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G = C102order 100 = 22·52

Abelian group of type [10,10]

direct product, abelian, monomial

Aliases: C102, SmallGroup(100,16)

Series: Derived Chief Lower central Upper central

C1 — C102
C1C5C52C5×C10 — 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 32 57 97 11 78 42 64 85 28)(2 33 58 98 12 79 43 65 86 29)(3 34 59 99 13 80 44 66 87 30)(4 35 60 100 14 71 45 67 88 21)(5 36 51 91 15 72 46 68 89 22)(6 37 52 92 16 73 47 69 90 23)(7 38 53 93 17 74 48 70 81 24)(8 39 54 94 18 75 49 61 82 25)(9 40 55 95 19 76 50 62 83 26)(10 31 56 96 20 77 41 63 84 27)

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), (1,32,57,97,11,78,42,64,85,28)(2,33,58,98,12,79,43,65,86,29)(3,34,59,99,13,80,44,66,87,30)(4,35,60,100,14,71,45,67,88,21)(5,36,51,91,15,72,46,68,89,22)(6,37,52,92,16,73,47,69,90,23)(7,38,53,93,17,74,48,70,81,24)(8,39,54,94,18,75,49,61,82,25)(9,40,55,95,19,76,50,62,83,26)(10,31,56,96,20,77,41,63,84,27)>;

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), (1,32,57,97,11,78,42,64,85,28)(2,33,58,98,12,79,43,65,86,29)(3,34,59,99,13,80,44,66,87,30)(4,35,60,100,14,71,45,67,88,21)(5,36,51,91,15,72,46,68,89,22)(6,37,52,92,16,73,47,69,90,23)(7,38,53,93,17,74,48,70,81,24)(8,39,54,94,18,75,49,61,82,25)(9,40,55,95,19,76,50,62,83,26)(10,31,56,96,20,77,41,63,84,27) );

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)], [(1,32,57,97,11,78,42,64,85,28),(2,33,58,98,12,79,43,65,86,29),(3,34,59,99,13,80,44,66,87,30),(4,35,60,100,14,71,45,67,88,21),(5,36,51,91,15,72,46,68,89,22),(6,37,52,92,16,73,47,69,90,23),(7,38,53,93,17,74,48,70,81,24),(8,39,54,94,18,75,49,61,82,25),(9,40,55,95,19,76,50,62,83,26),(10,31,56,96,20,77,41,63,84,27)])

C102 is a maximal subgroup of   C527D4  C52⋊A4

100 conjugacy classes

class 1 2A2B2C5A···5X10A···10BT
order12225···510···10
size11111···11···1

100 irreducible representations

dim1111
type++
imageC1C2C5C10
kernelC102C5×C10C2×C10C10
# reps132472

Matrix representation of C102 in GL2(𝔽11) generated by

30
010
,
80
05
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

Export

Subgroup lattice of C102 in TeX

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