direct product, p-group, abelian, monomial
Aliases: C4×C16, SmallGroup(64,26)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
| C1 — C4×C16 |
| C1 — C4×C16 |
| C1 — C4×C16 |
Generators and relations for C4×C16
G = < a,b | a4=b16=1, ab=ba >
Smallest permutation representation of C4×C16
►Regular action on 64 points
Generators in S64
(1 38 23 63)(2 39 24 64)(3 40 25 49)(4 41 26 50)(5 42 27 51)(6 43 28 52)(7 44 29 53)(8 45 30 54)(9 46 31 55)(10 47 32 56)(11 48 17 57)(12 33 18 58)(13 34 19 59)(14 35 20 60)(15 36 21 61)(16 37 22 62)
(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)| (1,38,23,63)(2,39,24,64)(3,40,25,49)(4,41,26,50)(5,42,27,51)(6,43,28,52)(7,44,29,53)(8,45,30,54)(9,46,31,55)(10,47,32,56)(11,48,17,57)(12,33,18,58)(13,34,19,59)(14,35,20,60)(15,36,21,61)(16,37,22,62), (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( (1,38,23,63)(2,39,24,64)(3,40,25,49)(4,41,26,50)(5,42,27,51)(6,43,28,52)(7,44,29,53)(8,45,30,54)(9,46,31,55)(10,47,32,56)(11,48,17,57)(12,33,18,58)(13,34,19,59)(14,35,20,60)(15,36,21,61)(16,37,22,62), (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([[(1,38,23,63),(2,39,24,64),(3,40,25,49),(4,41,26,50),(5,42,27,51),(6,43,28,52),(7,44,29,53),(8,45,30,54),(9,46,31,55),(10,47,32,56),(11,48,17,57),(12,33,18,58),(13,34,19,59),(14,35,20,60),(15,36,21,61),(16,37,22,62)], [(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)]])
C4×C16 is a maximal subgroup of
C16⋊5C8 C8⋊C16 D4⋊C16 C4.16D16 Q16⋊1C8 Q8⋊C16 C16⋊3C8 C16⋊4C8 C16.3C8 C32⋊5C4 C4⋊C32 C8.C16 C16○2M5(2) C42.13C8 C8.12M4(2) C16⋊6D4 C16○D8 C8○D16 C16⋊4Q8 C4.4D16 C4.SD32 C8.22SD16 C4⋊D16 C4⋊Q32 C16⋊5D4 C8.21D8 C16⋊2Q8 C16.5Q8 C16⋊3Q8 Dic5⋊C16
C4×C16 is a maximal quotient of
C8⋊C16 C22.7M5(2) C32⋊5C4 Dic5⋊C16
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4L | 8A | ··· | 8P | 16A | ··· | 16AF |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 16 | ··· | 16 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | + | ||||||
| image | C1 | C2 | C2 | C4 | C4 | C4 | C8 | C8 | C16 |
| kernel | C4×C16 | C4×C8 | C2×C16 | C16 | C42 | C2×C8 | C8 | C2×C4 | C4 |
| # reps | 1 | 1 | 2 | 8 | 2 | 2 | 8 | 8 | 32 |
Matrix representation of C4×C16 ►in GL2(𝔽17) generated by
| 13 | 0 |
| 0 | 1 |
| 11 | 0 |
| 0 | 10 |
G:=sub<GL(2,GF(17))| [13,0,0,1],[11,0,0,10] >;
C4×C16 in GAP, Magma, Sage, TeX
C_4\times C_{16} % in TeX
G:=Group("C4xC16"); // GroupNames label
G:=SmallGroup(64,26);
// by ID
G=gap.SmallGroup(64,26);
# by ID
G:=PCGroup([6,-2,2,-2,2,-2,-2,24,55,86,88]);
// Polycyclic
G:=Group<a,b|a^4=b^16=1,a*b=b*a>;
// generators/relations
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