Copied to
clipboard

G = C122  order 144 = 24·32

Abelian group of type [12,12]

direct product, abelian, monomial

Aliases: C122, SmallGroup(144,101)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C122
Chief series
C1 — C2 — C22 — C2×C6 — C62 — C6×C12 — C122
Lower central
C1 — C122
Upper central
C1 — C122

Generators and relations for C122
 G = < a,b | a12=b12=1, ab=ba >

Subgroups: 90, all normal (6 characteristic)
C1, C2, C3, C4, C22, C6, C2×C4, C32, C12, C2×C6, C42, C3×C6, C2×C12, C3×C12, C62, C4×C12, C6×C12, C122
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C32, C12, C2×C6, C42, C3×C6, C2×C12, C3×C12, C62, C4×C12, C6×C12, C122

Smallest permutation representation of C122
►Regular action on 144 points
Generators in S144
(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)(121 122 123 124 125 126 127 128 129 130 131 132)(133 134 135 136 137 138 139 140 141 142 143 144)

(1 144 109 65 52 94 36 84 126 48 98 15)(2 133 110 66 53 95 25 73 127 37 99 16)(3 134 111 67 54 96 26 74 128 38 100 17)(4 135 112 68 55 85 27 75 129 39 101 18)(5 136 113 69 56 86 28 76 130 40 102 19)(6 137 114 70 57 87 29 77 131 41 103 20)(7 138 115 71 58 88 30 78 132 42 104 21)(8 139 116 72 59 89 31 79 121 43 105 22)(9 140 117 61 60 90 32 80 122 44 106 23)(10 141 118 62 49 91 33 81 123 45 107 24)(11 142 119 63 50 92 34 82 124 46 108 13)(12 143 120 64 51 93 35 83 125 47 97 14)
 
G:=sub<Sym(144)| (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)(121,122,123,124,125,126,127,128,129,130,131,132)(133,134,135,136,137,138,139,140,141,142,143,144), (1,144,109,65,52,94,36,84,126,48,98,15)(2,133,110,66,53,95,25,73,127,37,99,16)(3,134,111,67,54,96,26,74,128,38,100,17)(4,135,112,68,55,85,27,75,129,39,101,18)(5,136,113,69,56,86,28,76,130,40,102,19)(6,137,114,70,57,87,29,77,131,41,103,20)(7,138,115,71,58,88,30,78,132,42,104,21)(8,139,116,72,59,89,31,79,121,43,105,22)(9,140,117,61,60,90,32,80,122,44,106,23)(10,141,118,62,49,91,33,81,123,45,107,24)(11,142,119,63,50,92,34,82,124,46,108,13)(12,143,120,64,51,93,35,83,125,47,97,14)>;
 
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)(121,122,123,124,125,126,127,128,129,130,131,132)(133,134,135,136,137,138,139,140,141,142,143,144), (1,144,109,65,52,94,36,84,126,48,98,15)(2,133,110,66,53,95,25,73,127,37,99,16)(3,134,111,67,54,96,26,74,128,38,100,17)(4,135,112,68,55,85,27,75,129,39,101,18)(5,136,113,69,56,86,28,76,130,40,102,19)(6,137,114,70,57,87,29,77,131,41,103,20)(7,138,115,71,58,88,30,78,132,42,104,21)(8,139,116,72,59,89,31,79,121,43,105,22)(9,140,117,61,60,90,32,80,122,44,106,23)(10,141,118,62,49,91,33,81,123,45,107,24)(11,142,119,63,50,92,34,82,124,46,108,13)(12,143,120,64,51,93,35,83,125,47,97,14) );
 
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),(121,122,123,124,125,126,127,128,129,130,131,132),(133,134,135,136,137,138,139,140,141,142,143,144)], [(1,144,109,65,52,94,36,84,126,48,98,15),(2,133,110,66,53,95,25,73,127,37,99,16),(3,134,111,67,54,96,26,74,128,38,100,17),(4,135,112,68,55,85,27,75,129,39,101,18),(5,136,113,69,56,86,28,76,130,40,102,19),(6,137,114,70,57,87,29,77,131,41,103,20),(7,138,115,71,58,88,30,78,132,42,104,21),(8,139,116,72,59,89,31,79,121,43,105,22),(9,140,117,61,60,90,32,80,122,44,106,23),(10,141,118,62,49,91,33,81,123,45,107,24),(11,142,119,63,50,92,34,82,124,46,108,13),(12,143,120,64,51,93,35,83,125,47,97,14)]])
 

C122 is a maximal subgroup of
 C122.C2  C12.57D12  C122⋊C2  C12⋊6Dic6  C12.25Dic6  C122⋊16C2  C12⋊4D12  C122⋊6C2  C122⋊2C2  C122.C3  C42⋊He3

144 conjugacy classes

class 1 2A2B2C3A···3H4A···4L6A···6X12A···12CR
order12223···34···46···612···12
size11111···11···11···11···1

144 irreducible representations

dim111111
type++
imageC1C2C3C4C6C12
kernelC122C6×C12C4×C12C3×C12C2×C12C12
# reps138122496

Matrix representation of C122 ►in GL2(𝔽13) generated by

40
06
,
70
011
G:=sub<GL(2,GF(13))| [4,0,0,6],[7,0,0,11] >;
 

C122 in GAP, Magma, Sage, TeX 

C_{12}^2
 
% in TeX
 
G:=Group("C12^2");
 
// GroupNames label
 
G:=SmallGroup(144,101);
 
// by ID
 
G=gap.SmallGroup(144,101);
 
# by ID
 
G:=PCGroup([6,-2,-2,-3,-3,-2,-2,216,439]);
 
// Polycyclic
 
G:=Group<a,b|a^12=b^12=1,a*b=b*a>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