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G = C31⋊C8order 248 = 23·31

The semidirect product of C31 and C8 acting via C8/C4=C2

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C31⋊C8, C62.C4, C4.2D31, C2.Dic31, C124.2C2, SmallGroup(248,1)

Series: Derived Chief Lower central Upper central

C1C31 — C31⋊C8
C1C31C62C124 — C31⋊C8
C31 — C31⋊C8
C1C4

Generators and relations for C31⋊C8
 G = < a,b | a31=b8=1, bab-1=a-1 >

31C8

Smallest permutation representation of C31⋊C8
Regular action 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)
(1 218 107 156 37 187 73 125)(2 248 108 186 38 217 74 155)(3 247 109 185 39 216 75 154)(4 246 110 184 40 215 76 153)(5 245 111 183 41 214 77 152)(6 244 112 182 42 213 78 151)(7 243 113 181 43 212 79 150)(8 242 114 180 44 211 80 149)(9 241 115 179 45 210 81 148)(10 240 116 178 46 209 82 147)(11 239 117 177 47 208 83 146)(12 238 118 176 48 207 84 145)(13 237 119 175 49 206 85 144)(14 236 120 174 50 205 86 143)(15 235 121 173 51 204 87 142)(16 234 122 172 52 203 88 141)(17 233 123 171 53 202 89 140)(18 232 124 170 54 201 90 139)(19 231 94 169 55 200 91 138)(20 230 95 168 56 199 92 137)(21 229 96 167 57 198 93 136)(22 228 97 166 58 197 63 135)(23 227 98 165 59 196 64 134)(24 226 99 164 60 195 65 133)(25 225 100 163 61 194 66 132)(26 224 101 162 62 193 67 131)(27 223 102 161 32 192 68 130)(28 222 103 160 33 191 69 129)(29 221 104 159 34 190 70 128)(30 220 105 158 35 189 71 127)(31 219 106 157 36 188 72 126)

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

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

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

C31⋊C8 is a maximal subgroup of   C8×D31  C8⋊D31  C4.Dic31  D4⋊D31  D4.D31  Q8⋊D31  C31⋊Q16
C31⋊C8 is a maximal quotient of   C31⋊C16

68 conjugacy classes

class 1  2 4A4B8A8B8C8D31A···31O62A···62O124A···124AD
order1244888831···3162···62124···124
size1111313131312···22···22···2

68 irreducible representations

dim1111222
type+++-
imageC1C2C4C8D31Dic31C31⋊C8
kernelC31⋊C8C124C62C31C4C2C1
# reps1124151530

Matrix representation of C31⋊C8 in GL2(𝔽1489) generated by

01
14881013
,
4171085
6301072
G:=sub<GL(2,GF(1489))| [0,1488,1,1013],[417,630,1085,1072] >;

C31⋊C8 in GAP, Magma, Sage, TeX

C_{31}\rtimes C_8
% in TeX

G:=Group("C31:C8");
// GroupNames label

G:=SmallGroup(248,1);
// by ID

G=gap.SmallGroup(248,1);
# by ID

G:=PCGroup([4,-2,-2,-2,-31,8,21,3843]);
// Polycyclic

G:=Group<a,b|a^31=b^8=1,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of C31⋊C8 in TeX

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