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G = C22⋊Dic26order 416 = 25·13

The semidirect product of C22 and Dic26 acting via Dic26/Dic13=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C221Dic26, C23.12D26, Dic13.14D4, (C2×C26)⋊Q8, C523C42C2, (C2×C4).5D26, C2.6(D4×D13), C26.4(C2×Q8), C26.16(C2×D4), C131(C22⋊Q8), (C2×Dic26)⋊2C2, C26.D44C2, (C2×C52).1C22, C22⋊C4.1D13, C2.6(C2×Dic26), C26.21(C4○D4), (C2×C26).19C23, C23.D13.2C2, C2.6(D42D13), (C22×C26).8C22, (C2×Dic13).5C22, (C22×Dic13).3C2, C22.39(C22×D13), (C13×C22⋊C4).1C2, SmallGroup(416,99)

Series: Derived Chief Lower central Upper central

C1C2×C26 — C22⋊Dic26
C1C13C26C2×C26C2×Dic13C22×Dic13 — C22⋊Dic26
C13C2×C26 — C22⋊Dic26
C1C22C22⋊C4

Generators and relations for C22⋊Dic26
 G = < a,b,c,d | a2=b2=c52=1, d2=c26, cac-1=ab=ba, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 424 in 74 conjugacy classes, 35 normal (29 characteristic)
C1, C2 [×3], C2 [×2], C4 [×7], C22, C22 [×2], C22 [×2], C2×C4 [×2], C2×C4 [×6], Q8 [×2], C23, C13, C22⋊C4, C22⋊C4, C4⋊C4 [×3], C22×C4, C2×Q8, C26 [×3], C26 [×2], C22⋊Q8, Dic13 [×2], Dic13 [×3], C52 [×2], C2×C26, C2×C26 [×2], C2×C26 [×2], Dic26 [×2], C2×Dic13 [×4], C2×Dic13 [×2], C2×C52 [×2], C22×C26, C26.D4 [×2], C523C4, C23.D13, C13×C22⋊C4, C2×Dic26, C22×Dic13, C22⋊Dic26
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], Q8 [×2], C23, C2×D4, C2×Q8, C4○D4, D13, C22⋊Q8, D26 [×3], Dic26 [×2], C22×D13, C2×Dic26, D4×D13, D42D13, C22⋊Dic26

