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G = C22.D52order 416 = 25·13

3rd non-split extension by C22 of D52 acting via D52/D26=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C22.4D52, C23.16D26, C523C45C2, C2.8(C2×D52), (C2×C26).4D4, C26.6(C2×D4), (C2×C4).9D26, C22⋊C46D13, D26⋊C47C2, (C2×C52).3C22, C26.23(C4○D4), (C2×C26).27C23, (C22×Dic13)⋊2C2, C132(C22.D4), C2.10(D42D13), (C22×C26).16C22, (C22×D13).5C22, C22.45(C22×D13), (C2×Dic13).31C22, (C13×C22⋊C4)⋊4C2, (C2×C13⋊D4).5C2, SmallGroup(416,107)

Series: Derived Chief Lower central Upper central

C1C2×C26 — C22.D52
C1C13C26C2×C26C22×D13C2×C13⋊D4 — C22.D52
C13C2×C26 — C22.D52
C1C22C22⋊C4

Generators and relations for C22.D52
 G = < a,b,c,d | a2=b2=c52=1, d2=b, cac-1=ab=ba, ad=da, bc=cb, bd=db, dcd-1=bc-1 >

Subgroups: 536 in 78 conjugacy classes, 33 normal (15 characteristic)
C1, C2, C2 [×2], C2 [×3], C4 [×5], C22, C22 [×2], C22 [×5], C2×C4 [×2], C2×C4 [×5], D4 [×2], C23, C23, C13, C22⋊C4, C22⋊C4 [×2], C4⋊C4 [×2], C22×C4, C2×D4, D13, C26, C26 [×2], C26 [×2], C22.D4, Dic13 [×3], C52 [×2], D26 [×3], C2×C26, C2×C26 [×2], C2×C26 [×2], C2×Dic13, C2×Dic13 [×2], C2×Dic13 [×2], C13⋊D4 [×2], C2×C52 [×2], C22×D13, C22×C26, C523C4 [×2], D26⋊C4 [×2], C13×C22⋊C4, C22×Dic13, C2×C13⋊D4, C22.D52
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, C2×D4, C4○D4 [×2], D13, C22.D4, D26 [×3], D52 [×2], C22×D13, C2×D52, D42D13 [×2], C22.D52

Smallest permutation representation of C22.D52
On 208 points
Generators in S208
(2 152)(4 154)(6 156)(8 106)(10 108)(12 110)(14 112)(16 114)(18 116)(20 118)(22 120)(24 122)(26 124)(28 126)(30 128)(32 130)(34 132)(36 134)(38 136)(40 138)(42 140)(44 142)(46 144)(48 146)(50 148)(52 150)(54 162)(56 164)(58 166)(60 168)(62 170)(64 172)(66 174)(68 176)(70 178)(72 180)(74 182)(76 184)(78 186)(80 188)(82 190)(84 192)(86 194)(88 196)(90 198)(92 200)(94 202)(96 204)(98 206)(100 208)(102 158)(104 160)
(1 151)(2 152)(3 153)(4 154)(5 155)(6 156)(7 105)(8 106)(9 107)(10 108)(11 109)(12 110)(13 111)(14 112)(15 113)(16 114)(17 115)(18 116)(19 117)(20 118)(21 119)(22 120)(23 121)(24 122)(25 123)(26 124)(27 125)(28 126)(29 127)(30 128)(31 129)(32 130)(33 131)(34 132)(35 133)(36 134)(37 135)(38 136)(39 137)(40 138)(41 139)(42 140)(43 141)(44 142)(45 143)(46 144)(47 145)(48 146)(49 147)(50 148)(51 149)(52 150)(53 161)(54 162)(55 163)(56 164)(57 165)(58 166)(59 167)(60 168)(61 169)(62 170)(63 171)(64 172)(65 173)(66 174)(67 175)(68 176)(69 177)(70 178)(71 179)(72 180)(73 181)(74 182)(75 183)(76 184)(77 185)(78 186)(79 187)(80 188)(81 189)(82 190)(83 191)(84 192)(85 193)(86 194)(87 195)(88 196)(89 197)(90 198)(91 199)(92 200)(93 201)(94 202)(95 203)(96 204)(97 205)(98 206)(99 207)(100 208)(101 157)(102 158)(103 159)(104 160)
(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)
(1 201 151 93)(2 92 152 200)(3 199 153 91)(4 90 154 198)(5 197 155 89)(6 88 156 196)(7 195 105 87)(8 86 106 194)(9 193 107 85)(10 84 108 192)(11 191 109 83)(12 82 110 190)(13 189 111 81)(14 80 112 188)(15 187 113 79)(16 78 114 186)(17 185 115 77)(18 76 116 184)(19 183 117 75)(20 74 118 182)(21 181 119 73)(22 72 120 180)(23 179 121 71)(24 70 122 178)(25 177 123 69)(26 68 124 176)(27 175 125 67)(28 66 126 174)(29 173 127 65)(30 64 128 172)(31 171 129 63)(32 62 130 170)(33 169 131 61)(34 60 132 168)(35 167 133 59)(36 58 134 166)(37 165 135 57)(38 56 136 164)(39 163 137 55)(40 54 138 162)(41 161 139 53)(42 104 140 160)(43 159 141 103)(44 102 142 158)(45 157 143 101)(46 100 144 208)(47 207 145 99)(48 98 146 206)(49 205 147 97)(50 96 148 204)(51 203 149 95)(52 94 150 202)

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

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

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

74 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G13A···13F26A···26R26S···26AD52A···52X
order1222222444444413···1326···2626···2652···52
size111122524426262626522···22···24···44···4

74 irreducible representations

dim1111112222224
type+++++++++++-
imageC1C2C2C2C2C2D4C4○D4D13D26D26D52D42D13
kernelC22.D52C523C4D26⋊C4C13×C22⋊C4C22×Dic13C2×C13⋊D4C2×C26C26C22⋊C4C2×C4C23C22C2
# reps1221112461262412

Matrix representation of C22.D52 in GL4(𝔽53) generated by

1000
0100
0010
00052
,
1000
0100
00520
00052
,
30200
443300
0001
00520
,
241400
462900
00230
00023
G:=sub<GL(4,GF(53))| [1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,52],[1,0,0,0,0,1,0,0,0,0,52,0,0,0,0,52],[30,44,0,0,2,33,0,0,0,0,0,52,0,0,1,0],[24,46,0,0,14,29,0,0,0,0,23,0,0,0,0,23] >;

C22.D52 in GAP, Magma, Sage, TeX

C_2^2.D_{52}
% in TeX

G:=Group("C2^2.D52");
// GroupNames label

G:=SmallGroup(416,107);
// by ID

G=gap.SmallGroup(416,107);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,217,218,188,122,13829]);
// Polycyclic

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

׿
×
𝔽