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G = M4(2)×C29order 464 = 24·29

Direct product of C29 and M4(2)

direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary

Aliases: M4(2)×C29, C83C58, C4.C116, C2327C2, C116.7C4, C22.C116, C116.22C22, (C2×C4).2C58, C4.6(C2×C58), (C2×C58).3C4, C58.19(C2×C4), (C2×C116).8C2, C2.3(C2×C116), SmallGroup(464,24)

Series: Derived Chief Lower central Upper central

C1C2 — M4(2)×C29
C1C2C4C116C232 — M4(2)×C29
C1C2 — M4(2)×C29
C1C116 — M4(2)×C29

Generators and relations for M4(2)×C29
 G = < a,b,c | a29=b8=c2=1, ab=ba, ac=ca, cbc=b5 >

2C2
2C58

Smallest permutation representation of M4(2)×C29
On 232 points
Generators in S232
(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)(204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232)
(1 44 208 163 201 83 101 128)(2 45 209 164 202 84 102 129)(3 46 210 165 203 85 103 130)(4 47 211 166 175 86 104 131)(5 48 212 167 176 87 105 132)(6 49 213 168 177 59 106 133)(7 50 214 169 178 60 107 134)(8 51 215 170 179 61 108 135)(9 52 216 171 180 62 109 136)(10 53 217 172 181 63 110 137)(11 54 218 173 182 64 111 138)(12 55 219 174 183 65 112 139)(13 56 220 146 184 66 113 140)(14 57 221 147 185 67 114 141)(15 58 222 148 186 68 115 142)(16 30 223 149 187 69 116 143)(17 31 224 150 188 70 88 144)(18 32 225 151 189 71 89 145)(19 33 226 152 190 72 90 117)(20 34 227 153 191 73 91 118)(21 35 228 154 192 74 92 119)(22 36 229 155 193 75 93 120)(23 37 230 156 194 76 94 121)(24 38 231 157 195 77 95 122)(25 39 232 158 196 78 96 123)(26 40 204 159 197 79 97 124)(27 41 205 160 198 80 98 125)(28 42 206 161 199 81 99 126)(29 43 207 162 200 82 100 127)
(30 69)(31 70)(32 71)(33 72)(34 73)(35 74)(36 75)(37 76)(38 77)(39 78)(40 79)(41 80)(42 81)(43 82)(44 83)(45 84)(46 85)(47 86)(48 87)(49 59)(50 60)(51 61)(52 62)(53 63)(54 64)(55 65)(56 66)(57 67)(58 68)(117 152)(118 153)(119 154)(120 155)(121 156)(122 157)(123 158)(124 159)(125 160)(126 161)(127 162)(128 163)(129 164)(130 165)(131 166)(132 167)(133 168)(134 169)(135 170)(136 171)(137 172)(138 173)(139 174)(140 146)(141 147)(142 148)(143 149)(144 150)(145 151)

G:=sub<Sym(232)| (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)(204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232), (1,44,208,163,201,83,101,128)(2,45,209,164,202,84,102,129)(3,46,210,165,203,85,103,130)(4,47,211,166,175,86,104,131)(5,48,212,167,176,87,105,132)(6,49,213,168,177,59,106,133)(7,50,214,169,178,60,107,134)(8,51,215,170,179,61,108,135)(9,52,216,171,180,62,109,136)(10,53,217,172,181,63,110,137)(11,54,218,173,182,64,111,138)(12,55,219,174,183,65,112,139)(13,56,220,146,184,66,113,140)(14,57,221,147,185,67,114,141)(15,58,222,148,186,68,115,142)(16,30,223,149,187,69,116,143)(17,31,224,150,188,70,88,144)(18,32,225,151,189,71,89,145)(19,33,226,152,190,72,90,117)(20,34,227,153,191,73,91,118)(21,35,228,154,192,74,92,119)(22,36,229,155,193,75,93,120)(23,37,230,156,194,76,94,121)(24,38,231,157,195,77,95,122)(25,39,232,158,196,78,96,123)(26,40,204,159,197,79,97,124)(27,41,205,160,198,80,98,125)(28,42,206,161,199,81,99,126)(29,43,207,162,200,82,100,127), (30,69)(31,70)(32,71)(33,72)(34,73)(35,74)(36,75)(37,76)(38,77)(39,78)(40,79)(41,80)(42,81)(43,82)(44,83)(45,84)(46,85)(47,86)(48,87)(49,59)(50,60)(51,61)(52,62)(53,63)(54,64)(55,65)(56,66)(57,67)(58,68)(117,152)(118,153)(119,154)(120,155)(121,156)(122,157)(123,158)(124,159)(125,160)(126,161)(127,162)(128,163)(129,164)(130,165)(131,166)(132,167)(133,168)(134,169)(135,170)(136,171)(137,172)(138,173)(139,174)(140,146)(141,147)(142,148)(143,149)(144,150)(145,151)>;

