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G = C52.C23order 416 = 25·13

18th non-split extension by C52 of C23 acting via C23/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4.8D26, C52.57D4, Q8.8D26, C52.18C23, D52.12C22, Dic26.11C22, D4⋊D137C2, C4○D42D13, C135(C4○D8), Q8⋊D137C2, (C2×C26).9D4, D4.D137C2, C13⋊Q167C2, C26.60(C2×D4), (C2×C4).59D26, D525C24C2, C4.32(C13⋊D4), (C2×C52).43C22, (D4×C13).8C22, C4.18(C22×D13), (Q8×C13).8C22, C132C8.10C22, C22.1(C13⋊D4), (C2×C132C8)⋊8C2, (C13×C4○D4)⋊2C2, C2.24(C2×C13⋊D4), SmallGroup(416,171)

Series: Derived Chief Lower central Upper central

C1C52 — C52.C23
C1C13C26C52D52D525C2 — C52.C23
C13C26C52 — C52.C23
C1C4C2×C4C4○D4

Generators and relations for C52.C23
 G = < a,b,c,d | a52=b2=d2=1, c2=a26, bab=a-1, ac=ca, dad=a27, bc=cb, dbd=a13b, cd=dc >

Subgroups: 368 in 62 conjugacy classes, 29 normal (all characteristic)
C1, C2, C2 [×3], C4 [×2], C4 [×2], C22, C22 [×2], C8 [×2], C2×C4, C2×C4 [×2], D4, D4 [×3], Q8, Q8, C13, C2×C8, D8, SD16 [×2], Q16, C4○D4, C4○D4, D13, C26, C26 [×2], C4○D8, Dic13, C52 [×2], C52, D26, C2×C26, C2×C26, C132C8 [×2], Dic26, C4×D13, D52, C13⋊D4, C2×C52, C2×C52, D4×C13, D4×C13, Q8×C13, C2×C132C8, D4⋊D13, D4.D13, Q8⋊D13, C13⋊Q16, D525C2, C13×C4○D4, C52.C23
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, C2×D4, D13, C4○D8, D26 [×3], C13⋊D4 [×2], C22×D13, C2×C13⋊D4, C52.C23

