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G = C3×C75order 225 = 32·52

Abelian group of type [3,75]

direct product, abelian, monomial, 3-elementary

Aliases: C3×C75, SmallGroup(225,2)

Series: Derived Chief Lower central Upper central

C1 — C3×C75
C1C5C25C75 — C3×C75
C1 — C3×C75
C1 — C3×C75

Generators and relations for C3×C75
 G = < a,b | a3=b75=1, ab=ba >


Smallest permutation representation of C3×C75
Regular action on 225 points
Generators in S225
(1 185 107)(2 186 108)(3 187 109)(4 188 110)(5 189 111)(6 190 112)(7 191 113)(8 192 114)(9 193 115)(10 194 116)(11 195 117)(12 196 118)(13 197 119)(14 198 120)(15 199 121)(16 200 122)(17 201 123)(18 202 124)(19 203 125)(20 204 126)(21 205 127)(22 206 128)(23 207 129)(24 208 130)(25 209 131)(26 210 132)(27 211 133)(28 212 134)(29 213 135)(30 214 136)(31 215 137)(32 216 138)(33 217 139)(34 218 140)(35 219 141)(36 220 142)(37 221 143)(38 222 144)(39 223 145)(40 224 146)(41 225 147)(42 151 148)(43 152 149)(44 153 150)(45 154 76)(46 155 77)(47 156 78)(48 157 79)(49 158 80)(50 159 81)(51 160 82)(52 161 83)(53 162 84)(54 163 85)(55 164 86)(56 165 87)(57 166 88)(58 167 89)(59 168 90)(60 169 91)(61 170 92)(62 171 93)(63 172 94)(64 173 95)(65 174 96)(66 175 97)(67 176 98)(68 177 99)(69 178 100)(70 179 101)(71 180 102)(72 181 103)(73 182 104)(74 183 105)(75 184 106)
(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)(151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225)

G:=sub<Sym(225)| (1,185,107)(2,186,108)(3,187,109)(4,188,110)(5,189,111)(6,190,112)(7,191,113)(8,192,114)(9,193,115)(10,194,116)(11,195,117)(12,196,118)(13,197,119)(14,198,120)(15,199,121)(16,200,122)(17,201,123)(18,202,124)(19,203,125)(20,204,126)(21,205,127)(22,206,128)(23,207,129)(24,208,130)(25,209,131)(26,210,132)(27,211,133)(28,212,134)(29,213,135)(30,214,136)(31,215,137)(32,216,138)(33,217,139)(34,218,140)(35,219,141)(36,220,142)(37,221,143)(38,222,144)(39,223,145)(40,224,146)(41,225,147)(42,151,148)(43,152,149)(44,153,150)(45,154,76)(46,155,77)(47,156,78)(48,157,79)(49,158,80)(50,159,81)(51,160,82)(52,161,83)(53,162,84)(54,163,85)(55,164,86)(56,165,87)(57,166,88)(58,167,89)(59,168,90)(60,169,91)(61,170,92)(62,171,93)(63,172,94)(64,173,95)(65,174,96)(66,175,97)(67,176,98)(68,177,99)(69,178,100)(70,179,101)(71,180,102)(72,181,103)(73,182,104)(74,183,105)(75,184,106), (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)(151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225)>;

G:=Group( (1,185,107)(2,186,108)(3,187,109)(4,188,110)(5,189,111)(6,190,112)(7,191,113)(8,192,114)(9,193,115)(10,194,116)(11,195,117)(12,196,118)(13,197,119)(14,198,120)(15,199,121)(16,200,122)(17,201,123)(18,202,124)(19,203,125)(20,204,126)(21,205,127)(22,206,128)(23,207,129)(24,208,130)(25,209,131)(26,210,132)(27,211,133)(28,212,134)(29,213,135)(30,214,136)(31,215,137)(32,216,138)(33,217,139)(34,218,140)(35,219,141)(36,220,142)(37,221,143)(38,222,144)(39,223,145)(40,224,146)(41,225,147)(42,151,148)(43,152,149)(44,153,150)(45,154,76)(46,155,77)(47,156,78)(48,157,79)(49,158,80)(50,159,81)(51,160,82)(52,161,83)(53,162,84)(54,163,85)(55,164,86)(56,165,87)(57,166,88)(58,167,89)(59,168,90)(60,169,91)(61,170,92)(62,171,93)(63,172,94)(64,173,95)(65,174,96)(66,175,97)(67,176,98)(68,177,99)(69,178,100)(70,179,101)(71,180,102)(72,181,103)(73,182,104)(74,183,105)(75,184,106), (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)(151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225) );

G=PermutationGroup([(1,185,107),(2,186,108),(3,187,109),(4,188,110),(5,189,111),(6,190,112),(7,191,113),(8,192,114),(9,193,115),(10,194,116),(11,195,117),(12,196,118),(13,197,119),(14,198,120),(15,199,121),(16,200,122),(17,201,123),(18,202,124),(19,203,125),(20,204,126),(21,205,127),(22,206,128),(23,207,129),(24,208,130),(25,209,131),(26,210,132),(27,211,133),(28,212,134),(29,213,135),(30,214,136),(31,215,137),(32,216,138),(33,217,139),(34,218,140),(35,219,141),(36,220,142),(37,221,143),(38,222,144),(39,223,145),(40,224,146),(41,225,147),(42,151,148),(43,152,149),(44,153,150),(45,154,76),(46,155,77),(47,156,78),(48,157,79),(49,158,80),(50,159,81),(51,160,82),(52,161,83),(53,162,84),(54,163,85),(55,164,86),(56,165,87),(57,166,88),(58,167,89),(59,168,90),(60,169,91),(61,170,92),(62,171,93),(63,172,94),(64,173,95),(65,174,96),(66,175,97),(67,176,98),(68,177,99),(69,178,100),(70,179,101),(71,180,102),(72,181,103),(73,182,104),(74,183,105),(75,184,106)], [(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),(151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225)])

C3×C75 is a maximal subgroup of   C3⋊D75

225 conjugacy classes

class 1 3A···3H5A5B5C5D15A···15AF25A···25T75A···75FD
order13···3555515···1525···2575···75
size11···111111···11···11···1

225 irreducible representations

dim111111
type+
imageC1C3C5C15C25C75
kernelC3×C75C75C3×C15C15C32C3
# reps1843220160

Matrix representation of C3×C75 in GL2(𝔽151) generated by

10
032
,
490
050
G:=sub<GL(2,GF(151))| [1,0,0,32],[49,0,0,50] >;

C3×C75 in GAP, Magma, Sage, TeX

C_3\times C_{75}
% in TeX

G:=Group("C3xC75");
// GroupNames label

G:=SmallGroup(225,2);
// by ID

G=gap.SmallGroup(225,2);
# by ID

G:=PCGroup([4,-3,-3,-5,-5,70]);
// Polycyclic

G:=Group<a,b|a^3=b^75=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C3×C75 in TeX

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