Smallest permutation representation of C22⋊Dic26
On 208 points
Generators in S208
(1 27)(2 100)(3 29)(4 102)(5 31)(6 104)(7 33)(8 54)(9 35)(10 56)(11 37)(12 58)(13 39)(14 60)(15 41)(16 62)(17 43)(18 64)(19 45)(20 66)(21 47)(22 68)(23 49)(24 70)(25 51)(26 72)(28 74)(30 76)(32 78)(34 80)(36 82)(38 84)(40 86)(42 88)(44 90)(46 92)(48 94)(50 96)(52 98)(53 79)(55 81)(57 83)(59 85)(61 87)(63 89)(65 91)(67 93)(69 95)(71 97)(73 99)(75 101)(77 103)(105 158)(106 132)(107 160)(108 134)(109 162)(110 136)(111 164)(112 138)(113 166)(114 140)(115 168)(116 142)(117 170)(118 144)(119 172)(120 146)(121 174)(122 148)(123 176)(124 150)(125 178)(126 152)(127 180)(128 154)(129 182)(130 156)(131 184)(133 186)(135 188)(137 190)(139 192)(141 194)(143 196)(145 198)(147 200)(149 202)(151 204)(153 206)(155 208)(157 183)(159 185)(161 187)(163 189)(165 191)(167 193)(169 195)(171 197)(173 199)(175 201)(177 203)(179 205)(181 207)
(1 73)(2 74)(3 75)(4 76)(5 77)(6 78)(7 79)(8 80)(9 81)(10 82)(11 83)(12 84)(13 85)(14 86)(15 87)(16 88)(17 89)(18 90)(19 91)(20 92)(21 93)(22 94)(23 95)(24 96)(25 97)(26 98)(27 99)(28 100)(29 101)(30 102)(31 103)(32 104)(33 53)(34 54)(35 55)(36 56)(37 57)(38 58)(39 59)(40 60)(41 61)(42 62)(43 63)(44 64)(45 65)(46 66)(47 67)(48 68)(49 69)(50 70)(51 71)(52 72)(105 184)(106 185)(107 186)(108 187)(109 188)(110 189)(111 190)(112 191)(113 192)(114 193)(115 194)(116 195)(117 196)(118 197)(119 198)(120 199)(121 200)(122 201)(123 202)(124 203)(125 204)(126 205)(127 206)(128 207)(129 208)(130 157)(131 158)(132 159)(133 160)(134 161)(135 162)(136 163)(137 164)(138 165)(139 166)(140 167)(141 168)(142 169)(143 170)(144 171)(145 172)(146 173)(147 174)(148 175)(149 176)(150 177)(151 178)(152 179)(153 180)(154 181)(155 182)(156 183)
(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)
(1 197 27 171)(2 196 28 170)(3 195 29 169)(4 194 30 168)(5 193 31 167)(6 192 32 166)(7 191 33 165)(8 190 34 164)(9 189 35 163)(10 188 36 162)(11 187 37 161)(12 186 38 160)(13 185 39 159)(14 184 40 158)(15 183 41 157)(16 182 42 208)(17 181 43 207)(18 180 44 206)(19 179 45 205)(20 178 46 204)(21 177 47 203)(22 176 48 202)(23 175 49 201)(24 174 50 200)(25 173 51 199)(26 172 52 198)(53 138 79 112)(54 137 80 111)(55 136 81 110)(56 135 82 109)(57 134 83 108)(58 133 84 107)(59 132 85 106)(60 131 86 105)(61 130 87 156)(62 129 88 155)(63 128 89 154)(64 127 90 153)(65 126 91 152)(66 125 92 151)(67 124 93 150)(68 123 94 149)(69 122 95 148)(70 121 96 147)(71 120 97 146)(72 119 98 145)(73 118 99 144)(74 117 100 143)(75 116 101 142)(76 115 102 141)(77 114 103 140)(78 113 104 139)

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

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

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

74 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H13A···13F26A···26R26S···26AD52A···52X
order1222224444444413···1326···2626···2652···52
size111122442626262652522···22···24···44···4

74 irreducible representations

dim1111111222222244
type++++++++-+++-+-
imageC1C2C2C2C2C2C2D4Q8C4○D4D13D26D26Dic26D4×D13D42D13
kernelC22⋊Dic26C26.D4C523C4C23.D13C13×C22⋊C4C2×Dic26C22×Dic13Dic13C2×C26C26C22⋊C4C2×C4C23C22C2C2
# reps121111122261262466

Matrix representation of C22⋊Dic26 in GL6(𝔽53)

100000
010000
0052000
0005200
0000520
000001
,
100000
010000
001000
000100
0000520
0000052
,
9520000
51180000
006800
0024700
000001
0000520
,
45100000
5280000
0021500
0073200
0000230
0000030

G:=sub<GL(6,GF(53))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,52,0,0,0,0,0,0,52,0,0,0,0,0,0,52,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,52,0,0,0,0,0,0,52],[9,51,0,0,0,0,52,18,0,0,0,0,0,0,6,2,0,0,0,0,8,47,0,0,0,0,0,0,0,52,0,0,0,0,1,0],[45,52,0,0,0,0,10,8,0,0,0,0,0,0,21,7,0,0,0,0,5,32,0,0,0,0,0,0,23,0,0,0,0,0,0,30] >;

C22⋊Dic26 in GAP, Magma, Sage, TeX

C_2^2\rtimes {\rm Dic}_{26}
% in TeX

G:=Group("C2^2:Dic26");
// GroupNames label

G:=SmallGroup(416,99);
// by ID

G=gap.SmallGroup(416,99);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,48,218,188,50,13829]);
// Polycyclic

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

׿
×
𝔽