Smallest permutation representation of C52.C23
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)
(1 135)(2 134)(3 133)(4 132)(5 131)(6 130)(7 129)(8 128)(9 127)(10 126)(11 125)(12 124)(13 123)(14 122)(15 121)(16 120)(17 119)(18 118)(19 117)(20 116)(21 115)(22 114)(23 113)(24 112)(25 111)(26 110)(27 109)(28 108)(29 107)(30 106)(31 105)(32 156)(33 155)(34 154)(35 153)(36 152)(37 151)(38 150)(39 149)(40 148)(41 147)(42 146)(43 145)(44 144)(45 143)(46 142)(47 141)(48 140)(49 139)(50 138)(51 137)(52 136)(53 178)(54 177)(55 176)(56 175)(57 174)(58 173)(59 172)(60 171)(61 170)(62 169)(63 168)(64 167)(65 166)(66 165)(67 164)(68 163)(69 162)(70 161)(71 160)(72 159)(73 158)(74 157)(75 208)(76 207)(77 206)(78 205)(79 204)(80 203)(81 202)(82 201)(83 200)(84 199)(85 198)(86 197)(87 196)(88 195)(89 194)(90 193)(91 192)(92 191)(93 190)(94 189)(95 188)(96 187)(97 186)(98 185)(99 184)(100 183)(101 182)(102 181)(103 180)(104 179)
(1 40 27 14)(2 41 28 15)(3 42 29 16)(4 43 30 17)(5 44 31 18)(6 45 32 19)(7 46 33 20)(8 47 34 21)(9 48 35 22)(10 49 36 23)(11 50 37 24)(12 51 38 25)(13 52 39 26)(53 66 79 92)(54 67 80 93)(55 68 81 94)(56 69 82 95)(57 70 83 96)(58 71 84 97)(59 72 85 98)(60 73 86 99)(61 74 87 100)(62 75 88 101)(63 76 89 102)(64 77 90 103)(65 78 91 104)(105 118 131 144)(106 119 132 145)(107 120 133 146)(108 121 134 147)(109 122 135 148)(110 123 136 149)(111 124 137 150)(112 125 138 151)(113 126 139 152)(114 127 140 153)(115 128 141 154)(116 129 142 155)(117 130 143 156)(157 196 183 170)(158 197 184 171)(159 198 185 172)(160 199 186 173)(161 200 187 174)(162 201 188 175)(163 202 189 176)(164 203 190 177)(165 204 191 178)(166 205 192 179)(167 206 193 180)(168 207 194 181)(169 208 195 182)
(1 68)(2 95)(3 70)(4 97)(5 72)(6 99)(7 74)(8 101)(9 76)(10 103)(11 78)(12 53)(13 80)(14 55)(15 82)(16 57)(17 84)(18 59)(19 86)(20 61)(21 88)(22 63)(23 90)(24 65)(25 92)(26 67)(27 94)(28 69)(29 96)(30 71)(31 98)(32 73)(33 100)(34 75)(35 102)(36 77)(37 104)(38 79)(39 54)(40 81)(41 56)(42 83)(43 58)(44 85)(45 60)(46 87)(47 62)(48 89)(49 64)(50 91)(51 66)(52 93)(105 172)(106 199)(107 174)(108 201)(109 176)(110 203)(111 178)(112 205)(113 180)(114 207)(115 182)(116 157)(117 184)(118 159)(119 186)(120 161)(121 188)(122 163)(123 190)(124 165)(125 192)(126 167)(127 194)(128 169)(129 196)(130 171)(131 198)(132 173)(133 200)(134 175)(135 202)(136 177)(137 204)(138 179)(139 206)(140 181)(141 208)(142 183)(143 158)(144 185)(145 160)(146 187)(147 162)(148 189)(149 164)(150 191)(151 166)(152 193)(153 168)(154 195)(155 170)(156 197)

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

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

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

74 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E8A8B8C8D13A···13F26A···26F26G···26X52A···52L52M···52AD
order1222244444888813···1326···2626···2652···5252···52
size112452112452262626262···22···24···42···24···4

74 irreducible representations

dim111111112222222224
type++++++++++++++
imageC1C2C2C2C2C2C2C2D4D4D13C4○D8D26D26D26C13⋊D4C13⋊D4C52.C23
kernelC52.C23C2×C132C8D4⋊D13D4.D13Q8⋊D13C13⋊Q16D525C2C13×C4○D4C52C2×C26C4○D4C13C2×C4D4Q8C4C22C1
# reps111111111164666121212

Matrix representation of C52.C23 in GL4(𝔽313) generated by

15128800
13423900
00250
0018288
,
14616500
7016700
005160
00172308
,
1000
0100
002880
000288
,
1000
0100
00312281
0001
G:=sub<GL(4,GF(313))| [151,134,0,0,288,239,0,0,0,0,25,18,0,0,0,288],[146,70,0,0,165,167,0,0,0,0,5,172,0,0,160,308],[1,0,0,0,0,1,0,0,0,0,288,0,0,0,0,288],[1,0,0,0,0,1,0,0,0,0,312,0,0,0,281,1] >;

C52.C23 in GAP, Magma, Sage, TeX

C_{52}.C_2^3
% in TeX

G:=Group("C52.C2^3");
// GroupNames label

G:=SmallGroup(416,171);
// by ID

G=gap.SmallGroup(416,171);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,103,218,579,159,69,13829]);
// Polycyclic

G:=Group<a,b,c,d|a^52=b^2=d^2=1,c^2=a^26,b*a*b=a^-1,a*c=c*a,d*a*d=a^27,b*c=c*b,d*b*d=a^13*b,c*d=d*c>;
// generators/relations

׿
×
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