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G = C142order 196 = 22·72

Abelian group of type [14,14]

direct product, abelian, monomial

Aliases: C142, SmallGroup(196,12)

Series: Derived Chief Lower central Upper central

C1 — C142
C1C7C72C7×C14 — C142
C1 — C142
C1 — C142

Generators and relations for C142
 G = < a,b | a14=b14=1, ab=ba >


Smallest permutation representation of C142
Regular action on 196 points
Generators in S196
(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)(141 142 143 144 145 146 147 148 149 150 151 152 153 154)(155 156 157 158 159 160 161 162 163 164 165 166 167 168)(169 170 171 172 173 174 175 176 177 178 179 180 181 182)(183 184 185 186 187 188 189 190 191 192 193 194 195 196)
(1 108 121 33 73 44 162 17 196 130 172 91 66 152)(2 109 122 34 74 45 163 18 183 131 173 92 67 153)(3 110 123 35 75 46 164 19 184 132 174 93 68 154)(4 111 124 36 76 47 165 20 185 133 175 94 69 141)(5 112 125 37 77 48 166 21 186 134 176 95 70 142)(6 99 126 38 78 49 167 22 187 135 177 96 57 143)(7 100 113 39 79 50 168 23 188 136 178 97 58 144)(8 101 114 40 80 51 155 24 189 137 179 98 59 145)(9 102 115 41 81 52 156 25 190 138 180 85 60 146)(10 103 116 42 82 53 157 26 191 139 181 86 61 147)(11 104 117 29 83 54 158 27 192 140 182 87 62 148)(12 105 118 30 84 55 159 28 193 127 169 88 63 149)(13 106 119 31 71 56 160 15 194 128 170 89 64 150)(14 107 120 32 72 43 161 16 195 129 171 90 65 151)

G:=sub<Sym(196)| (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)(141,142,143,144,145,146,147,148,149,150,151,152,153,154)(155,156,157,158,159,160,161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176,177,178,179,180,181,182)(183,184,185,186,187,188,189,190,191,192,193,194,195,196), (1,108,121,33,73,44,162,17,196,130,172,91,66,152)(2,109,122,34,74,45,163,18,183,131,173,92,67,153)(3,110,123,35,75,46,164,19,184,132,174,93,68,154)(4,111,124,36,76,47,165,20,185,133,175,94,69,141)(5,112,125,37,77,48,166,21,186,134,176,95,70,142)(6,99,126,38,78,49,167,22,187,135,177,96,57,143)(7,100,113,39,79,50,168,23,188,136,178,97,58,144)(8,101,114,40,80,51,155,24,189,137,179,98,59,145)(9,102,115,41,81,52,156,25,190,138,180,85,60,146)(10,103,116,42,82,53,157,26,191,139,181,86,61,147)(11,104,117,29,83,54,158,27,192,140,182,87,62,148)(12,105,118,30,84,55,159,28,193,127,169,88,63,149)(13,106,119,31,71,56,160,15,194,128,170,89,64,150)(14,107,120,32,72,43,161,16,195,129,171,90,65,151)>;

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)(141,142,143,144,145,146,147,148,149,150,151,152,153,154)(155,156,157,158,159,160,161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176,177,178,179,180,181,182)(183,184,185,186,187,188,189,190,191,192,193,194,195,196), (1,108,121,33,73,44,162,17,196,130,172,91,66,152)(2,109,122,34,74,45,163,18,183,131,173,92,67,153)(3,110,123,35,75,46,164,19,184,132,174,93,68,154)(4,111,124,36,76,47,165,20,185,133,175,94,69,141)(5,112,125,37,77,48,166,21,186,134,176,95,70,142)(6,99,126,38,78,49,167,22,187,135,177,96,57,143)(7,100,113,39,79,50,168,23,188,136,178,97,58,144)(8,101,114,40,80,51,155,24,189,137,179,98,59,145)(9,102,115,41,81,52,156,25,190,138,180,85,60,146)(10,103,116,42,82,53,157,26,191,139,181,86,61,147)(11,104,117,29,83,54,158,27,192,140,182,87,62,148)(12,105,118,30,84,55,159,28,193,127,169,88,63,149)(13,106,119,31,71,56,160,15,194,128,170,89,64,150)(14,107,120,32,72,43,161,16,195,129,171,90,65,151) );

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),(141,142,143,144,145,146,147,148,149,150,151,152,153,154),(155,156,157,158,159,160,161,162,163,164,165,166,167,168),(169,170,171,172,173,174,175,176,177,178,179,180,181,182),(183,184,185,186,187,188,189,190,191,192,193,194,195,196)], [(1,108,121,33,73,44,162,17,196,130,172,91,66,152),(2,109,122,34,74,45,163,18,183,131,173,92,67,153),(3,110,123,35,75,46,164,19,184,132,174,93,68,154),(4,111,124,36,76,47,165,20,185,133,175,94,69,141),(5,112,125,37,77,48,166,21,186,134,176,95,70,142),(6,99,126,38,78,49,167,22,187,135,177,96,57,143),(7,100,113,39,79,50,168,23,188,136,178,97,58,144),(8,101,114,40,80,51,155,24,189,137,179,98,59,145),(9,102,115,41,81,52,156,25,190,138,180,85,60,146),(10,103,116,42,82,53,157,26,191,139,181,86,61,147),(11,104,117,29,83,54,158,27,192,140,182,87,62,148),(12,105,118,30,84,55,159,28,193,127,169,88,63,149),(13,106,119,31,71,56,160,15,194,128,170,89,64,150),(14,107,120,32,72,43,161,16,195,129,171,90,65,151)])

C142 is a maximal subgroup of   C727D4

196 conjugacy classes

class 1 2A2B2C7A···7AV14A···14EN
order12227···714···14
size11111···11···1

196 irreducible representations

dim1111
type++
imageC1C2C7C14
kernelC142C7×C14C2×C14C14
# reps1348144

Matrix representation of C142 in GL2(𝔽29) generated by

250
05
,
90
01
G:=sub<GL(2,GF(29))| [25,0,0,5],[9,0,0,1] >;

C142 in GAP, Magma, Sage, TeX

C_{14}^2
% in TeX

G:=Group("C14^2");
// GroupNames label

G:=SmallGroup(196,12);
// by ID

G=gap.SmallGroup(196,12);
# by ID

G:=PCGroup([4,-2,-2,-7,-7]);
// Polycyclic

G:=Group<a,b|a^14=b^14=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C142 in TeX

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