direct product, abelian, monomial
Aliases: C102, SmallGroup(100,16)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C102 |
| C1 — C102 |
| C1 — C102 |
Generators and relations for C102
G = < a,b | a10=b10=1, ab=ba >
Smallest permutation representation of C102
►Regular action on 100 points
Generators in S100
(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)(81 82 83 84 85 86 87 88 89 90)(91 92 93 94 95 96 97 98 99 100)
(1 97 17 28 38 89 45 64 76 53)(2 98 18 29 39 90 46 65 77 54)(3 99 19 30 40 81 47 66 78 55)(4 100 20 21 31 82 48 67 79 56)(5 91 11 22 32 83 49 68 80 57)(6 92 12 23 33 84 50 69 71 58)(7 93 13 24 34 85 41 70 72 59)(8 94 14 25 35 86 42 61 73 60)(9 95 15 26 36 87 43 62 74 51)(10 96 16 27 37 88 44 63 75 52)
G:=sub<Sym(100)| (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)(81,82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99,100), (1,97,17,28,38,89,45,64,76,53)(2,98,18,29,39,90,46,65,77,54)(3,99,19,30,40,81,47,66,78,55)(4,100,20,21,31,82,48,67,79,56)(5,91,11,22,32,83,49,68,80,57)(6,92,12,23,33,84,50,69,71,58)(7,93,13,24,34,85,41,70,72,59)(8,94,14,25,35,86,42,61,73,60)(9,95,15,26,36,87,43,62,74,51)(10,96,16,27,37,88,44,63,75,52)>;
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)(61,62,63,64,65,66,67,68,69,70)(71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99,100), (1,97,17,28,38,89,45,64,76,53)(2,98,18,29,39,90,46,65,77,54)(3,99,19,30,40,81,47,66,78,55)(4,100,20,21,31,82,48,67,79,56)(5,91,11,22,32,83,49,68,80,57)(6,92,12,23,33,84,50,69,71,58)(7,93,13,24,34,85,41,70,72,59)(8,94,14,25,35,86,42,61,73,60)(9,95,15,26,36,87,43,62,74,51)(10,96,16,27,37,88,44,63,75,52) );
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),(61,62,63,64,65,66,67,68,69,70),(71,72,73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88,89,90),(91,92,93,94,95,96,97,98,99,100)], [(1,97,17,28,38,89,45,64,76,53),(2,98,18,29,39,90,46,65,77,54),(3,99,19,30,40,81,47,66,78,55),(4,100,20,21,31,82,48,67,79,56),(5,91,11,22,32,83,49,68,80,57),(6,92,12,23,33,84,50,69,71,58),(7,93,13,24,34,85,41,70,72,59),(8,94,14,25,35,86,42,61,73,60),(9,95,15,26,36,87,43,62,74,51),(10,96,16,27,37,88,44,63,75,52)]])
C102 is a maximal subgroup of
C52⋊7D4 C52⋊A4
100 conjugacy classes
| class | 1 | 2A | 2B | 2C | 5A | ··· | 5X | 10A | ··· | 10BT |
| order | 1 | 2 | 2 | 2 | 5 | ··· | 5 | 10 | ··· | 10 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
100 irreducible representations
| dim | 1 | 1 | 1 | 1 |
| type | + | + | ||
| image | C1 | C2 | C5 | C10 |
| kernel | C102 | C5×C10 | C2×C10 | C10 |
| # reps | 1 | 3 | 24 | 72 |
Matrix representation of C102 ►in GL2(𝔽11) generated by
| 3 | 0 |
| 0 | 10 |
| 8 | 0 |
| 0 | 5 |
G:=sub<GL(2,GF(11))| [3,0,0,10],[8,0,0,5] >;
C102 in GAP, Magma, Sage, TeX
C_{10}^2 % in TeX
G:=Group("C10^2"); // GroupNames label
G:=SmallGroup(100,16);
// by ID
G=gap.SmallGroup(100,16);
# by ID
G:=PCGroup([4,-2,-2,-5,-5]);
// Polycyclic
G:=Group<a,b|a^10=b^10=1,a*b=b*a>;
// generators/relations
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