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G = C112order 121 = 112

Elementary abelian group of type [11,11]

direct product, p-group, elementary abelian, monomial

Aliases: C112, SmallGroup(121,2)

Series: Derived Chief Lower central Upper central Jennings

C1 — C112
C1C11 — C112
C1 — C112
C1 — C112
C1 — C112

Generators and relations for C112
 G = < a,b | a11=b11=1, ab=ba >


Smallest permutation representation of C112
Regular action on 121 points
Generators in S121
(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)(100 101 102 103 104 105 106 107 108 109 110)(111 112 113 114 115 116 117 118 119 120 121)
(1 121 21 85 109 65 32 76 98 54 43)(2 111 22 86 110 66 33 77 99 55 44)(3 112 12 87 100 56 23 67 89 45 34)(4 113 13 88 101 57 24 68 90 46 35)(5 114 14 78 102 58 25 69 91 47 36)(6 115 15 79 103 59 26 70 92 48 37)(7 116 16 80 104 60 27 71 93 49 38)(8 117 17 81 105 61 28 72 94 50 39)(9 118 18 82 106 62 29 73 95 51 40)(10 119 19 83 107 63 30 74 96 52 41)(11 120 20 84 108 64 31 75 97 53 42)

G:=sub<Sym(121)| (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)(100,101,102,103,104,105,106,107,108,109,110)(111,112,113,114,115,116,117,118,119,120,121), (1,121,21,85,109,65,32,76,98,54,43)(2,111,22,86,110,66,33,77,99,55,44)(3,112,12,87,100,56,23,67,89,45,34)(4,113,13,88,101,57,24,68,90,46,35)(5,114,14,78,102,58,25,69,91,47,36)(6,115,15,79,103,59,26,70,92,48,37)(7,116,16,80,104,60,27,71,93,49,38)(8,117,17,81,105,61,28,72,94,50,39)(9,118,18,82,106,62,29,73,95,51,40)(10,119,19,83,107,63,30,74,96,52,41)(11,120,20,84,108,64,31,75,97,53,42)>;

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)(100,101,102,103,104,105,106,107,108,109,110)(111,112,113,114,115,116,117,118,119,120,121), (1,121,21,85,109,65,32,76,98,54,43)(2,111,22,86,110,66,33,77,99,55,44)(3,112,12,87,100,56,23,67,89,45,34)(4,113,13,88,101,57,24,68,90,46,35)(5,114,14,78,102,58,25,69,91,47,36)(6,115,15,79,103,59,26,70,92,48,37)(7,116,16,80,104,60,27,71,93,49,38)(8,117,17,81,105,61,28,72,94,50,39)(9,118,18,82,106,62,29,73,95,51,40)(10,119,19,83,107,63,30,74,96,52,41)(11,120,20,84,108,64,31,75,97,53,42) );

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),(100,101,102,103,104,105,106,107,108,109,110),(111,112,113,114,115,116,117,118,119,120,121)], [(1,121,21,85,109,65,32,76,98,54,43),(2,111,22,86,110,66,33,77,99,55,44),(3,112,12,87,100,56,23,67,89,45,34),(4,113,13,88,101,57,24,68,90,46,35),(5,114,14,78,102,58,25,69,91,47,36),(6,115,15,79,103,59,26,70,92,48,37),(7,116,16,80,104,60,27,71,93,49,38),(8,117,17,81,105,61,28,72,94,50,39),(9,118,18,82,106,62,29,73,95,51,40),(10,119,19,83,107,63,30,74,96,52,41),(11,120,20,84,108,64,31,75,97,53,42)])

C112 is a maximal subgroup of   C11⋊D11  C112⋊C3

121 conjugacy classes

class 1 11A···11DP
order111···11
size11···1

121 irreducible representations

dim11
type+
imageC1C11
kernelC112C11
# reps1120

Matrix representation of C112 in GL2(𝔽23) generated by

130
01
,
80
018
G:=sub<GL(2,GF(23))| [13,0,0,1],[8,0,0,18] >;

C112 in GAP, Magma, Sage, TeX

C_{11}^2
% in TeX

G:=Group("C11^2");
// GroupNames label

G:=SmallGroup(121,2);
// by ID

G=gap.SmallGroup(121,2);
# by ID

G:=PCGroup([2,-11,11]:ExponentLimit:=1);
// Polycyclic

G:=Group<a,b|a^11=b^11=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C112 in TeX

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