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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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C13×C8.C22
 Chief series C1 — C2 — C4 — C52 — D4×C13 — C13×SD16 — C13×C8.C22
 Lower central C1 — C2 — C4 — C13×C8.C22
 Upper central C1 — C26 — C2×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, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, Q8, C13, M4(2), SD16, Q16, C2×Q8, C4○D4, C26, C26, C8.C22, C52, C52, C2×C26, C2×C26, C104, C2×C52, C2×C52, D4×C13, D4×C13, Q8×C13, Q8×C13, Q8×C13, C13×M4(2), C13×SD16, C13×Q16, Q8×C26, C13×C4○D4, C13×C8.C22
Quotients: C1, C2, C22, D4, C23, C13, C2×D4, C26, C8.C22, C2×C26, D4×C13, 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)
(1 129 175 76 187 52 168 85)(2 130 176 77 188 40 169 86)(3 118 177 78 189 41 157 87)(4 119 178 66 190 42 158 88)(5 120 179 67 191 43 159 89)(6 121 180 68 192 44 160 90)(7 122 181 69 193 45 161 91)(8 123 182 70 194 46 162 79)(9 124 170 71 195 47 163 80)(10 125 171 72 183 48 164 81)(11 126 172 73 184 49 165 82)(12 127 173 74 185 50 166 83)(13 128 174 75 186 51 167 84)(14 152 116 30 99 201 137 60)(15 153 117 31 100 202 138 61)(16 154 105 32 101 203 139 62)(17 155 106 33 102 204 140 63)(18 156 107 34 103 205 141 64)(19 144 108 35 104 206 142 65)(20 145 109 36 92 207 143 53)(21 146 110 37 93 208 131 54)(22 147 111 38 94 196 132 55)(23 148 112 39 95 197 133 56)(24 149 113 27 96 198 134 57)(25 150 114 28 97 199 135 58)(26 151 115 29 98 200 136 59)
(1 61)(2 62)(3 63)(4 64)(5 65)(6 53)(7 54)(8 55)(9 56)(10 57)(11 58)(12 59)(13 60)(14 75)(15 76)(16 77)(17 78)(18 66)(19 67)(20 68)(21 69)(22 70)(23 71)(24 72)(25 73)(26 74)(27 183)(28 184)(29 185)(30 186)(31 187)(32 188)(33 189)(34 190)(35 191)(36 192)(37 193)(38 194)(39 195)(40 139)(41 140)(42 141)(43 142)(44 143)(45 131)(46 132)(47 133)(48 134)(49 135)(50 136)(51 137)(52 138)(79 94)(80 95)(81 96)(82 97)(83 98)(84 99)(85 100)(86 101)(87 102)(88 103)(89 104)(90 92)(91 93)(105 130)(106 118)(107 119)(108 120)(109 121)(110 122)(111 123)(112 124)(113 125)(114 126)(115 127)(116 128)(117 129)(144 159)(145 160)(146 161)(147 162)(148 163)(149 164)(150 165)(151 166)(152 167)(153 168)(154 169)(155 157)(156 158)(170 197)(171 198)(172 199)(173 200)(174 201)(175 202)(176 203)(177 204)(178 205)(179 206)(180 207)(181 208)(182 196)
(27 57)(28 58)(29 59)(30 60)(31 61)(32 62)(33 63)(34 64)(35 65)(36 53)(37 54)(38 55)(39 56)(40 130)(41 118)(42 119)(43 120)(44 121)(45 122)(46 123)(47 124)(48 125)(49 126)(50 127)(51 128)(52 129)(66 88)(67 89)(68 90)(69 91)(70 79)(71 80)(72 81)(73 82)(74 83)(75 84)(76 85)(77 86)(78 87)(144 206)(145 207)(146 208)(147 196)(148 197)(149 198)(150 199)(151 200)(152 201)(153 202)(154 203)(155 204)(156 205)

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

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

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

143 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 8A 8B 13A ··· 13L 26A ··· 26L 26M ··· 26X 26Y ··· 26AJ 52A ··· 52X 52Y ··· 52BH 104A ··· 104X order 1 2 2 2 4 4 4 4 4 8 8 13 ··· 13 26 ··· 26 26 ··· 26 26 ··· 26 52 ··· 52 52 ··· 52 104 ··· 104 size 1 1 2 4 2 2 4 4 4 4 4 1 ··· 1 1 ··· 1 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4 4 ··· 4

143 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + + + - image C1 C2 C2 C2 C2 C2 C13 C26 C26 C26 C26 C26 D4 D4 D4×C13 D4×C13 C8.C22 C13×C8.C22 kernel C13×C8.C22 C13×M4(2) C13×SD16 C13×Q16 Q8×C26 C13×C4○D4 C8.C22 M4(2) SD16 Q16 C2×Q8 C4○D4 C52 C2×C26 C4 C22 C13 C1 # reps 1 1 2 2 1 1 12 12 24 24 12 12 1 1 12 12 1 12

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

 103 0 0 0 0 103 0 0 0 0 103 0 0 0 0 103
,
 110 149 304 44 250 83 293 288 255 22 271 138 211 233 159 162
,
 138 279 115 30 47 16 176 267 311 312 308 256 0 1 180 164
,
 1 0 138 206 0 1 47 15 0 0 312 0 0 0 0 312
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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