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G = D528C4order 416 = 25·13

5th semidirect product of D52 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D528C4, Dic135D4, C134(C4×D4), C525(C2×C4), C4⋊C48D13, C41(C4×D13), D264(C2×C4), C2.4(D4×D13), (C2×D52).8C2, C26.24(C2×D4), (C2×C4).31D26, (C4×Dic13)⋊3C2, D26⋊C412C2, C26.33(C4○D4), C26.24(C22×C4), (C2×C26).34C23, (C2×C52).24C22, C2.2(D52⋊C2), C22.18(C22×D13), (C2×Dic13).63C22, (C22×D13).23C22, (C13×C4⋊C4)⋊4C2, (C2×C4×D13)⋊12C2, C2.13(C2×C4×D13), SmallGroup(416,114)

Series: Derived Chief Lower central Upper central

C1C26 — D528C4
C1C13C26C2×C26C22×D13C2×D52 — D528C4
C13C26 — D528C4
C1C22C4⋊C4

Generators and relations for D528C4
 G = < a,b,c | a52=b2=c4=1, bab=a-1, cac-1=a27, cbc-1=a26b >

Subgroups: 688 in 94 conjugacy classes, 43 normal (19 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×5], C22, C22 [×8], C2×C4, C2×C4 [×2], C2×C4 [×6], D4 [×4], C23 [×2], C13, C42, C22⋊C4 [×2], C4⋊C4, C22×C4 [×2], C2×D4, D13 [×4], C26 [×3], C4×D4, Dic13 [×2], Dic13, C52 [×2], C52 [×2], D26 [×4], D26 [×4], C2×C26, C4×D13 [×4], D52 [×4], C2×Dic13 [×2], C2×C52, C2×C52 [×2], C22×D13 [×2], C4×Dic13, D26⋊C4 [×2], C13×C4⋊C4, C2×C4×D13 [×2], C2×D52, D528C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, C22×C4, C2×D4, C4○D4, D13, C4×D4, D26 [×3], C4×D13 [×2], C22×D13, C2×C4×D13, D4×D13, D52⋊C2, D528C4

Smallest permutation representation of D528C4
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 56)(54 55)(57 104)(58 103)(59 102)(60 101)(61 100)(62 99)(63 98)(64 97)(65 96)(66 95)(67 94)(68 93)(69 92)(70 91)(71 90)(72 89)(73 88)(74 87)(75 86)(76 85)(77 84)(78 83)(79 82)(80 81)(105 124)(106 123)(107 122)(108 121)(109 120)(110 119)(111 118)(112 117)(113 116)(114 115)(125 156)(126 155)(127 154)(128 153)(129 152)(130 151)(131 150)(132 149)(133 148)(134 147)(135 146)(136 145)(137 144)(138 143)(139 142)(140 141)(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)(172 181)(173 180)(174 179)(175 178)(176 177)(197 208)(198 207)(199 206)(200 205)(201 204)(202 203)
(1 203 81 115)(2 178 82 142)(3 205 83 117)(4 180 84 144)(5 207 85 119)(6 182 86 146)(7 157 87 121)(8 184 88 148)(9 159 89 123)(10 186 90 150)(11 161 91 125)(12 188 92 152)(13 163 93 127)(14 190 94 154)(15 165 95 129)(16 192 96 156)(17 167 97 131)(18 194 98 106)(19 169 99 133)(20 196 100 108)(21 171 101 135)(22 198 102 110)(23 173 103 137)(24 200 104 112)(25 175 53 139)(26 202 54 114)(27 177 55 141)(28 204 56 116)(29 179 57 143)(30 206 58 118)(31 181 59 145)(32 208 60 120)(33 183 61 147)(34 158 62 122)(35 185 63 149)(36 160 64 124)(37 187 65 151)(38 162 66 126)(39 189 67 153)(40 164 68 128)(41 191 69 155)(42 166 70 130)(43 193 71 105)(44 168 72 132)(45 195 73 107)(46 170 74 134)(47 197 75 109)(48 172 76 136)(49 199 77 111)(50 174 78 138)(51 201 79 113)(52 176 80 140)

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

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

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

80 conjugacy classes

class 1 2A2B2C2D2E2F2G4A···4F4G4H4I4J4K4L13A···13F26A···26R52A···52AJ
order122222224···444444413···1326···2652···52
size1111262626262···21313131326262···22···24···4

80 irreducible representations

dim11111112222244
type+++++++++++
imageC1C2C2C2C2C2C4D4C4○D4D13D26C4×D13D4×D13D52⋊C2
kernelD528C4C4×Dic13D26⋊C4C13×C4⋊C4C2×C4×D13C2×D52D52Dic13C26C4⋊C4C2×C4C4C2C2
# reps1121218226182466

Matrix representation of D528C4 in GL4(𝔽53) generated by

491000
143100
001819
00835
,
92700
524400
001819
003635
,
30000
03000
00300
003823
G:=sub<GL(4,GF(53))| [49,14,0,0,10,31,0,0,0,0,18,8,0,0,19,35],[9,52,0,0,27,44,0,0,0,0,18,36,0,0,19,35],[30,0,0,0,0,30,0,0,0,0,30,38,0,0,0,23] >;

D528C4 in GAP, Magma, Sage, TeX

D_{52}\rtimes_8C_4
% in TeX

G:=Group("D52:8C4");
// GroupNames label

G:=SmallGroup(416,114);
// by ID

G=gap.SmallGroup(416,114);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,217,103,188,50,13829]);
// 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^26*b>;
// generators/relations

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