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G = D526C4order 416 = 25·13

3rd semidirect product of D52 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D526C4, C52.1D4, C4.9D52, C26.7D8, C26.7SD16, C4⋊C41D13, C4.1(C4×D13), C52.24(C2×C4), (C2×D52).6C2, (C2×C4).35D26, (C2×C26).30D4, C132(D4⋊C4), C2.2(D4⋊D13), C2.2(Q8⋊D13), (C2×C52).10C22, C2.5(D26⋊C4), C26.14(C22⋊C4), C22.14(C13⋊D4), (C13×C4⋊C4)⋊1C2, (C2×C132C8)⋊1C2, SmallGroup(416,16)

Series: Derived Chief Lower central Upper central

C1C52 — D526C4
C1C13C26C2×C26C2×C52C2×D52 — D526C4
C13C26C52 — D526C4
C1C22C2×C4C4⋊C4

Generators and relations for D526C4
 G = < a,b,c | a52=b2=c4=1, bab=a-1, cac-1=a27, cbc-1=a39b >

52C2
52C2
4C4
26C22
26C22
52C22
52C22
4D13
4D13
2C2×C4
13D4
13D4
26C8
26D4
26C23
2D26
2D26
4D26
4D26
4C52
13C2×C8
13C2×D4
2C132C8
2C2×C52
2C22×D13
2D52
13D4⋊C4

Smallest permutation representation of D526C4
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 52)(2 51)(3 50)(4 49)(5 48)(6 47)(7 46)(8 45)(9 44)(10 43)(11 42)(12 41)(13 40)(14 39)(15 38)(16 37)(17 36)(18 35)(19 34)(20 33)(21 32)(22 31)(23 30)(24 29)(25 28)(26 27)(53 62)(54 61)(55 60)(56 59)(57 58)(63 104)(64 103)(65 102)(66 101)(67 100)(68 99)(69 98)(70 97)(71 96)(72 95)(73 94)(74 93)(75 92)(76 91)(77 90)(78 89)(79 88)(80 87)(81 86)(82 85)(83 84)(106 156)(107 155)(108 154)(109 153)(110 152)(111 151)(112 150)(113 149)(114 148)(115 147)(116 146)(117 145)(118 144)(119 143)(120 142)(121 141)(122 140)(123 139)(124 138)(125 137)(126 136)(127 135)(128 134)(129 133)(130 132)(157 171)(158 170)(159 169)(160 168)(161 167)(162 166)(163 165)(172 208)(173 207)(174 206)(175 205)(176 204)(177 203)(178 202)(179 201)(180 200)(181 199)(182 198)(183 197)(184 196)(185 195)(186 194)(187 193)(188 192)(189 191)
(1 184 84 151)(2 159 85 126)(3 186 86 153)(4 161 87 128)(5 188 88 155)(6 163 89 130)(7 190 90 105)(8 165 91 132)(9 192 92 107)(10 167 93 134)(11 194 94 109)(12 169 95 136)(13 196 96 111)(14 171 97 138)(15 198 98 113)(16 173 99 140)(17 200 100 115)(18 175 101 142)(19 202 102 117)(20 177 103 144)(21 204 104 119)(22 179 53 146)(23 206 54 121)(24 181 55 148)(25 208 56 123)(26 183 57 150)(27 158 58 125)(28 185 59 152)(29 160 60 127)(30 187 61 154)(31 162 62 129)(32 189 63 156)(33 164 64 131)(34 191 65 106)(35 166 66 133)(36 193 67 108)(37 168 68 135)(38 195 69 110)(39 170 70 137)(40 197 71 112)(41 172 72 139)(42 199 73 114)(43 174 74 141)(44 201 75 116)(45 176 76 143)(46 203 77 118)(47 178 78 145)(48 205 79 120)(49 180 80 147)(50 207 81 122)(51 182 82 149)(52 157 83 124)

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

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

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

74 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D8A8B8C8D13A···13F26A···26R52A···52AJ
order1222224444888813···1326···2652···52
size111152522244262626262···22···24···4

74 irreducible representations

dim1111122222222244
type++++++++++++
imageC1C2C2C2C4D4D4D8SD16D13D26C4×D13D52C13⋊D4D4⋊D13Q8⋊D13
kernelD526C4C2×C132C8C13×C4⋊C4C2×D52D52C52C2×C26C26C26C4⋊C4C2×C4C4C4C22C2C2
# reps1111411226612121266

Matrix representation of D526C4 in GL4(𝔽313) generated by

21128800
772500
009152
00238222
,
29331200
862000
009152
00226222
,
3074200
96600
0011410
00109199
G:=sub<GL(4,GF(313))| [211,77,0,0,288,25,0,0,0,0,91,238,0,0,52,222],[293,86,0,0,312,20,0,0,0,0,91,226,0,0,52,222],[307,96,0,0,42,6,0,0,0,0,114,109,0,0,10,199] >;

D526C4 in GAP, Magma, Sage, TeX

D_{52}\rtimes_6C_4
% in TeX

G:=Group("D52:6C4");
// GroupNames label

G:=SmallGroup(416,16);
// by ID

G=gap.SmallGroup(416,16);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,121,31,579,297,69,13829]);
// Polycyclic

G:=Group<a,b,c|a^52=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^27,c*b*c^-1=a^39*b>;
// generators/relations

Export

Subgroup lattice of D526C4 in TeX

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