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G = M4(2)×C31order 496 = 24·31

Direct product of C31 and M4(2)

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

Aliases: M4(2)×C31, C83C62, C4.C124, C2487C2, C124.4C4, C22.C124, C124.22C22, (C2×C62).1C4, (C2×C4).2C62, C4.6(C2×C62), C62.12(C2×C4), (C2×C124).8C2, C2.3(C2×C124), SmallGroup(496,23)

Series: Derived Chief Lower central Upper central

C1C2 — M4(2)×C31
C1C2C4C124C248 — M4(2)×C31
C1C2 — M4(2)×C31
C1C124 — M4(2)×C31

Generators and relations for M4(2)×C31
 G = < a,b,c | a31=b8=c2=1, ab=ba, ac=ca, cbc=b5 >

2C2
2C62

Smallest permutation representation of M4(2)×C31
On 248 points
Generators in S248
(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)(218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248)
(1 222 101 48 183 153 213 72)(2 223 102 49 184 154 214 73)(3 224 103 50 185 155 215 74)(4 225 104 51 186 125 216 75)(5 226 105 52 156 126 217 76)(6 227 106 53 157 127 187 77)(7 228 107 54 158 128 188 78)(8 229 108 55 159 129 189 79)(9 230 109 56 160 130 190 80)(10 231 110 57 161 131 191 81)(11 232 111 58 162 132 192 82)(12 233 112 59 163 133 193 83)(13 234 113 60 164 134 194 84)(14 235 114 61 165 135 195 85)(15 236 115 62 166 136 196 86)(16 237 116 32 167 137 197 87)(17 238 117 33 168 138 198 88)(18 239 118 34 169 139 199 89)(19 240 119 35 170 140 200 90)(20 241 120 36 171 141 201 91)(21 242 121 37 172 142 202 92)(22 243 122 38 173 143 203 93)(23 244 123 39 174 144 204 63)(24 245 124 40 175 145 205 64)(25 246 94 41 176 146 206 65)(26 247 95 42 177 147 207 66)(27 248 96 43 178 148 208 67)(28 218 97 44 179 149 209 68)(29 219 98 45 180 150 210 69)(30 220 99 46 181 151 211 70)(31 221 100 47 182 152 212 71)
(32 87)(33 88)(34 89)(35 90)(36 91)(37 92)(38 93)(39 63)(40 64)(41 65)(42 66)(43 67)(44 68)(45 69)(46 70)(47 71)(48 72)(49 73)(50 74)(51 75)(52 76)(53 77)(54 78)(55 79)(56 80)(57 81)(58 82)(59 83)(60 84)(61 85)(62 86)(125 225)(126 226)(127 227)(128 228)(129 229)(130 230)(131 231)(132 232)(133 233)(134 234)(135 235)(136 236)(137 237)(138 238)(139 239)(140 240)(141 241)(142 242)(143 243)(144 244)(145 245)(146 246)(147 247)(148 248)(149 218)(150 219)(151 220)(152 221)(153 222)(154 223)(155 224)

G:=sub<Sym(248)| (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)(218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248), (1,222,101,48,183,153,213,72)(2,223,102,49,184,154,214,73)(3,224,103,50,185,155,215,74)(4,225,104,51,186,125,216,75)(5,226,105,52,156,126,217,76)(6,227,106,53,157,127,187,77)(7,228,107,54,158,128,188,78)(8,229,108,55,159,129,189,79)(9,230,109,56,160,130,190,80)(10,231,110,57,161,131,191,81)(11,232,111,58,162,132,192,82)(12,233,112,59,163,133,193,83)(13,234,113,60,164,134,194,84)(14,235,114,61,165,135,195,85)(15,236,115,62,166,136,196,86)(16,237,116,32,167,137,197,87)(17,238,117,33,168,138,198,88)(18,239,118,34,169,139,199,89)(19,240,119,35,170,140,200,90)(20,241,120,36,171,141,201,91)(21,242,121,37,172,142,202,92)(22,243,122,38,173,143,203,93)(23,244,123,39,174,144,204,63)(24,245,124,40,175,145,205,64)(25,246,94,41,176,146,206,65)(26,247,95,42,177,147,207,66)(27,248,96,43,178,148,208,67)(28,218,97,44,179,149,209,68)(29,219,98,45,180,150,210,69)(30,220,99,46,181,151,211,70)(31,221,100,47,182,152,212,71), (32,87)(33,88)(34,89)(35,90)(36,91)(37,92)(38,93)(39,63)(40,64)(41,65)(42,66)(43,67)(44,68)(45,69)(46,70)(47,71)(48,72)(49,73)(50,74)(51,75)(52,76)(53,77)(54,78)(55,79)(56,80)(57,81)(58,82)(59,83)(60,84)(61,85)(62,86)(125,225)(126,226)(127,227)(128,228)(129,229)(130,230)(131,231)(132,232)(133,233)(134,234)(135,235)(136,236)(137,237)(138,238)(139,239)(140,240)(141,241)(142,242)(143,243)(144,244)(145,245)(146,246)(147,247)(148,248)(149,218)(150,219)(151,220)(152,221)(153,222)(154,223)(155,224)>;

