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G = D52.3C4order 416 = 25·13

The non-split extension by D52 of C4 acting through Inn(D52)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D52.3C4, C8.18D26, C52.37C23, Dic26.3C4, C104.23C22, (C2×C8)⋊7D13, (C8×D13)⋊6C2, C133(C8○D4), (C2×C104)⋊10C2, C8⋊D137C2, C52.41(C2×C4), D26.3(C2×C4), (C2×C4).78D26, C4.10(C4×D13), C13⋊D4.3C4, C52.4C411C2, C22.2(C4×D13), D525C2.6C2, (C2×C52).98C22, C26.27(C22×C4), Dic13.5(C2×C4), C4.37(C22×D13), C132C8.11C22, (C4×D13).23C22, C2.15(C2×C4×D13), (C2×C26).36(C2×C4), SmallGroup(416,122)

Series: Derived Chief Lower central Upper central

C1C26 — D52.3C4
C1C13C26C52C4×D13D525C2 — D52.3C4
C13C26 — D52.3C4
C1C8C2×C8

Generators and relations for D52.3C4
 G = < a,b,c | a52=b2=1, c4=a26, bab=a-1, ac=ca, bc=cb >

Subgroups: 344 in 62 conjugacy classes, 37 normal (23 characteristic)
C1, C2, C2 [×3], C4 [×2], C4 [×2], C22, C22 [×2], C8 [×2], C8 [×2], C2×C4, C2×C4 [×2], D4 [×3], Q8, C13, C2×C8, C2×C8 [×2], M4(2) [×3], C4○D4, D13 [×2], C26, C26, C8○D4, Dic13 [×2], C52 [×2], D26 [×2], C2×C26, C132C8 [×2], C104 [×2], Dic26, C4×D13 [×2], D52, C13⋊D4 [×2], C2×C52, C8×D13 [×2], C8⋊D13 [×2], C52.4C4, C2×C104, D525C2, D52.3C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], C23, C22×C4, D13, C8○D4, D26 [×3], C4×D13 [×2], C22×D13, C2×C4×D13, D52.3C4

Smallest permutation representation of D52.3C4
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)(53 104)(54 103)(55 102)(56 101)(57 100)(58 99)(59 98)(60 97)(61 96)(62 95)(63 94)(64 93)(65 92)(66 91)(67 90)(68 89)(69 88)(70 87)(71 86)(72 85)(73 84)(74 83)(75 82)(76 81)(77 80)(78 79)(105 140)(106 139)(107 138)(108 137)(109 136)(110 135)(111 134)(112 133)(113 132)(114 131)(115 130)(116 129)(117 128)(118 127)(119 126)(120 125)(121 124)(122 123)(141 156)(142 155)(143 154)(144 153)(145 152)(146 151)(147 150)(148 149)(157 208)(158 207)(159 206)(160 205)(161 204)(162 203)(163 202)(164 201)(165 200)(166 199)(167 198)(168 197)(169 196)(170 195)(171 194)(172 193)(173 192)(174 191)(175 190)(176 189)(177 188)(178 187)(179 186)(180 185)(181 184)(182 183)
(1 53 149 157 27 79 123 183)(2 54 150 158 28 80 124 184)(3 55 151 159 29 81 125 185)(4 56 152 160 30 82 126 186)(5 57 153 161 31 83 127 187)(6 58 154 162 32 84 128 188)(7 59 155 163 33 85 129 189)(8 60 156 164 34 86 130 190)(9 61 105 165 35 87 131 191)(10 62 106 166 36 88 132 192)(11 63 107 167 37 89 133 193)(12 64 108 168 38 90 134 194)(13 65 109 169 39 91 135 195)(14 66 110 170 40 92 136 196)(15 67 111 171 41 93 137 197)(16 68 112 172 42 94 138 198)(17 69 113 173 43 95 139 199)(18 70 114 174 44 96 140 200)(19 71 115 175 45 97 141 201)(20 72 116 176 46 98 142 202)(21 73 117 177 47 99 143 203)(22 74 118 178 48 100 144 204)(23 75 119 179 49 101 145 205)(24 76 120 180 50 102 146 206)(25 77 121 181 51 103 147 207)(26 78 122 182 52 104 148 208)

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

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

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

116 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E8A8B8C8D8E8F8G8H8I8J13A···13F26A···26R52A···52X104A···104AV
order1222244444888888888813···1326···2652···52104···104
size11226261122626111122262626262···22···22···22···2

116 irreducible representations

dim1111111112222222
type+++++++++
imageC1C2C2C2C2C2C4C4C4D13C8○D4D26D26C4×D13C4×D13D52.3C4
kernelD52.3C4C8×D13C8⋊D13C52.4C4C2×C104D525C2Dic26D52C13⋊D4C2×C8C13C8C2×C4C4C22C1
# reps12211122464126121248

Matrix representation of D52.3C4 in GL2(𝔽313) generated by

14269
244134
,
14269
48171
,
1880
0188
G:=sub<GL(2,GF(313))| [142,244,69,134],[142,48,69,171],[188,0,0,188] >;

D52.3C4 in GAP, Magma, Sage, TeX

D_{52}._3C_4
% in TeX

G:=Group("D52.3C4");
// GroupNames label

G:=SmallGroup(416,122);
// by ID

G=gap.SmallGroup(416,122);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,217,50,69,13829]);
// Polycyclic

G:=Group<a,b,c|a^52=b^2=1,c^4=a^26,b*a*b=a^-1,a*c=c*a,b*c=c*b>;
// generators/relations

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