Copied to
clipboard

G = C57⋊D4order 456 = 23·3·19

1st semidirect product of C57 and D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C571D4, D61D19, D381S3, C6.4D38, C38.4D6, Dic574C2, C114.4C22, (S3×C38)⋊1C2, (C6×D19)⋊1C2, C192(C3⋊D4), C32(C19⋊D4), C2.4(S3×D19), SmallGroup(456,15)

Series: Derived Chief Lower central Upper central

C1C114 — C57⋊D4
C1C19C57C114C6×D19 — C57⋊D4
C57C114 — C57⋊D4
C1C2

Generators and relations for C57⋊D4
 G = < a,b,c | a57=b4=c2=1, bab-1=a-1, cac=a37, cbc=b-1 >

6C2
38C2
3C22
19C22
57C4
2S3
38C6
2D19
6C38
57D4
19C2×C6
19Dic3
3Dic19
3C2×C38
2S3×C19
2C3×D19
19C3⋊D4
3C19⋊D4

Smallest permutation representation of C57⋊D4
On 228 points
Generators in S228
(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)
(1 119 88 179)(2 118 89 178)(3 117 90 177)(4 116 91 176)(5 115 92 175)(6 171 93 174)(7 170 94 173)(8 169 95 172)(9 168 96 228)(10 167 97 227)(11 166 98 226)(12 165 99 225)(13 164 100 224)(14 163 101 223)(15 162 102 222)(16 161 103 221)(17 160 104 220)(18 159 105 219)(19 158 106 218)(20 157 107 217)(21 156 108 216)(22 155 109 215)(23 154 110 214)(24 153 111 213)(25 152 112 212)(26 151 113 211)(27 150 114 210)(28 149 58 209)(29 148 59 208)(30 147 60 207)(31 146 61 206)(32 145 62 205)(33 144 63 204)(34 143 64 203)(35 142 65 202)(36 141 66 201)(37 140 67 200)(38 139 68 199)(39 138 69 198)(40 137 70 197)(41 136 71 196)(42 135 72 195)(43 134 73 194)(44 133 74 193)(45 132 75 192)(46 131 76 191)(47 130 77 190)(48 129 78 189)(49 128 79 188)(50 127 80 187)(51 126 81 186)(52 125 82 185)(53 124 83 184)(54 123 84 183)(55 122 85 182)(56 121 86 181)(57 120 87 180)
(2 38)(3 18)(4 55)(5 35)(6 15)(7 52)(8 32)(9 12)(10 49)(11 29)(13 46)(14 26)(16 43)(17 23)(19 40)(21 57)(22 37)(24 54)(25 34)(27 51)(28 31)(30 48)(33 45)(36 42)(41 56)(44 53)(47 50)(58 61)(59 98)(60 78)(62 95)(63 75)(64 112)(65 92)(66 72)(67 109)(68 89)(70 106)(71 86)(73 103)(74 83)(76 100)(77 80)(79 97)(81 114)(82 94)(84 111)(85 91)(87 108)(90 105)(93 102)(96 99)(101 113)(104 110)(115 202)(116 182)(117 219)(118 199)(119 179)(120 216)(121 196)(122 176)(123 213)(124 193)(125 173)(126 210)(127 190)(128 227)(129 207)(130 187)(131 224)(132 204)(133 184)(134 221)(135 201)(136 181)(137 218)(138 198)(139 178)(140 215)(141 195)(142 175)(143 212)(144 192)(145 172)(146 209)(147 189)(148 226)(149 206)(150 186)(151 223)(152 203)(153 183)(154 220)(155 200)(156 180)(157 217)(158 197)(159 177)(160 214)(161 194)(162 174)(163 211)(164 191)(165 228)(166 208)(167 188)(168 225)(169 205)(170 185)(171 222)

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

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

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

63 conjugacy classes

class 1 2A2B2C 3  4 6A6B6C19A···19I38A···38I38J···38AA57A···57I114A···114I
order12223466619···1938···3838···3857···57114···114
size116382114238382···22···26···64···44···4

63 irreducible representations

dim1111222222244
type++++++++++-
imageC1C2C2C2S3D4D6C3⋊D4D19D38C19⋊D4S3×D19C57⋊D4
kernelC57⋊D4Dic57C6×D19S3×C38D38C57C38C19D6C6C3C2C1
# reps11111112991899

Matrix representation of C57⋊D4 in GL4(𝔽229) generated by

5721700
204000
00940
0017134
,
1172500
11211200
0018204
0013211
,
1172500
11211200
0010
00148228
G:=sub<GL(4,GF(229))| [57,20,0,0,217,40,0,0,0,0,94,17,0,0,0,134],[117,112,0,0,25,112,0,0,0,0,18,13,0,0,204,211],[117,112,0,0,25,112,0,0,0,0,1,148,0,0,0,228] >;

C57⋊D4 in GAP, Magma, Sage, TeX

C_{57}\rtimes D_4
% in TeX

G:=Group("C57:D4");
// GroupNames label

G:=SmallGroup(456,15);
// by ID

G=gap.SmallGroup(456,15);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-19,61,168,10804]);
// Polycyclic

G:=Group<a,b,c|a^57=b^4=c^2=1,b*a*b^-1=a^-1,c*a*c=a^37,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C57⋊D4 in TeX

׿
×
𝔽