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G = C13×C8.C22order 416 = 25·13

Direct product of C13 and C8.C22

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Aliases: C13×C8.C22, Q162C26, C52.64D4, SD162C26, M4(2)⋊2C26, C52.49C23, C104.13C22, C8.(C2×C26), (C2×Q8)⋊4C26, (C13×Q16)⋊6C2, (Q8×C26)⋊11C2, C4○D4.2C26, D4.3(C2×C26), (C2×C26).25D4, C2.16(D4×C26), C4.15(D4×C13), C26.79(C2×D4), Q8.3(C2×C26), (C13×SD16)⋊6C2, C4.6(C22×C26), C22.6(D4×C13), (C13×M4(2))⋊6C2, (C2×C52).70C22, (D4×C13).13C22, (Q8×C13).14C22, (C2×C4).11(C2×C26), (C13×C4○D4).5C2, SmallGroup(416,198)

Series: Derived Chief Lower central Upper central

C1C4 — C13×C8.C22
C1C2C4C52D4×C13C13×SD16 — C13×C8.C22
C1C2C4 — C13×C8.C22
C1C26C2×C52 — C13×C8.C22

Generators and relations for C13×C8.C22
 G = < a,b,c,d | a13=b8=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b3, dbd=b5, dcd=b4c >

Subgroups: 84 in 60 conjugacy classes, 40 normal (24 characteristic)
C1, C2, C2 [×2], C4 [×2], C4 [×3], C22, C22, C8 [×2], C2×C4, C2×C4 [×2], D4, D4, Q8, Q8 [×2], Q8, C13, M4(2), SD16 [×2], Q16 [×2], C2×Q8, C4○D4, C26, C26 [×2], C8.C22, C52 [×2], C52 [×3], C2×C26, C2×C26, C104 [×2], C2×C52, C2×C52 [×2], D4×C13, D4×C13, Q8×C13, Q8×C13 [×2], Q8×C13, C13×M4(2), C13×SD16 [×2], C13×Q16 [×2], Q8×C26, C13×C4○D4, C13×C8.C22
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, C13, C2×D4, C26 [×7], C8.C22, C2×C26 [×7], D4×C13 [×2], C22×C26, D4×C26, C13×C8.C22

Smallest permutation representation of C13×C8.C22
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)
(1 107 159 180 201 81 50 132)(2 108 160 181 202 82 51 133)(3 109 161 182 203 83 52 134)(4 110 162 170 204 84 40 135)(5 111 163 171 205 85 41 136)(6 112 164 172 206 86 42 137)(7 113 165 173 207 87 43 138)(8 114 166 174 208 88 44 139)(9 115 167 175 196 89 45 140)(10 116 168 176 197 90 46 141)(11 117 169 177 198 91 47 142)(12 105 157 178 199 79 48 143)(13 106 158 179 200 80 49 131)(14 190 155 73 122 58 104 31)(15 191 156 74 123 59 92 32)(16 192 144 75 124 60 93 33)(17 193 145 76 125 61 94 34)(18 194 146 77 126 62 95 35)(19 195 147 78 127 63 96 36)(20 183 148 66 128 64 97 37)(21 184 149 67 129 65 98 38)(22 185 150 68 130 53 99 39)(23 186 151 69 118 54 100 27)(24 187 152 70 119 55 101 28)(25 188 153 71 120 56 102 29)(26 189 154 72 121 57 103 30)
(1 34)(2 35)(3 36)(4 37)(5 38)(6 39)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 33)(14 177)(15 178)(16 179)(17 180)(18 181)(19 182)(20 170)(21 171)(22 172)(23 173)(24 174)(25 175)(26 176)(40 183)(41 184)(42 185)(43 186)(44 187)(45 188)(46 189)(47 190)(48 191)(49 192)(50 193)(51 194)(52 195)(53 164)(54 165)(55 166)(56 167)(57 168)(58 169)(59 157)(60 158)(61 159)(62 160)(63 161)(64 162)(65 163)(66 204)(67 205)(68 206)(69 207)(70 208)(71 196)(72 197)(73 198)(74 199)(75 200)(76 201)(77 202)(78 203)(79 92)(80 93)(81 94)(82 95)(83 96)(84 97)(85 98)(86 99)(87 100)(88 101)(89 102)(90 103)(91 104)(105 156)(106 144)(107 145)(108 146)(109 147)(110 148)(111 149)(112 150)(113 151)(114 152)(115 153)(116 154)(117 155)(118 138)(119 139)(120 140)(121 141)(122 142)(123 143)(124 131)(125 132)(126 133)(127 134)(128 135)(129 136)(130 137)
(27 69)(28 70)(29 71)(30 72)(31 73)(32 74)(33 75)(34 76)(35 77)(36 78)(37 66)(38 67)(39 68)(53 185)(54 186)(55 187)(56 188)(57 189)(58 190)(59 191)(60 192)(61 193)(62 194)(63 195)(64 183)(65 184)(79 105)(80 106)(81 107)(82 108)(83 109)(84 110)(85 111)(86 112)(87 113)(88 114)(89 115)(90 116)(91 117)(131 179)(132 180)(133 181)(134 182)(135 170)(136 171)(137 172)(138 173)(139 174)(140 175)(141 176)(142 177)(143 178)

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

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

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

143 conjugacy classes

class 1 2A2B2C4A4B4C4D4E8A8B13A···13L26A···26L26M···26X26Y···26AJ52A···52X52Y···52BH104A···104X
order1222444448813···1326···2626···2626···2652···5252···52104···104
size112422444441···11···12···24···42···24···44···4

143 irreducible representations

dim111111111111222244
type++++++++-
imageC1C2C2C2C2C2C13C26C26C26C26C26D4D4D4×C13D4×C13C8.C22C13×C8.C22
kernelC13×C8.C22C13×M4(2)C13×SD16C13×Q16Q8×C26C13×C4○D4C8.C22M4(2)SD16Q16C2×Q8C4○D4C52C2×C26C4C22C13C1
# reps112211121224241212111212112

Matrix representation of C13×C8.C22 in GL4(𝔽313) generated by

103000
010300
001030
000103
,
11014930444
25083293288
25522271138
211233159162
,
13827911530
4716176267
311312308256
01180164
,
10138206
014715
003120
000312
G:=sub<GL(4,GF(313))| [103,0,0,0,0,103,0,0,0,0,103,0,0,0,0,103],[110,250,255,211,149,83,22,233,304,293,271,159,44,288,138,162],[138,47,311,0,279,16,312,1,115,176,308,180,30,267,256,164],[1,0,0,0,0,1,0,0,138,47,312,0,206,15,0,312] >;

C13×C8.C22 in GAP, Magma, Sage, TeX

C_{13}\times C_8.C_2^2
% in TeX

G:=Group("C13xC8.C2^2");
// GroupNames label

G:=SmallGroup(416,198);
// by ID

G=gap.SmallGroup(416,198);
# by ID

G:=PCGroup([6,-2,-2,-2,-13,-2,-2,1273,1255,3818,9364,4690,88]);
// Polycyclic

G:=Group<a,b,c,d|a^13=b^8=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^3,d*b*d=b^5,d*c*d=b^4*c>;
// generators/relations

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