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)(218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248), (1,222,101,48,183,153,213,72)(2,223,102,49,184,154,214,73)(3,224,103,50,185,155,215,74)(4,225,104,51,186,125,216,75)(5,226,105,52,156,126,217,76)(6,227,106,53,157,127,187,77)(7,228,107,54,158,128,188,78)(8,229,108,55,159,129,189,79)(9,230,109,56,160,130,190,80)(10,231,110,57,161,131,191,81)(11,232,111,58,162,132,192,82)(12,233,112,59,163,133,193,83)(13,234,113,60,164,134,194,84)(14,235,114,61,165,135,195,85)(15,236,115,62,166,136,196,86)(16,237,116,32,167,137,197,87)(17,238,117,33,168,138,198,88)(18,239,118,34,169,139,199,89)(19,240,119,35,170,140,200,90)(20,241,120,36,171,141,201,91)(21,242,121,37,172,142,202,92)(22,243,122,38,173,143,203,93)(23,244,123,39,174,144,204,63)(24,245,124,40,175,145,205,64)(25,246,94,41,176,146,206,65)(26,247,95,42,177,147,207,66)(27,248,96,43,178,148,208,67)(28,218,97,44,179,149,209,68)(29,219,98,45,180,150,210,69)(30,220,99,46,181,151,211,70)(31,221,100,47,182,152,212,71), (32,87)(33,88)(34,89)(35,90)(36,91)(37,92)(38,93)(39,63)(40,64)(41,65)(42,66)(43,67)(44,68)(45,69)(46,70)(47,71)(48,72)(49,73)(50,74)(51,75)(52,76)(53,77)(54,78)(55,79)(56,80)(57,81)(58,82)(59,83)(60,84)(61,85)(62,86)(125,225)(126,226)(127,227)(128,228)(129,229)(130,230)(131,231)(132,232)(133,233)(134,234)(135,235)(136,236)(137,237)(138,238)(139,239)(140,240)(141,241)(142,242)(143,243)(144,244)(145,245)(146,246)(147,247)(148,248)(149,218)(150,219)(151,220)(152,221)(153,222)(154,223)(155,224) );

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),(218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248)], [(1,222,101,48,183,153,213,72),(2,223,102,49,184,154,214,73),(3,224,103,50,185,155,215,74),(4,225,104,51,186,125,216,75),(5,226,105,52,156,126,217,76),(6,227,106,53,157,127,187,77),(7,228,107,54,158,128,188,78),(8,229,108,55,159,129,189,79),(9,230,109,56,160,130,190,80),(10,231,110,57,161,131,191,81),(11,232,111,58,162,132,192,82),(12,233,112,59,163,133,193,83),(13,234,113,60,164,134,194,84),(14,235,114,61,165,135,195,85),(15,236,115,62,166,136,196,86),(16,237,116,32,167,137,197,87),(17,238,117,33,168,138,198,88),(18,239,118,34,169,139,199,89),(19,240,119,35,170,140,200,90),(20,241,120,36,171,141,201,91),(21,242,121,37,172,142,202,92),(22,243,122,38,173,143,203,93),(23,244,123,39,174,144,204,63),(24,245,124,40,175,145,205,64),(25,246,94,41,176,146,206,65),(26,247,95,42,177,147,207,66),(27,248,96,43,178,148,208,67),(28,218,97,44,179,149,209,68),(29,219,98,45,180,150,210,69),(30,220,99,46,181,151,211,70),(31,221,100,47,182,152,212,71)], [(32,87),(33,88),(34,89),(35,90),(36,91),(37,92),(38,93),(39,63),(40,64),(41,65),(42,66),(43,67),(44,68),(45,69),(46,70),(47,71),(48,72),(49,73),(50,74),(51,75),(52,76),(53,77),(54,78),(55,79),(56,80),(57,81),(58,82),(59,83),(60,84),(61,85),(62,86),(125,225),(126,226),(127,227),(128,228),(129,229),(130,230),(131,231),(132,232),(133,233),(134,234),(135,235),(136,236),(137,237),(138,238),(139,239),(140,240),(141,241),(142,242),(143,243),(144,244),(145,245),(146,246),(147,247),(148,248),(149,218),(150,219),(151,220),(152,221),(153,222),(154,223),(155,224)])

310 conjugacy classes

class 1 2A2B4A4B4C8A8B8C8D31A···31AD62A···62AD62AE···62BH124A···124BH124BI···124CL248A···248DP
order122444888831···3162···6262···62124···124124···124248···248
size11211222221···11···12···21···12···22···2

310 irreducible representations

dim111111111122
type+++
imageC1C2C2C4C4C31C62C62C124C124M4(2)M4(2)×C31
kernelM4(2)×C31C248C2×C124C124C2×C62M4(2)C8C2×C4C4C22C31C1
# reps121223060306060260

Matrix representation of M4(2)×C31 in GL2(𝔽1489) generated by

9920
0992
,
01
12640
,
10
01488
G:=sub<GL(2,GF(1489))| [992,0,0,992],[0,1264,1,0],[1,0,0,1488] >;

M4(2)×C31 in GAP, Magma, Sage, TeX

M_4(2)\times C_{31}
% in TeX

G:=Group("M4(2)xC31");
// GroupNames label

G:=SmallGroup(496,23);
// by ID

G=gap.SmallGroup(496,23);
# by ID

G:=PCGroup([5,-2,-2,-31,-2,-2,620,2501,58]);
// Polycyclic

G:=Group<a,b,c|a^31=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)×C31 in TeX

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