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G = D525C4order 416 = 25·13

2nd semidirect product of D52 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D525C4, C26.5D8, C52.45D4, C2.2D104, C26.3SD16, C22.10D52, (C2×C8)⋊2D13, (C2×C104)⋊2C2, C523C41C2, C4.8(C4×D13), C52.39(C2×C4), (C2×D52).1C2, (C2×C4).71D26, (C2×C26).15D4, C133(D4⋊C4), C2.3(C104⋊C2), C4.20(C13⋊D4), (C2×C52).83C22, C2.8(D26⋊C4), C26.18(C22⋊C4), SmallGroup(416,28)

Series: Derived Chief Lower central Upper central

C1C52 — D525C4
C1C13C26C52C2×C52C2×D52 — D525C4
C13C26C52 — D525C4
C1C22C2×C4C2×C8

Generators and relations for D525C4
 G = < a,b,c | a52=b2=c4=1, bab=cac-1=a-1, cbc-1=a11b >

52C2
52C2
26C22
26C22
52C22
52C22
52C4
4D13
4D13
2C8
13D4
13D4
26C2×C4
26C23
26D4
2D26
2D26
4Dic13
4D26
4D26
13C4⋊C4
13C2×D4
2C22×D13
2C2×Dic13
2C104
2D52
13D4⋊C4

Smallest permutation representation of D525C4
On 208 points
Generators in S208
(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)
(1 65)(2 64)(3 63)(4 62)(5 61)(6 60)(7 59)(8 58)(9 57)(10 56)(11 55)(12 54)(13 53)(14 104)(15 103)(16 102)(17 101)(18 100)(19 99)(20 98)(21 97)(22 96)(23 95)(24 94)(25 93)(26 92)(27 91)(28 90)(29 89)(30 88)(31 87)(32 86)(33 85)(34 84)(35 83)(36 82)(37 81)(38 80)(39 79)(40 78)(41 77)(42 76)(43 75)(44 74)(45 73)(46 72)(47 71)(48 70)(49 69)(50 68)(51 67)(52 66)(105 164)(106 163)(107 162)(108 161)(109 160)(110 159)(111 158)(112 157)(113 208)(114 207)(115 206)(116 205)(117 204)(118 203)(119 202)(120 201)(121 200)(122 199)(123 198)(124 197)(125 196)(126 195)(127 194)(128 193)(129 192)(130 191)(131 190)(132 189)(133 188)(134 187)(135 186)(136 185)(137 184)(138 183)(139 182)(140 181)(141 180)(142 179)(143 178)(144 177)(145 176)(146 175)(147 174)(148 173)(149 172)(150 171)(151 170)(152 169)(153 168)(154 167)(155 166)(156 165)
(1 145 92 190)(2 144 93 189)(3 143 94 188)(4 142 95 187)(5 141 96 186)(6 140 97 185)(7 139 98 184)(8 138 99 183)(9 137 100 182)(10 136 101 181)(11 135 102 180)(12 134 103 179)(13 133 104 178)(14 132 53 177)(15 131 54 176)(16 130 55 175)(17 129 56 174)(18 128 57 173)(19 127 58 172)(20 126 59 171)(21 125 60 170)(22 124 61 169)(23 123 62 168)(24 122 63 167)(25 121 64 166)(26 120 65 165)(27 119 66 164)(28 118 67 163)(29 117 68 162)(30 116 69 161)(31 115 70 160)(32 114 71 159)(33 113 72 158)(34 112 73 157)(35 111 74 208)(36 110 75 207)(37 109 76 206)(38 108 77 205)(39 107 78 204)(40 106 79 203)(41 105 80 202)(42 156 81 201)(43 155 82 200)(44 154 83 199)(45 153 84 198)(46 152 85 197)(47 151 86 196)(48 150 87 195)(49 149 88 194)(50 148 89 193)(51 147 90 192)(52 146 91 191)

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

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

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

110 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D8A8B8C8D13A···13F26A···26R52A···52X104A···104AV
order1222224444888813···1326···2652···52104···104
size1111525222525222222···22···22···22···2

110 irreducible representations

dim1111122222222222
type+++++++++++
imageC1C2C2C2C4D4D4D8SD16D13D26C4×D13C13⋊D4D52C104⋊C2D104
kernelD525C4C523C4C2×C104C2×D52D52C52C2×C26C26C26C2×C8C2×C4C4C4C22C2C2
# reps111141122661212122424

Matrix representation of D525C4 in GL3(𝔽313) generated by

100
015569
0172121
,
100
0204202
0217109
,
2500
0116272
0290197
G:=sub<GL(3,GF(313))| [1,0,0,0,155,172,0,69,121],[1,0,0,0,204,217,0,202,109],[25,0,0,0,116,290,0,272,197] >;

D525C4 in GAP, Magma, Sage, TeX

D_{52}\rtimes_5C_4
% in TeX

G:=Group("D52:5C4");
// GroupNames label

G:=SmallGroup(416,28);
// by ID

G=gap.SmallGroup(416,28);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,73,79,362,86,13829]);
// Polycyclic

G:=Group<a,b,c|a^52=b^2=c^4=1,b*a*b=c*a*c^-1=a^-1,c*b*c^-1=a^11*b>;
// generators/relations

Export

Subgroup lattice of D525C4 in TeX

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