5,39,27,53)(6,40,28,54)(7,41,29,55)(8,42,30,56)(9,43,31,57)(10,44,32,58)(11,45,17,59)(12,46,18,60)(13,47,19,61)(14,48,20,62)(15,33,21,63)(16,34,22,64), (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), (1,23)(2,22)(3,21)(4,20)(5,19)(6,18)(7,17)(8,32)(9,31)(10,30)(11,29)(12,28)(13,27)(14,26)(15,25)(16,24)(33,51)(34,50)(35,49)(36,64)(37,63)(38,62)(39,61)(40,60)(41,59)(42,58)(43,57)(44,56)(45,55)(46,54)(47,53)(48,52)>;
G:=Group( (1,35,23,49)(2,36,24,50)(3,37,25,51)(4,38,26,52)(5,39,27,53)(6,40,28,54)(7,41,29,55)(8,42,30,56)(9,43,31,57)(10,44,32,58)(11,45,17,59)(12,46,18,60)(13,47,19,61)(14,48,20,62)(15,33,21,63)(16,34,22,64), (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), (1,23)(2,22)(3,21)(4,20)(5,19)(6,18)(7,17)(8,32)(9,31)(10,30)(11,29)(12,28)(13,27)(14,26)(15,25)(16,24)(33,51)(34,50)(35,49)(36,64)(37,63)(38,62)(39,61)(40,60)(41,59)(42,58)(43,57)(44,56)(45,55)(46,54)(47,53)(48,52) );
G=PermutationGroup([[(1,35,23,49),(2,36,24,50),(3,37,25,51),(4,38,26,52),(5,39,27,53),(6,40,28,54),(7,41,29,55),(8,42,30,56),(9,43,31,57),(10,44,32,58),(11,45,17,59),(12,46,18,60),(13,47,19,61),(14,48,20,62),(15,33,21,63),(16,34,22,64)], [(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)], [(1,23),(2,22),(3,21),(4,20),(5,19),(6,18),(7,17),(8,32),(9,31),(10,30),(11,29),(12,28),(13,27),(14,26),(15,25),(16,24),(33,51),(34,50),(35,49),(36,64),(37,63),(38,62),(39,61),(40,60),(41,59),(42,58),(43,57),(44,56),(45,55),(46,54),(47,53),(48,52)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 8A | ··· | 8H | 16A | ··· | 16P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 16 | ··· | 16 |
| size | 1 | 1 | 1 | 1 | 8 | 8 | 8 | 8 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 2 | ··· | 2 | 2 | ··· | 2 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | C4○D4 | D8 | D16 | C4○D8 | C4○D16 |
| kernel | C4×D16 | C4×C16 | C2.D16 | C16⋊3C4 | C4×D8 | C2×D16 | D16 | C42 | C2×C8 | C8 | C2×C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 8 | 1 | 1 | 2 | 4 | 8 | 4 | 8 |
Matrix representation of C4×D16 ►in GL3(𝔽17) generated by
| 13 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 16 | 0 | 0 |
| 0 | 13 | 11 |
| 0 | 6 | 13 |
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 16 |
G:=sub<GL(3,GF(17))| [13,0,0,0,1,0,0,0,1],[16,0,0,0,13,6,0,11,13],[1,0,0,0,1,0,0,0,16] >;
C4×D16 in GAP, Magma, Sage, TeX
C_4\times D_{16} % in TeX
G:=Group("C4xD16"); // GroupNames label
G:=SmallGroup(128,904);
// by ID
G=gap.SmallGroup(128,904);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,100,1123,570,360,4037,2028,124]);
// Polycyclic
G:=Group<a,b,c|a^4=b^16=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations