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G = C7×C28order 196 = 22·72

Abelian group of type [7,28]

direct product, abelian, monomial, 7-elementary

Aliases: C7×C28, SmallGroup(196,7)

Series: Derived Chief Lower central Upper central

C1 — C7×C28
C1C2C14C7×C14 — C7×C28
C1 — C7×C28
C1 — C7×C28

Generators and relations for C7×C28
 G = < a,b | a7=b28=1, ab=ba >


Smallest permutation representation of C7×C28
Regular action on 196 points
Generators in S196
(1 37 165 88 65 181 132)(2 38 166 89 66 182 133)(3 39 167 90 67 183 134)(4 40 168 91 68 184 135)(5 41 141 92 69 185 136)(6 42 142 93 70 186 137)(7 43 143 94 71 187 138)(8 44 144 95 72 188 139)(9 45 145 96 73 189 140)(10 46 146 97 74 190 113)(11 47 147 98 75 191 114)(12 48 148 99 76 192 115)(13 49 149 100 77 193 116)(14 50 150 101 78 194 117)(15 51 151 102 79 195 118)(16 52 152 103 80 196 119)(17 53 153 104 81 169 120)(18 54 154 105 82 170 121)(19 55 155 106 83 171 122)(20 56 156 107 84 172 123)(21 29 157 108 57 173 124)(22 30 158 109 58 174 125)(23 31 159 110 59 175 126)(24 32 160 111 60 176 127)(25 33 161 112 61 177 128)(26 34 162 85 62 178 129)(27 35 163 86 63 179 130)(28 36 164 87 64 180 131)
(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)

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

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

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

C7×C28 is a maximal subgroup of   C724C8  C724Q8  C28⋊D7

196 conjugacy classes

class 1  2 4A4B7A···7AV14A···14AV28A···28CR
order12447···714···1428···28
size11111···11···11···1

196 irreducible representations

dim111111
type++
imageC1C2C4C7C14C28
kernelC7×C28C7×C14C72C28C14C7
# reps112484896

Matrix representation of C7×C28 in GL2(𝔽29) generated by

250
024
,
120
021
G:=sub<GL(2,GF(29))| [25,0,0,24],[12,0,0,21] >;

C7×C28 in GAP, Magma, Sage, TeX

C_7\times C_{28}
% in TeX

G:=Group("C7xC28");
// GroupNames label

G:=SmallGroup(196,7);
// by ID

G=gap.SmallGroup(196,7);
# by ID

G:=PCGroup([4,-2,-7,-7,-2,392]);
// Polycyclic

G:=Group<a,b|a^7=b^28=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C7×C28 in TeX

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