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G = C53⋊D4order 424 = 23·53

The semidirect product of C53 and D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C532D4, C22⋊D53, Dic53⋊C2, D1062C2, C2.5D106, C106.5C22, (C2×C106)⋊2C2, SmallGroup(424,8)

Series: Derived Chief Lower central Upper central

C1C106 — C53⋊D4
C1C53C106D106 — C53⋊D4
C53C106 — C53⋊D4
C1C2C22

Generators and relations for C53⋊D4
 G = < a,b,c | a53=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >

2C2
106C2
53C4
53C22
2D53
2C106
53D4

Smallest permutation representation of C53⋊D4
On 212 points
Generators in S212
(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)
(1 132 75 205)(2 131 76 204)(3 130 77 203)(4 129 78 202)(5 128 79 201)(6 127 80 200)(7 126 81 199)(8 125 82 198)(9 124 83 197)(10 123 84 196)(11 122 85 195)(12 121 86 194)(13 120 87 193)(14 119 88 192)(15 118 89 191)(16 117 90 190)(17 116 91 189)(18 115 92 188)(19 114 93 187)(20 113 94 186)(21 112 95 185)(22 111 96 184)(23 110 97 183)(24 109 98 182)(25 108 99 181)(26 107 100 180)(27 159 101 179)(28 158 102 178)(29 157 103 177)(30 156 104 176)(31 155 105 175)(32 154 106 174)(33 153 54 173)(34 152 55 172)(35 151 56 171)(36 150 57 170)(37 149 58 169)(38 148 59 168)(39 147 60 167)(40 146 61 166)(41 145 62 165)(42 144 63 164)(43 143 64 163)(44 142 65 162)(45 141 66 161)(46 140 67 160)(47 139 68 212)(48 138 69 211)(49 137 70 210)(50 136 71 209)(51 135 72 208)(52 134 73 207)(53 133 74 206)
(2 53)(3 52)(4 51)(5 50)(6 49)(7 48)(8 47)(9 46)(10 45)(11 44)(12 43)(13 42)(14 41)(15 40)(16 39)(17 38)(18 37)(19 36)(20 35)(21 34)(22 33)(23 32)(24 31)(25 30)(26 29)(27 28)(54 96)(55 95)(56 94)(57 93)(58 92)(59 91)(60 90)(61 89)(62 88)(63 87)(64 86)(65 85)(66 84)(67 83)(68 82)(69 81)(70 80)(71 79)(72 78)(73 77)(74 76)(97 106)(98 105)(99 104)(100 103)(101 102)(107 177)(108 176)(109 175)(110 174)(111 173)(112 172)(113 171)(114 170)(115 169)(116 168)(117 167)(118 166)(119 165)(120 164)(121 163)(122 162)(123 161)(124 160)(125 212)(126 211)(127 210)(128 209)(129 208)(130 207)(131 206)(132 205)(133 204)(134 203)(135 202)(136 201)(137 200)(138 199)(139 198)(140 197)(141 196)(142 195)(143 194)(144 193)(145 192)(146 191)(147 190)(148 189)(149 188)(150 187)(151 186)(152 185)(153 184)(154 183)(155 182)(156 181)(157 180)(158 179)(159 178)

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

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

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

109 conjugacy classes

class 1 2A2B2C 4 53A···53Z106A···106BZ
order1222453···53106···106
size1121061062···22···2

109 irreducible representations

dim11112222
type+++++++
imageC1C2C2C2D4D53D106C53⋊D4
kernelC53⋊D4Dic53D106C2×C106C53C22C2C1
# reps11111262652

Matrix representation of C53⋊D4 in GL2(𝔽1061) generated by

01
10601012
,
198138
1046863
,
10
10121060
G:=sub<GL(2,GF(1061))| [0,1060,1,1012],[198,1046,138,863],[1,1012,0,1060] >;

C53⋊D4 in GAP, Magma, Sage, TeX

C_{53}\rtimes D_4
% in TeX

G:=Group("C53:D4");
// GroupNames label

G:=SmallGroup(424,8);
// by ID

G=gap.SmallGroup(424,8);
# by ID

G:=PCGroup([4,-2,-2,-2,-53,49,6659]);
// Polycyclic

G:=Group<a,b,c|a^53=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 C53⋊D4 in TeX

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