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G = C4×D52order 416 = 25·13

Direct product of C4 and D52

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4×D52, C525D4, C424D13, C132(C4×D4), (C4×C52)⋊7C2, C527(C2×C4), C42(C4×D13), D262(C2×C4), C2.1(C2×D52), C26.2(C2×D4), C523C416C2, (C2×C4).75D26, (C2×D52).11C2, C26.4(C4○D4), D26⋊C417C2, (C2×C26).14C23, (C2×C52).86C22, C26.17(C22×C4), C2.3(D525C2), C22.11(C22×D13), (C2×Dic13).28C22, (C22×D13).18C22, (C2×C4×D13)⋊7C2, C2.6(C2×C4×D13), SmallGroup(416,94)

Series: Derived Chief Lower central Upper central

C1C26 — C4×D52
C1C13C26C2×C26C22×D13C2×D52 — C4×D52
C13C26 — C4×D52
C1C2×C4C42

Generators and relations for C4×D52
 G = < a,b,c | a4=b52=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 688 in 94 conjugacy classes, 45 normal (21 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×3], C22, C22 [×8], C2×C4 [×3], 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], C52 [×4], C52, D26 [×4], D26 [×4], C2×C26, C4×D13 [×4], D52 [×4], C2×Dic13 [×2], C2×C52 [×3], C22×D13 [×2], C523C4, D26⋊C4 [×2], C4×C52, C2×C4×D13 [×2], C2×D52, C4×D52
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], D52 [×2], C22×D13, C2×C4×D13, C2×D52, D525C2, C4×D52

Smallest permutation representation of C4×D52
On 208 points
Generators in S208
(1 203 105 86)(2 204 106 87)(3 205 107 88)(4 206 108 89)(5 207 109 90)(6 208 110 91)(7 157 111 92)(8 158 112 93)(9 159 113 94)(10 160 114 95)(11 161 115 96)(12 162 116 97)(13 163 117 98)(14 164 118 99)(15 165 119 100)(16 166 120 101)(17 167 121 102)(18 168 122 103)(19 169 123 104)(20 170 124 53)(21 171 125 54)(22 172 126 55)(23 173 127 56)(24 174 128 57)(25 175 129 58)(26 176 130 59)(27 177 131 60)(28 178 132 61)(29 179 133 62)(30 180 134 63)(31 181 135 64)(32 182 136 65)(33 183 137 66)(34 184 138 67)(35 185 139 68)(36 186 140 69)(37 187 141 70)(38 188 142 71)(39 189 143 72)(40 190 144 73)(41 191 145 74)(42 192 146 75)(43 193 147 76)(44 194 148 77)(45 195 149 78)(46 196 150 79)(47 197 151 80)(48 198 152 81)(49 199 153 82)(50 200 154 83)(51 201 155 84)(52 202 156 85)
(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 39)(2 38)(3 37)(4 36)(5 35)(6 34)(7 33)(8 32)(9 31)(10 30)(11 29)(12 28)(13 27)(14 26)(15 25)(16 24)(17 23)(18 22)(19 21)(40 52)(41 51)(42 50)(43 49)(44 48)(45 47)(54 104)(55 103)(56 102)(57 101)(58 100)(59 99)(60 98)(61 97)(62 96)(63 95)(64 94)(65 93)(66 92)(67 91)(68 90)(69 89)(70 88)(71 87)(72 86)(73 85)(74 84)(75 83)(76 82)(77 81)(78 80)(105 143)(106 142)(107 141)(108 140)(109 139)(110 138)(111 137)(112 136)(113 135)(114 134)(115 133)(116 132)(117 131)(118 130)(119 129)(120 128)(121 127)(122 126)(123 125)(144 156)(145 155)(146 154)(147 153)(148 152)(149 151)(157 183)(158 182)(159 181)(160 180)(161 179)(162 178)(163 177)(164 176)(165 175)(166 174)(167 173)(168 172)(169 171)(184 208)(185 207)(186 206)(187 205)(188 204)(189 203)(190 202)(191 201)(192 200)(193 199)(194 198)(195 197)

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

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

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

116 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L13A···13F26A···26R52A···52BT
order1222222244444444444413···1326···2652···52
size11112626262611112222262626262···22···22···2

116 irreducible representations

dim11111112222222
type++++++++++
imageC1C2C2C2C2C2C4D4C4○D4D13D26C4×D13D52D525C2
kernelC4×D52C523C4D26⋊C4C4×C52C2×C4×D13C2×D52D52C52C26C42C2×C4C4C4C2
# reps112121822618242424

Matrix representation of C4×D52 in GL3(𝔽53) generated by

3000
010
001
,
5200
0492
05114
,
5200
0926
0144
G:=sub<GL(3,GF(53))| [30,0,0,0,1,0,0,0,1],[52,0,0,0,49,51,0,2,14],[52,0,0,0,9,1,0,26,44] >;

C4×D52 in GAP, Magma, Sage, TeX

C_4\times D_{52}
% in TeX

G:=Group("C4xD52");
// GroupNames label

G:=SmallGroup(416,94);
// by ID

G=gap.SmallGroup(416,94);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,217,103,50,13829]);
// Polycyclic

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

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