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)(175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203)(204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232), (1,44,208,163,201,83,101,128)(2,45,209,164,202,84,102,129)(3,46,210,165,203,85,103,130)(4,47,211,166,175,86,104,131)(5,48,212,167,176,87,105,132)(6,49,213,168,177,59,106,133)(7,50,214,169,178,60,107,134)(8,51,215,170,179,61,108,135)(9,52,216,171,180,62,109,136)(10,53,217,172,181,63,110,137)(11,54,218,173,182,64,111,138)(12,55,219,174,183,65,112,139)(13,56,220,146,184,66,113,140)(14,57,221,147,185,67,114,141)(15,58,222,148,186,68,115,142)(16,30,223,149,187,69,116,143)(17,31,224,150,188,70,88,144)(18,32,225,151,189,71,89,145)(19,33,226,152,190,72,90,117)(20,34,227,153,191,73,91,118)(21,35,228,154,192,74,92,119)(22,36,229,155,193,75,93,120)(23,37,230,156,194,76,94,121)(24,38,231,157,195,77,95,122)(25,39,232,158,196,78,96,123)(26,40,204,159,197,79,97,124)(27,41,205,160,198,80,98,125)(28,42,206,161,199,81,99,126)(29,43,207,162,200,82,100,127), (30,69)(31,70)(32,71)(33,72)(34,73)(35,74)(36,75)(37,76)(38,77)(39,78)(40,79)(41,80)(42,81)(43,82)(44,83)(45,84)(46,85)(47,86)(48,87)(49,59)(50,60)(51,61)(52,62)(53,63)(54,64)(55,65)(56,66)(57,67)(58,68)(117,152)(118,153)(119,154)(120,155)(121,156)(122,157)(123,158)(124,159)(125,160)(126,161)(127,162)(128,163)(129,164)(130,165)(131,166)(132,167)(133,168)(134,169)(135,170)(136,171)(137,172)(138,173)(139,174)(140,146)(141,147)(142,148)(143,149)(144,150)(145,151) );

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),(175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203),(204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232)], [(1,44,208,163,201,83,101,128),(2,45,209,164,202,84,102,129),(3,46,210,165,203,85,103,130),(4,47,211,166,175,86,104,131),(5,48,212,167,176,87,105,132),(6,49,213,168,177,59,106,133),(7,50,214,169,178,60,107,134),(8,51,215,170,179,61,108,135),(9,52,216,171,180,62,109,136),(10,53,217,172,181,63,110,137),(11,54,218,173,182,64,111,138),(12,55,219,174,183,65,112,139),(13,56,220,146,184,66,113,140),(14,57,221,147,185,67,114,141),(15,58,222,148,186,68,115,142),(16,30,223,149,187,69,116,143),(17,31,224,150,188,70,88,144),(18,32,225,151,189,71,89,145),(19,33,226,152,190,72,90,117),(20,34,227,153,191,73,91,118),(21,35,228,154,192,74,92,119),(22,36,229,155,193,75,93,120),(23,37,230,156,194,76,94,121),(24,38,231,157,195,77,95,122),(25,39,232,158,196,78,96,123),(26,40,204,159,197,79,97,124),(27,41,205,160,198,80,98,125),(28,42,206,161,199,81,99,126),(29,43,207,162,200,82,100,127)], [(30,69),(31,70),(32,71),(33,72),(34,73),(35,74),(36,75),(37,76),(38,77),(39,78),(40,79),(41,80),(42,81),(43,82),(44,83),(45,84),(46,85),(47,86),(48,87),(49,59),(50,60),(51,61),(52,62),(53,63),(54,64),(55,65),(56,66),(57,67),(58,68),(117,152),(118,153),(119,154),(120,155),(121,156),(122,157),(123,158),(124,159),(125,160),(126,161),(127,162),(128,163),(129,164),(130,165),(131,166),(132,167),(133,168),(134,169),(135,170),(136,171),(137,172),(138,173),(139,174),(140,146),(141,147),(142,148),(143,149),(144,150),(145,151)])

290 conjugacy classes

class 1 2A2B4A4B4C8A8B8C8D29A···29AB58A···58AB58AC···58BD116A···116BD116BE···116CF232A···232DH
order122444888829···2958···5858···58116···116116···116232···232
size11211222221···11···12···21···12···22···2

290 irreducible representations

dim111111111122
type+++
imageC1C2C2C4C4C29C58C58C116C116M4(2)M4(2)×C29
kernelM4(2)×C29C232C2×C116C116C2×C58M4(2)C8C2×C4C4C22C29C1
# reps121222856285656256

Matrix representation of M4(2)×C29 in GL2(𝔽233) generated by

630
063
,
140197
17893
,
1111
0232
G:=sub<GL(2,GF(233))| [63,0,0,63],[140,178,197,93],[1,0,111,232] >;

M4(2)×C29 in GAP, Magma, Sage, TeX

M_4(2)\times C_{29}
% in TeX

G:=Group("M4(2)xC29");
// GroupNames label

G:=SmallGroup(464,24);
// by ID

G=gap.SmallGroup(464,24);
# by ID

G:=PCGroup([5,-2,-2,-29,-2,-2,580,2341,58]);
// Polycyclic

G:=Group<a,b,c|a^29=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^5>;
// generators/relations

Export

Subgroup lattice of M4(2)×C29 in TeX

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