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G = D4×C57order 456 = 23·3·19

Direct product of C57 and D4

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

Aliases: D4×C57, C4⋊C114, C767C6, C123C38, C2287C2, C222C114, C114.23C22, (C2×C38)⋊9C6, (C2×C6)⋊1C38, (C2×C114)⋊1C2, C6.6(C2×C38), C38.14(C2×C6), C2.1(C2×C114), SmallGroup(456,40)

Series: Derived Chief Lower central Upper central

C1C2 — D4×C57
C1C2C38C114C2×C114 — D4×C57
C1C2 — D4×C57
C1C114 — D4×C57

Generators and relations for D4×C57
 G = < a,b,c | a57=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >

2C2
2C2
2C6
2C6
2C38
2C38
2C114
2C114

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

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

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

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

285 conjugacy classes

class 1 2A2B2C3A3B 4 6A6B6C6D6E6F12A12B19A···19R38A···38R38S···38BB57A···57AJ76A···76R114A···114AJ114AK···114DD228A···228AJ
order1222334666666121219···1938···3838···3857···5776···76114···114114···114228···228
size1122112112222221···11···12···21···12···21···12···22···2

285 irreducible representations

dim1111111111112222
type++++
imageC1C2C2C3C6C6C19C38C38C57C114C114D4C3×D4D4×C19D4×C57
kernelD4×C57C228C2×C114D4×C19C76C2×C38C3×D4C12C2×C6D4C4C22C57C19C3C1
# reps112224181836363672121836

Matrix representation of D4×C57 in GL3(𝔽229) generated by

13400
0610
0061
,
22800
010163
0227128
,
22800
01101
00228
G:=sub<GL(3,GF(229))| [134,0,0,0,61,0,0,0,61],[228,0,0,0,101,227,0,63,128],[228,0,0,0,1,0,0,101,228] >;

D4×C57 in GAP, Magma, Sage, TeX

D_4\times C_{57}
% in TeX

G:=Group("D4xC57");
// GroupNames label

G:=SmallGroup(456,40);
// by ID

G=gap.SmallGroup(456,40);
# by ID

G:=PCGroup([5,-2,-2,-3,-19,-2,2301]);
// Polycyclic

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

Export

Subgroup lattice of D4×C57 in TeX

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