metabelian, supersoluble, monomial, A-group
Aliases: C52⋊3C16, C20.13F5, C20.2Dic5, C5⋊3(C5⋊C16), C5⋊(C5⋊2C16), C10.5(C5⋊C8), C10.(C5⋊2C8), (C5×C20).6C4, (C5×C10).3C8, C5⋊2C8.2D5, C2.(C52⋊3C8), C4.2(D5.D5), (C5×C5⋊2C8).3C2, SmallGroup(400,57)
Series: Derived ►Chief ►Lower central ►Upper central
| C52 — C52⋊3C16 |
Generators and relations for C52⋊3C16
G = < a,b,c | a5=b5=c16=1, ab=ba, cac-1=a-1, cbc-1=b2 >
Smallest permutation representation of C52⋊3C16
►On 80 points
Generators in S80
(1 80 29 44 58)(2 59 45 30 65)(3 66 31 46 60)(4 61 47 32 67)(5 68 17 48 62)(6 63 33 18 69)(7 70 19 34 64)(8 49 35 20 71)(9 72 21 36 50)(10 51 37 22 73)(11 74 23 38 52)(12 53 39 24 75)(13 76 25 40 54)(14 55 41 26 77)(15 78 27 42 56)(16 57 43 28 79)
(1 58 44 29 80)(2 45 65 59 30)(3 66 31 46 60)(4 32 61 67 47)(5 62 48 17 68)(6 33 69 63 18)(7 70 19 34 64)(8 20 49 71 35)(9 50 36 21 72)(10 37 73 51 22)(11 74 23 38 52)(12 24 53 75 39)(13 54 40 25 76)(14 41 77 55 26)(15 78 27 42 56)(16 28 57 79 43)
(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)| (1,80,29,44,58)(2,59,45,30,65)(3,66,31,46,60)(4,61,47,32,67)(5,68,17,48,62)(6,63,33,18,69)(7,70,19,34,64)(8,49,35,20,71)(9,72,21,36,50)(10,51,37,22,73)(11,74,23,38,52)(12,53,39,24,75)(13,76,25,40,54)(14,55,41,26,77)(15,78,27,42,56)(16,57,43,28,79), (1,58,44,29,80)(2,45,65,59,30)(3,66,31,46,60)(4,32,61,67,47)(5,62,48,17,68)(6,33,69,63,18)(7,70,19,34,64)(8,20,49,71,35)(9,50,36,21,72)(10,37,73,51,22)(11,74,23,38,52)(12,24,53,75,39)(13,54,40,25,76)(14,41,77,55,26)(15,78,27,42,56)(16,28,57,79,43), (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( (1,80,29,44,58)(2,59,45,30,65)(3,66,31,46,60)(4,61,47,32,67)(5,68,17,48,62)(6,63,33,18,69)(7,70,19,34,64)(8,49,35,20,71)(9,72,21,36,50)(10,51,37,22,73)(11,74,23,38,52)(12,53,39,24,75)(13,76,25,40,54)(14,55,41,26,77)(15,78,27,42,56)(16,57,43,28,79), (1,58,44,29,80)(2,45,65,59,30)(3,66,31,46,60)(4,32,61,67,47)(5,62,48,17,68)(6,33,69,63,18)(7,70,19,34,64)(8,20,49,71,35)(9,50,36,21,72)(10,37,73,51,22)(11,74,23,38,52)(12,24,53,75,39)(13,54,40,25,76)(14,41,77,55,26)(15,78,27,42,56)(16,28,57,79,43), (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([[(1,80,29,44,58),(2,59,45,30,65),(3,66,31,46,60),(4,61,47,32,67),(5,68,17,48,62),(6,63,33,18,69),(7,70,19,34,64),(8,49,35,20,71),(9,72,21,36,50),(10,51,37,22,73),(11,74,23,38,52),(12,53,39,24,75),(13,76,25,40,54),(14,55,41,26,77),(15,78,27,42,56),(16,57,43,28,79)], [(1,58,44,29,80),(2,45,65,59,30),(3,66,31,46,60),(4,32,61,67,47),(5,62,48,17,68),(6,33,69,63,18),(7,70,19,34,64),(8,20,49,71,35),(9,50,36,21,72),(10,37,73,51,22),(11,74,23,38,52),(12,24,53,75,39),(13,54,40,25,76),(14,41,77,55,26),(15,78,27,42,56),(16,28,57,79,43)], [(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)]])
52 conjugacy classes
| class | 1 | 2 | 4A | 4B | 5A | 5B | 5C | ··· | 5G | 8A | 8B | 8C | 8D | 10A | 10B | 10C | ··· | 10G | 16A | ··· | 16H | 20A | 20B | 20C | 20D | 20E | ··· | 20N | 40A | ··· | 40H |
| order | 1 | 2 | 4 | 4 | 5 | 5 | 5 | ··· | 5 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | ··· | 10 | 16 | ··· | 16 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 2 | 2 | 4 | ··· | 4 | 25 | ··· | 25 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 10 | ··· | 10 |
52 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | - | + | - | |||||||||
| image | C1 | C2 | C4 | C8 | C16 | D5 | Dic5 | C5⋊2C8 | C5⋊2C16 | F5 | C5⋊C8 | C5⋊C16 | D5.D5 | C52⋊3C8 | C52⋊3C16 |
| kernel | C52⋊3C16 | C5×C5⋊2C8 | C5×C20 | C5×C10 | C52 | C5⋊2C8 | C20 | C10 | C5 | C20 | C10 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 8 | 2 | 2 | 4 | 8 | 1 | 1 | 2 | 4 | 4 | 8 |
Matrix representation of C52⋊3C16 ►in GL4(𝔽241) generated by
| 91 | 0 | 0 | 0 |
| 0 | 91 | 0 | 0 |
| 104 | 0 | 98 | 0 |
| 137 | 0 | 0 | 98 |
| 98 | 0 | 0 | 0 |
| 92 | 91 | 0 | 0 |
| 112 | 0 | 87 | 0 |
| 235 | 0 | 0 | 205 |
| 64 | 0 | 97 | 0 |
| 0 | 0 | 1 | 1 |
| 0 | 1 | 177 | 0 |
| 105 | 0 | 64 | 0 |
G:=sub<GL(4,GF(241))| [91,0,104,137,0,91,0,0,0,0,98,0,0,0,0,98],[98,92,112,235,0,91,0,0,0,0,87,0,0,0,0,205],[64,0,0,105,0,0,1,0,97,1,177,64,0,1,0,0] >;
C52⋊3C16 in GAP, Magma, Sage, TeX
C_5^2\rtimes_3C_{16} % in TeX
G:=Group("C5^2:3C16"); // GroupNames label
G:=SmallGroup(400,57);
// by ID
G=gap.SmallGroup(400,57);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-5,-5,12,31,50,1924,8645,5771]);
// Polycyclic
G:=Group<a,b,c|a^5=b^5=c^16=1,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^2>;
// generators/relations
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