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G = C577D4order 456 = 23·3·19

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C577D4, D1142C2, C6.12D38, C38.12D6, C2.5D114, C222D57, Dic571C2, C114.12C22, (C2×C38)⋊4S3, (C2×C6)⋊2D19, (C2×C114)⋊2C2, C33(C19⋊D4), C193(C3⋊D4), SmallGroup(456,38)

Series: Derived Chief Lower central Upper central

C1C114 — C577D4
C1C19C57C114D114 — C577D4
C57C114 — C577D4
C1C2C22

Generators and relations for C577D4
 G = < a,b,c | a57=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >

2C2
114C2
57C22
57C4
2C6
38S3
2C38
6D19
57D4
19D6
19Dic3
3D38
3Dic19
2C114
2D57
19C3⋊D4
3C19⋊D4

Smallest permutation representation of C577D4
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)
(1 120 68 176)(2 119 69 175)(3 118 70 174)(4 117 71 173)(5 116 72 172)(6 115 73 228)(7 171 74 227)(8 170 75 226)(9 169 76 225)(10 168 77 224)(11 167 78 223)(12 166 79 222)(13 165 80 221)(14 164 81 220)(15 163 82 219)(16 162 83 218)(17 161 84 217)(18 160 85 216)(19 159 86 215)(20 158 87 214)(21 157 88 213)(22 156 89 212)(23 155 90 211)(24 154 91 210)(25 153 92 209)(26 152 93 208)(27 151 94 207)(28 150 95 206)(29 149 96 205)(30 148 97 204)(31 147 98 203)(32 146 99 202)(33 145 100 201)(34 144 101 200)(35 143 102 199)(36 142 103 198)(37 141 104 197)(38 140 105 196)(39 139 106 195)(40 138 107 194)(41 137 108 193)(42 136 109 192)(43 135 110 191)(44 134 111 190)(45 133 112 189)(46 132 113 188)(47 131 114 187)(48 130 58 186)(49 129 59 185)(50 128 60 184)(51 127 61 183)(52 126 62 182)(53 125 63 181)(54 124 64 180)(55 123 65 179)(56 122 66 178)(57 121 67 177)
(2 57)(3 56)(4 55)(5 54)(6 53)(7 52)(8 51)(9 50)(10 49)(11 48)(12 47)(13 46)(14 45)(15 44)(16 43)(17 42)(18 41)(19 40)(20 39)(21 38)(22 37)(23 36)(24 35)(25 34)(26 33)(27 32)(28 31)(29 30)(58 78)(59 77)(60 76)(61 75)(62 74)(63 73)(64 72)(65 71)(66 70)(67 69)(79 114)(80 113)(81 112)(82 111)(83 110)(84 109)(85 108)(86 107)(87 106)(88 105)(89 104)(90 103)(91 102)(92 101)(93 100)(94 99)(95 98)(96 97)(115 181)(116 180)(117 179)(118 178)(119 177)(120 176)(121 175)(122 174)(123 173)(124 172)(125 228)(126 227)(127 226)(128 225)(129 224)(130 223)(131 222)(132 221)(133 220)(134 219)(135 218)(136 217)(137 216)(138 215)(139 214)(140 213)(141 212)(142 211)(143 210)(144 209)(145 208)(146 207)(147 206)(148 205)(149 204)(150 203)(151 202)(152 201)(153 200)(154 199)(155 198)(156 197)(157 196)(158 195)(159 194)(160 193)(161 192)(162 191)(163 190)(164 189)(165 188)(166 187)(167 186)(168 185)(169 184)(170 183)(171 182)

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

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

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

117 conjugacy classes

class 1 2A2B2C 3  4 6A6B6C19A···19I38A···38AA57A···57R114A···114BB
order12223466619···1938···3857···57114···114
size11211421142222···22···22···22···2

117 irreducible representations

dim11112222222222
type+++++++++++
imageC1C2C2C2S3D4D6C3⋊D4D19D38D57C19⋊D4D114C577D4
kernelC577D4Dic57D114C2×C114C2×C38C57C38C19C2×C6C6C22C3C2C1
# reps111111129918181836

Matrix representation of C577D4 in GL2(𝔽229) generated by

156180
49196
,
82178
69147
,
10
102228
G:=sub<GL(2,GF(229))| [156,49,180,196],[82,69,178,147],[1,102,0,228] >;

C577D4 in GAP, Magma, Sage, TeX

C_{57}\rtimes_7D_4
% in TeX

G:=Group("C57:7D4");
// GroupNames label

G:=SmallGroup(456,38);
// by ID

G=gap.SmallGroup(456,38);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-19,61,323,10804]);
// Polycyclic

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

Export

Subgroup lattice of C577D4 in TeX

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