metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C5:2C16, C8.2D5, C10.2C8, C40.2C2, C20.5C4, C4.2Dic5, C2.(C5:2C8), SmallGroup(80,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — C5:2C16 |
Generators and relations for C5:2C16
G = < a,b | a5=b16=1, bab-1=a-1 >
Quotients: C1, C2, C4, C8, D5, C16, Dic5, C5:2C8, C5:2C16
Smallest permutation representation of C5:2C16
►Regular action on 80 points
Generators in S80
(1 18 78 47 61)(2 62 48 79 19)(3 20 80 33 63)(4 64 34 65 21)(5 22 66 35 49)(6 50 36 67 23)(7 24 68 37 51)(8 52 38 69 25)(9 26 70 39 53)(10 54 40 71 27)(11 28 72 41 55)(12 56 42 73 29)(13 30 74 43 57)(14 58 44 75 31)(15 32 76 45 59)(16 60 46 77 17)
(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)(65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80)
G:=sub<Sym(80)| (1,18,78,47,61)(2,62,48,79,19)(3,20,80,33,63)(4,64,34,65,21)(5,22,66,35,49)(6,50,36,67,23)(7,24,68,37,51)(8,52,38,69,25)(9,26,70,39,53)(10,54,40,71,27)(11,28,72,41,55)(12,56,42,73,29)(13,30,74,43,57)(14,58,44,75,31)(15,32,76,45,59)(16,60,46,77,17), (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)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)>;
G:=Group( (1,18,78,47,61)(2,62,48,79,19)(3,20,80,33,63)(4,64,34,65,21)(5,22,66,35,49)(6,50,36,67,23)(7,24,68,37,51)(8,52,38,69,25)(9,26,70,39,53)(10,54,40,71,27)(11,28,72,41,55)(12,56,42,73,29)(13,30,74,43,57)(14,58,44,75,31)(15,32,76,45,59)(16,60,46,77,17), (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)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80) );
G=PermutationGroup([[(1,18,78,47,61),(2,62,48,79,19),(3,20,80,33,63),(4,64,34,65,21),(5,22,66,35,49),(6,50,36,67,23),(7,24,68,37,51),(8,52,38,69,25),(9,26,70,39,53),(10,54,40,71,27),(11,28,72,41,55),(12,56,42,73,29),(13,30,74,43,57),(14,58,44,75,31),(15,32,76,45,59),(16,60,46,77,17)], [(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),(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)]])
C5:2C16 is a maximal subgroup of
C5:C32 D5xC16 C80:C2 C20.4C8 C5:D16 D8.D5 C5:SD32 C5:Q32 C15:3C16 C25:2C16 C52:7C16 C52:3C16
C5:2C16 is a maximal quotient of
C5:2C32 C15:3C16 C25:2C16 C52:7C16 C52:3C16
32 conjugacy classes
| class | 1 | 2 | 4A | 4B | 5A | 5B | 8A | 8B | 8C | 8D | 10A | 10B | 16A | ··· | 16H | 20A | 20B | 20C | 20D | 40A | ··· | 40H |
| order | 1 | 2 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | 10 | 16 | ··· | 16 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 5 | ··· | 5 | 2 | 2 | 2 | 2 | 2 | ··· | 2 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | |||||
| image | C1 | C2 | C4 | C8 | C16 | D5 | Dic5 | C5:2C8 | C5:2C16 |
| kernel | C5:2C16 | C40 | C20 | C10 | C5 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 8 | 2 | 2 | 4 | 8 |
Matrix representation of C5:2C16 ►in GL2(F41) generated by
| 40 | 18 |
| 39 | 35 |
| 29 | 23 |
| 15 | 12 |
G:=sub<GL(2,GF(41))| [40,39,18,35],[29,15,23,12] >;
C5:2C16 in GAP, Magma, Sage, TeX
C_5\rtimes_2C_{16} % in TeX
G:=Group("C5:2C16"); // GroupNames label
G:=SmallGroup(80,1);
// by ID
G=gap.SmallGroup(80,1);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-5,10,26,42,1604]);
// Polycyclic
G:=Group<a,b|a^5=b^16=1,b*a*b^-1=a^-1>;
// generators/relations
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